OPTIMIZATION MODELS FOR SERVICE AND DELIVERY PLANNING

Transcription

1 OPTIMIZATION MODELS FOR SERVICE AND DELIVERY PLANNING By SHUANG CHEN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011

2 c 2011 Shuang Chen 2

3 I dedicate this work to my family 3

4 ACKNOWLEDGMENTS First, I would like to thank my advisor, Dr. Joe Geunes, for all his guidance and continuous support. He paid attention to every step of this dissertation, allowed me the space to think creatively, motivated me to pursue quality research, helped me improve my writing and encouraged me to follow my interests. I feel so fortunate to have him as my advisor. Thanks to his thoughtfulness and understanding, I have led a happy and fulfilling life during my PhD program. I would like to acknowledge Dr. Sartaj Sahni, Dr. Yongpei Guan, Dr. Cole Smith and Dr. Anand Paul for participating in my dissertation committee and their insightful comments. Special thanks go to Dr. Cole Smith, for his passion at work impacts me professionally, and to Dr.Yongpei Guan, Dr. Anand Paul, Dr. Stan Uryasev and my advisor Dr. Joe Geunes for their invaluable advice during my job search process. I m indebted to my friends at the ISE Department, in my church and in Gainesville. Thank you for your friendship throughout these four years. I remain grateful for knowing that you are always there to lend a listening ear and to take away my blues. Moreover, I thank my parents and my grandparents. I owe a great deal to them for always believing in me and for their support at every step of my life. Last, but not least, I thank my beloved husband, Junjie Liu. He is my best friend and my biggest emotional support. He makes my home sweet. I thank him for his unwavering love, care and support. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION OPTIMAL ALLOCATION OF STOCK LEVELS AND STOCHASTIC CUSTOMER DEMANDS TO A CAPACITATED RESOURCE Motivation Problem Definition and Formulation Generalized Benders Decomposition for (P) Heuristic Solution Approach for (P) Computational Results Concluding Remarks ALGORITHMS FOR MULTI-ITEM PROCUREMENT PLANNING WITH CASE PACKS Motivation Related Literature Problem Definition and Formulation Solution Methods for Special Cases Single Case Pack Two Case Packs N Case Packs Strengthening the Formulation Heuristic Solution Approach Addressing Demand Uncertainty Computational Results Heuristic performance and comparison with CPLEX Sensitivity analysis of parameters Cost of requiring a case pack Concluding Remarks INTEGRATED CASE-PACK CONFIGURATION AND PROCUREMENT Motivation Related Literature

6 4.3 Problem Definition and Formulation Solution Methods Reformulation and linearization method An iterative heuristic An iterative heuristic with alternative case-pack sizes Geometric programming based iterative heuristic Computational Results Concluding Remarks CONCLUSIONS APPENDIX A SOLVING THE MULTI-ITEM NEWS VENDOR SUBPROBLEM B OBTAINING THE SUBGRADIENT OF v(.) AT X k REFERENCES BIOGRAPHICAL SKETCH

7 Table LIST OF TABLES page 2-1 Parameter distributions used in computational tests Levels used for analysis of the impact of capacity level V and customer revenue value ^π j Computational test results for average running time (second) and performance Computational test results for various levels of capacity and revenue Computational test results for average running time (second) and performance of the heuristic approach Example of composition of two case packs with four SKUs Parameter distributions used in computational tests The number of time windows and cuts used within relax-and-fix heuristics Computational test results for average running time (seconds) and optimality gap for solution approaches Number of times (out of ten) each solution method attained the lowest-cost solution Effects of the time-window length, α, on solution value for the medium-size problem set Effects of the length of fixing interval, β, on solution value for the medium-size problem set Parameter distributions used in computational tests Computational test results for average running time (seconds) and optimality gap The iteration number for iterative and integrated heuristics

8 Figure LIST OF FIGURES page 2-1 Number of iterations required for solving problems with different parameter levels GBD convergence illustration when m = 3, n = 10, V is at level 1, and ^π j is at level Single case pack problem network Layered network for (P 2 ), the problem with two case packs Effects of time window parameters on solution value. (T=30, N=5, M=6) Objective value vs iterations for the integrated approach (2nd instance M = 6, T = 20) Objective value vs R value for the iterative and integrated heuristics (5th instance M = 6, T = 20) Objective value vs R value for the Iter+R Enum (5th instance M = 6, T = 20) 112 8

9 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy OPTIMIZATION MODELS FOR SERVICE AND DELIVERY PLANNING Chair: Joseph P. Geunes Major: Industrial and Systems Engineering By Shuang Chen August 2011 This dissertation considers the development of efficient algorithms and mixed-integer programming models in the context of several specific real world applications, including service and retail operations. We present models of three specific systems in the service and delivery planning context. We first consider a new class of stochastic resource allocation problems that requires simultaneously determining the customers that a capacitated resource must serve and the stock levels of multiple items that may be used in meeting these customers demands. Our model considers a reward (revenue) for serving each assigned customer, a variable cost for allocating each item to the resource, and a shortage cost for each unit of unsatisfied customer demand in a single-period context. The model maximizes the expected profit resulting from the assignment of customers and items to the resource while obeying the resource capacity constraint. Our contribution includes providing an exact solution method for this mixed integer nonlinear optimization problem and presenting a family of efficient heuristic approaches based on applied probability. For these service supply chain operations, we demonstrate opportunities for strong financial returns and customer satisfaction by focusing on operational excellence. We then move to a new variant of the class of multi-item lot sizing problems where the planner must order in integer quantities of case packs. A distribution case pack contains an assortment of varying quantities of different stock keeping units (SKUs) packed together in a single box or pallet, with 9

10 a goal of reducing handling requirements in the distribution chain. While the inclusion of multiple items in one case pack reduces the number of touches individual items experience in the distribution chain, it can severely increase the complexity involved in retailer procurement planning. This complexity arises because different case packs (and the items within them) share certain (fixed) order costs, and the composition of the case pack requires ordering in defined combinations of multiple SKUs. Thus it is necessary to balance the tradeoff between the handling cost advantages and the reduced ordering flexibility and consequently increased probability of overstock. Since little past research in the literature has addressed this practice, which is widely accepted in industry, our contributions include successfully modeling and solving two retailer problems within this domain: (1) Procurement planning when a supplier of multiple items requires purchasing pre-defined case packs; (2) Integrated case pack configuration and procurement planning problem. The first problem focuses on the short-range decision to determine case pack orders given certain case pack configurations. As for the second long-range decision problem, we provide a model for jointly determining the retailer s optimal case-pack configuration and order timing decisions for a given demand stream over a finite horizon. 10

11 CHAPTER 1 INTRODUCTION With the growth of new products and markets, many suppliers are finding it challenging to provide optimal delivery of goods and services because of resource and cost limitations. Thus, the strategic planning of logistics operations is important in supply chain systems, which generally encompass inventory, transportation, information, and facilities as the key drivers of supply chain performance. This dissertation is aimed at relating mathematical models and methodologies to practice, and solving real-life, complicated problems in service and delivery planning contexts by exploiting insights and special mathematical structure. The optimization of service and delivery planning problems often comes along with optimal resource allocation, where the resource may correspond to an item (e.g. vehicle), a group of items which are in fixed proportion to each other (e.g. a pallet of packed merchandise in pre-defined combinations) or a mix of items whose composition needs to be designed (e.g. a module created to be used in different systems). The resource is usually of limited availability because of its capacity and cost. This dissertation presents mathematical decision models for three service and delivery planning problems where the three types of resources mentioned above are involved, respectively. Each chapter is a self-contained work that begins with a description of a planning problem. After developing the system model and studying relevant operations planning literature, we investigate novel solution procedures to solve the problems of interest. Two of the problems (the subjects of Chapter 2 and Chapter 4) are modeled as mixed integer nonlinear programming problems and the other one, in Chapter 3, as an integer linear programming problem. All of the problems we study are proved to be NP-hard. By exploiting the combinatorial structure and special structural properties of the problems, 11

12 we develop both exact algorithms and efficient heuristic approaches for solving them. We now summarize the three problems considered in greater detail. Many operations settings require the assignment of uncertain customer demands to a resource with limited capacity. One such problem is faced by sales personnel who travel to customers with a stock of goods for sale. That is, given a set of customers on whom the sales force may call, a sales manager must determine how to allocate visits to individual sales personnel, and which items each sales person should stock in their corresponding vehicles prior to making sales calls. Since the demand for each item is only revealed upon being visited by each sales person, we are facing a stochastic problem. Similar challenges arise in determining a subset of customers that a service technician will visit on a given service call, when the technician carries a stock of spare parts, and missing items needed for repair require a special unplanned trip. We thus first consider an important subproblem that arises in this context the allocation of customers and items to a single vehicle in a single-period context. As we will see in Chapter 2, this subproblem is both interesting and challenging by itself. The combined stochastic demand allocation and multi-item stocking decisions for each vehicle lead to extremely complex optimization problems. Chapter 2 considers this class of stochastic resource allocation problems that requires simultaneously determining the customers that a capacitated resource must serve and the stock levels of multiple items that may be used in meeting these customers demands. Our model considers a reward (revenue) for serving each assigned customer, a variable cost for allocating each item to the resource, and a shortage cost for each unit of unsatisfied customer demand in a single-period context. The model maximizes the expected profit resulting from the assignment of customers and items to the resource while obeying the resource capacity constraint. We provide an exact solution method for this mixed integer nonlinear optimization problem using a Generalized Benders Decomposition approach. This decomposition approach uses 12

13 Lagrangian relaxation to solve a constrained multi-item newsvendor subproblem and uses CPLEX to solve a mixed-integer linear Master Problem. We generate Benders cuts for the master problem by obtaining a series of subgradients of the subproblem s convex objective function. In addition, we present a family of heuristic solution approaches and compare our methods with several MINLP (Mixed-Integer Non-Linear Programming) commercial solvers in order to benchmark their efficiency and quality. The other real-life problem we consider is faced by retail chain stores in the fashion and apparel industry. Products such as apparel and shoes are characterized by different sizes. A mismatch between the sizes sent to a store and those desired by its customers can lead to lost sales in some sizes, while resulting in excess stocks and consequent markdowns in others. Retail chains sometimes receive bulk packages for each size and then break, sort, and repackage these to meet the demands of individual stores. However, breaking bulk typically involves a great deal of work at a distribution center and requires associated infrastructure and personnel; thus many retail chains prefer to receive case packs from vendors and then use combinations of these case packs to satisfy store demands. The case packs consist of varying quantities of different Stock Keeping Units (SKUs) packed together to form the lowest level of packaging hierarchy, and are designed to flow through from the vendor to the retail stores without being opened. Handling of these larger case packs rather than individual SKUs proves to be cheaper and faster at all touch points in the supply chain, often through the use of cross-docking. Thus, more and more retail chains are moving to the use of case packs to satisfy store demands instead of receiving bulk packages for each SKU at distribution centers (DCs). However, using case packs leads to decision complexity. A decision hierarchy may be applied that first considers long-term decisions, followed by those made in the short term. In the long term, the following question apply: (1) How many case packs should be used? (2) Which items should be combined into each case pack? In the short term, 13

14 given a set of defined case packs and the need to meet a set of demands, how often should a retailer order case packs, and in what quantities? In Chapter 3, we first model the short-range decisions and solve the resulting procurement planning problem, given a pre-defined number and composition of case packs. In Chapter 4, we integrate both the long- and short-range decisions into a single model and solve the resulting joint case-pack configuration and procurement planning problem. In Chapter 3, we consider a retailer s problem of procurement planning when a supplier of multiple items requires purchasing pre-defined case packs under dynamic, deterministic demands over a finite horizon. While the inclusion of multiple items in one case pack reduces the number of touches individual items experience in the distribution chain, it can severely increase the complexity of the retailer s procurement planning. This results because items within a case pack share certain (fixed) order costs and the composition of the case pack requires ordering in pre-defined combinations of multiple SKUs. Retailers need to solve such problems routinely in pre-season planning. To do this, they often use deterministic demand (fixed forecasts) to create a finite-horizon plan (for a season). However, little past research has addressed this need for planning using case packs. We thus study case pack procurement planning problems that address the tradeoff between reduced order handling costs and higher inventory-related costs. We first establish properties of optimal solutions for special cases involving one and two case packs and then exploit these properties to solve the problem via dynamic programming. For the general model with multiple case packs, which we show to be strongly NP-Hard, we generalize our exact approach to solve this problem in pseudo-polynomial time for a fixed number of case packs. In addition, for large-size problems, we strengthen the problem formulation using valid inequalities and design a family of heuristic solutions. Our computational tests show that these heuristic approaches perform very well compared to the commercial mixed integer programming solver CPLEX. In addition to providing detailed methods for solving problems with 14

15 deterministic demand, we also discuss strategies for addressing problems with uncertain demands. In Chapter 4, we suppose case pack compositions are not pre-defined and the modeling issues and formulation for the integrated problem are considered. To determine optimal case-pack configurations and make replenishment decisions jointly, we model the problem as a quadratically constrained quadratic program with integer variables, which is strongly NP-hard. We first consider a reformulation via linearization that permits solving small-sized problems using a mixed integer linear programming solver. Then, we discuss a fast iterative heuristic that exploits the problem s bilinear structure and further fine tune this approach to improve its performance with respect to the final optimality gap. We also consider a geometric programming based heuristic for the single-case-pack special case. A series of computational tests shows that our proposed approaches perform very well compared to the commercial nonlinear programming solver GAMS/BARON. My work contributes to the current literature by identifying new problems for service and delivery systems and developing mathematical models and novel solution approaches for optimizing such complicated real-life systems. Moreover, the thesis provides some insights and methods for allocating different types of resources in service and delivery planning contexts. With vehicle capacity constraints, decomposition and Lagrangian relaxation techniques are highly effective as discussed in Chapter 2. When the resource corresponds to a predefined case pack, as in Chapter 3, or any group of items with a fixed relative composition, we can take advantage of the fixed ratio between items for allocating such resource to different time periods in the production and replenishment systems. If the configuration of such a resource unit (e.g. a case pack) must be designed (in terms of the number and type of SKUs in each case pack), the resulting joint resource design and allocation problems will be extremely difficult to solve, likely containing a number of bilinear or quadratic terms in the model. In such 15

16 situations, the approaches presented in Chapter 4 will provide valuable guidelines for resource and operations planning managers. Finally, this work identifies new areas open for future research that have promising potential to address additional challenging and practical issues. 16

17 CHAPTER 2 OPTIMAL ALLOCATION OF STOCK LEVELS AND STOCHASTIC CUSTOMER DEMANDS TO A CAPACITATED RESOURCE 2.1 Motivation The optimal deployment of constrained resources lies at the heart of nearly all operations problems. Many operations settings require the assignment of uncertain customer demands to a resource with limited capacity. One such problem is faced by sales personnel who travel to customers with a stock of goods for sale. That is, given a set of customers on whom the sales force may call, a sales manager must determine how to allocate visits to individual sales personnel, and which items each sales person should stock in their corresponding vehicles prior to making sales calls. Because customers purchase preferences are not known with 100% certainty prior to the visit, the required items to stock are not known with certainty prior to dispatching the sales person. Although the degree of uncertainty in demand can be decreased via an initial telephone- or Internet-based pre-sales screening, the uncertainty in demand cannot be completely resolved prior to the sales visit. Combined demand allocation and multi-item part stocking decisions are therefore required for each sales vehicle prior to departure on a sequence of sales calls. Similar challenges arise in determining a subset of customers that a service technician will visit on a given service call, when the technician carries a stock of spare parts, and missing items needed for repair require a special unplanned trip. These customer assignment and part-stock-level decision problems lead to extremely complex optimization problems. Because of the level of complexity associated with this problem class, the literature on the exact solution of problems of realistic size is reasonably limited. In this chapter, we consider an important subproblem that arises in this context the allocation of customers and items to a single vehicle in a single-period context. As we will see, this subproblem is both interesting and challenging by itself, and may arise as a stand-alone problem when only a single resource is available. Moreover, 17

18 the ability to solve this class of subproblems exactly will facilitate effective decomposition approaches for handling the multi-resource planning problem. Demand allocation problems have been widely studied in the literature in the context of make-to-order systems under the assumption that demand streams can be split among the available facilities (resources). In situations where the demand for items from a customer cannot be split and the demand from each customer must therefore be assigned to a single resource, the problem becomes combinatorial in nature and the decision variables are discrete. The literature on this class of problems under stochastic demand is limited, as this class of problems is generally quite difficult to solve. For the problem class we consider, we will assume that a customer cannot be assigned to multiple vehicles, since customers generally prefer a single service visit or a visit by a single salesperson. This class of problems will therefore contain a substantial combinatorial component. From the point of view of allocating a customer subset to a capacity-constrained vehicle, this problem is closely related to certain classes of stochastic knapsack problems (SKP) (e.g., Kleywegt and Papastavrou [57], Dean et al. [27], Merzifonluoğlu et al. [66], and Ağralı and Geunes [2], Kosuch and Lisser [58]), with each customer s (uncertain) demand quantity corresponding to the typical item size in these knapsack problems. However, our problem generalizes this problem class, as it involves the additional dimension of limited item availability as a result of the item-to-vehicle stock level decisions. That is, the vehicle s capacity constrains the allocation of item stock levels to the vehicle, and the item stock levels within the vehicle constrain the ability to meet customer requirements during sales visits. As in several of these cited works on stochastic knapsack problems, we will focus on the case where item sizes (or customer parts requirements) are independent but not necessarily identically distributed normal random variables (the normal distribution parameters we use are such that the probability of negative demand is negligible). 18

19 To determine item stock level decisions for a given vehicle, observe that if we fix the customer-to-vehicle assignment decisions, we then face a constrained multiple-item newsvendor problem involving uncertain shortage costs, expected customer revenues from sales visits, and expected variable operating costs. Newsvendor problems have of course been well studied in the literature. Hadley and Whitin [49] discussed the constrained multi-item newsvendor model in detail. Another class of closely-related problems to the one we consider is known as the class of repair kit problems (e.g., Smith et al. [88], Teunter and Haneveld [92] [93] and Gorman and Ahire [44]). Most of these repair kit problems are aimed at minimizing expected holding costs under given service level requirements. In contrast, we consider the maximization of expected profit under an additional vehicle capacity constraint, using a shortage (penalty) cost in the objective (which, in turn, can be used to ensure desired service levels). The most closely related repair kit problem that considers multiple types of items and vehicle capacity is the one proposed by Gorman and Ahire [44], who developed a single-pass greedy heuristic approach for the multiple vehicle planning problem. Within the single-period context we consider, demand allocation and vehicle stocking decisions must be determined prior to actual customer demand realizations. Hence, we face a joint demand assignment and stock level problem with uncertain customer demands. Our goal is to simultaneously set stock levels for the multiple items (parts) within the vehicle and determine customer demand assignments in order to maximize expected profit, equal to the expected revenue from customer visits less expected shortage and variable costs. To the best of our knowledge, no existing work considers an exact solution for combined demand allocation and vehicle stocking problems for multiple items under uncertain demand, as we address in this chapter. Thus, our contributions include providing a new model for this problem class, as well as exact and heuristic solution procedures, which we demonstrate to be extremely effective when compared with a state-of-the-art mixed integer nonlinear optimization solver. 19

20 Our class of stochastic resource allocation problems has several variants relevant to various operations planning contexts. If we restrict our attention to only one type of item, we will have the static stochastic knapsack problem (Kleywegt and Papastavrou [57], Merzifonluoğlu et al. [66], Ağralı and Geunes [2], Kosuch and Lisser [58]). If there is no shared resource (vehicle) capacity constraint, we will have a stochastic multidimensional knapsack problem (MKP) (Kellerer et al. [56], Vasquez and Vimont [98], and Akçay et al. [5]). The MKP is a well-known NP-hard problem, which implies the NP-hardness of our problem under general demand distributions as well. Further, if we consider the multiple vehicle version of our problem with one item type and no shared resource capacity, the problem takes the form of the joint facility assignment and capacity acquisition problem (Taaffe et al. [91]), where the optimal stock level for the part corresponds to the facility capacity. Similarly, if the customer demand assignments are given, we only need to consider the stock levels for multiple items on the vehicle, which reduces to a set of constrained multi-item newsvendor problems. Within the context of make-to-stock queues, Benjaafar et al. [15] appear to be the first to consider the joint demand allocation and inventory control problem. They considered the long-run fraction of demand for each product i (in a set of products) allocated to each facility (from a set of facilities, analogous to the vehicles in the context we discussed above). They considered the minimum long-run expected cost per unit time under continuous assignment variables (the demand allocation problem, or DAP), and the demand partitioning problem (DPP) with binary assignment variables. They used convex optimization algorithms to solve the DAP and a branch and bound algorithm for the DPP. Federgruen and Zipkin [33] developed a combined routing and inventory allocation model for a single item under stochastic customer demands (Federgruen et al. [30], generalized this to account for perishable items when fresh and old stock of the item is available and out-of-date costs may exist). In their model, customer inventory 20

21 allocation decisions are made prior to assignment and routing decisions. Thus, customer-related inventory costs are separable. In our model, however, inventory is not allocated prior to customer assignment decisions. Therefore, individual demands for customers who share the resource capacity (i.e., customers visited by the same truck) create a single, pooled demand distribution for each item, and inventory costs are not separable by customer. Moreover, we consider a multi-item setting in which resource capacity is shared by different items. We note, however, that their model considered a multiple vehicle context, whereas we consider a single-resource problem. As we later discuss, under a normal demand distribution for each customer s parts requirements, a generalized Benders decomposition (GBD) approach is quite effective for this nonlinear mixed integer problem. GBD techniques, developed by Geoffrion [39], can be used to efficiently solve nonlinear programming problems with complicating variables by temporarily fixing these variables and dealing with the remaining problem, which is generally much easier to solve. To extend the applicability of the GBD approach, Geromel and Belloni [40] studied the differentiability of a set of related perturbation functions and proposed a method to handle problems of a more general class than Geoffrion [39] considered. We will employ these generalized methods of Geromel and Belloni [40] to generate Benders cuts. Many practical problems, such as multi-commodity network flows, quadratic assignment problems, and combined location-inventory problems have been efficiently solved using GBD techniques. In our approach, we first fix the complicating integer variables; that is, for a given feasible demand allocation, we solve the remaining multi-item constrained newsvendor subproblem using Lagrangian relaxation (Hadley and Whitin [49]). We then use the solution of this subproblem to add the corresponding support function (Benders cut) to the so-called master problem and solve a relaxed master problem. We repeat this process by using the demand allocation decision from the solution of the relaxed master 21

22 problem to solve a new version of the newsvendor subproblem. Benders cuts are then iteratively added to the master problem until an optimal solution is found. The remainder of this chapter is organized as follows. In Section 2.2 we define and formulate our stochastic demand allocation and inventory stock-level problem. We then analyze important properties of optimal solutions for our problem and develop a generalized Benders decomposition approach in Section 2.3. Section 2.4 presents a heuristic solution approach for solving large size problem instances that may preclude the use of exact methods. In Section 2.5 we discuss the results of a computational study, used to validate our solution methods and compare them with the results of three benchmark commercial solvers, GAMS/LINDOGlobal, GAMS/SBB and GAMS/CoinBonmin. Finally, concluding remarks are provided in Section Problem Definition and Formulation In this section, we present a single-period model to solve a joint customer demand allocation and multiple-item stock level problem for a resource that must respond to uncertain customer demands. For example, the resource may correspond to a vehicle (and corresponding sales person) on a given working day, and the items may correspond to items that are potentially needed on sales calls. We assume that customer demands are statistically independent and that the demands for items are revealed upon being visited by the sales person. While there are m potential customers, the supplier must choose which subset of customers to serve using a capacity constrained resource (e.g., a vehicle) and the optimal resource stock level for each item. This individual resource level problem may appear as a stand-alone problem for a manager of a single resource, or as a subproblem for a larger multiple resource assignment problem with multiple customer/item demands. We assume that during a given visit, any item requested that is not available results in an item-specific penalty cost for an inability to complete service (thus, individual item demands are independent of one another). For each item carried on the vehicle, 22

23 a variable cost is incurred (e.g., for loading/unloading and/or transporting the item). Because the vehicle stocking decisions must be determined prior to actual customer demand realizations, it may be practical in some contexts to consider a salvage value for each unused item carried on the vehicle (this might correspond to a reduction in future loading/unloading costs; alternatively, a negative value would correspond to an opportunity cost of the vehicle capacity usage). A customer-specific revenue is gained for each customer visit. The objective is to maximize the expected profit, or equivalently, to minimize the expected cost (equal to the negative of expected profit). Henceforth, we state our objective using this minimization form and refer to this objective as the expected cost. To formalize our model, we define following notation: Inputs and Parameters ^π j : i: item (part) index, i = 1,..., m. j: customer index, j = 1,..., n. expected revenue gained by serving customer j, i.e., by allocating customer j to the vehicle. ^e i : unit shortage cost incurred for not satisfying a unit of demand for item i. g i : unit salvage value incurred for an unused item i. ^c i : unit variable cost for carrying item i in the vehicle, where ^e i > ^c i > g i. s i : unit size of item i. V : d ij : vehicle capacity. random variable denoting the demand for item i by customer j, with mean and standard deviation µ ij and σ ij, respectively. Decision Variables x j : binary decision variable, equal to 1 if customer j is assigned to the vehicle, 0 otherwise. The vector ( column ) X = [x 1,, x n ] T characterizes the assignment of customers to the vehicle. y i : nonnegative decision variable equal to the number of units of item i carried in the vehicle. The vector Y = [y 1,, y m ] T determines the assignment of items to the vehicle. 23

24 We define D i = n j=1 d ijx j as the random variable for the aggregate demand of item i by all customers assigned to the vehicle (note that D i is a random variable with distribution parameters determined by the assignment variable values and the random variables for individual customer and item demands). The expected cost as a function of the part stock levels and customer assignments can be expressed as: P(Y, X ) [( m n ) + ] = ^e i E d ij x j y i i=1 j=1 [( m n ) + ] = e i E d ij x j y i + i=1 j=1 [( ) + ] m n g i E y i d ij x j + i=1 m c i y i i=1 j=1 m n ^c i y i ^π j x j j=1 i=1 j=1 n π j x j, (2 1) where [x] + = max{x, 0}, e i = ^e i g i, c i = ^c i g i, and π j = ^π j m i=1 g iµ ij denote the net penalty cost, variable cost, and revenue, respectively (^e i > ^c i > g i implies that both e i and c i are nonnegative for all i). Observe that, for any customer j, if the coefficient of the third term is such that π j = ^π j m i=1 g iµ ij 0, it is straightforward to show that we can set x j = 0 without loss of optimality. We therefore assume without loss of generality that π j > 0 for all j. We can now formulate our stochastic resource allocation problem with a resource capacity constraint as: (P) min P(Y, X ) m subject to: s i y i V, i=1 x j {0, 1}, j = 1,..., n, y i 0, i = 1,..., m. The constraint ensures that the vehicle s capacity is not violated. Observe that we permit the stock levels to take continuous values. For a given vector of customer 24

25 assignments, X, however, we can easily determine optimal integer stock levels for the associated multi-item constrained newsvendor subproblem (Hadley and Whitin [49]). 2.3 Generalized Benders Decomposition for (P) For a given vector X of customer assignments, problem (P) becomes a standard multi-item newsvendor problem with a single constraint: (MINV) min subject to: m { i=1 ci y i + e i E [ (D i y i ) +]} m i=1 s iy i V, (2 2) y i 0. It is straightforward to solve (MINV) using Lagrangian relaxation (Appendix A discusses a solution approach for MINV based on results from Hadley and Whitin [49]). Moreover, for a given vector of item stock levels Y, we have a 0-1 integer programming problem without explicit constraints. We thus consider a generalized Benders decomposition approach for solving this resource allocation problem. We first streamline the notation for problem (P) by defining: [( m n ) + ] H(X, Y ) = e i E d ij x j y i ; G 1 (Y ) = i=1 m c i y i ; i=1 j=1 n Q(X ) = π j x j ; G 2 (Y ) = j=1 m s i y i V. Using this notation, problem (P) can be formulated as: i=1 25

26 (P ) min H(X, Y ) + G 1 (Y ) Q(X ) (2 3) subject to: G 2 (Y ) 0, (2 4) X {0, 1}, (2 5) Y 0. (2 6) Alternatively, we can formulate problem (P ) in the space of the binary variables as follows: (P ) min v(x ) Q(X ) subject to: X {0, 1}, where, for a given vector X, the value function v(x ) is determined by solving the subproblem (SP), formulated as (SP) v(x ) = min H(X, Y ) + G 1 (Y ) (2 7) subject to: G 2 (Y ) 0, Y 0. Because the subproblem (SP) is convex in Y, a dual representation of v(x ) can be formulated, and its optimal dual solution value equals the optimal primal solution value (Theorem 2.3 in Geoffrion [39]). Defining the dual variable w 0 corresponding to the constraint in (SP), we can write the Lagrangian dual as { } v(x ) = max min [H(X, Y ) + G 1(Y ) + wg 2 (Y )]. (2 8) w 0 Y 0 Problem (P ) is therefore equivalent to the following Master Problem (MP): (MP) min θ Q(X ) subject to: θ min [H(X, Y ) + G 1(Y ) + wg 2 (Y )], w 0, Y 0 X {0, 1}. 26

27 Clearly we cannot enumerate all possible constraints in the above formulation for all possible values of w. Therefore, we begin with a relaxed formulation of the master problem and iteratively add violated Benders cuts. In order to find violated Benders cuts, we need to first characterize the support function of v(x ), which we denote by ξ(x ). For a given vector X k and the optimal dual solution w k of (SP) associated with the given X k, we know that v(x k [ ) = min H(X k, Y ) + G 1 (Y ) + w k G 2 (Y ) ], Y 0 and by the definition of the support function of v(x ) at the point X k, ξ(x k ) = v(x k ) and ξ(x ) v(x ), X X k. The following proposition and theorem will aid us in characterizing the support function. Lemma 1. P(Y, X ) in Equation (2 1) is a jointly convex function on R + n R + m. Proof. To show that (2 1) is a jointly convex function, we only need to show that ( n j=1 d ijx j y i ) + is a jointly convex function by limiting our attention to a given realization of the demand d ij, because the other terms in (2 1) are all linear, and the expectation of a convex function is a convex function. Note that ( n j=1 d ijx j y i ) + = max(0, n j=1 d ijx j y i ) is a maximum of two jointly convex functions and is, hence, jointly convex. Then, by the fact that the summation of jointly convex functions is jointly convex, the proof is complete. Theorem 2.1. v(x ) is a convex function, and a linear support function ξ k (X ) can be readily calculated as: ξ k (X ) = v(x k ) + δ(x k )(X X k ), where δ(x k ) v(x k ) is any available subgradient of v(.) at X k. One such subgradient of v(.) is: δ(x k ) = X { H(X, Y k ) + G 1 (Y k ) + w k G 2 (Y k ) } X=X k. 27

28 Proof. Using results from Heyman and Sobel ([53], p. 525), we can show that v(x ) in (2 7) is convex in x, since H(X, Y ) and G 1 (Y ) are convex in both x and y. Then, because v(x ) is convex, it is well known that ξ k (X ) serves as a linear support function at X k. Details regarding the correctness of this support function can be found in Geromel and Belloni [40] using Theorem 3 and formulas (31) (34). We omit the details for the sake of brevity. The details of how we compute ξ k (X ) are provided in Appendix B. We can now represent the Relaxed Master Problem (assuming K linear support functions have been generated) as: (RMP) min θ Q(X ) subject to: θ ξ k (X ), k = 1, 2,, K, X {0, 1}, θ R. Here θ ξ k (X ) is a Benders cut for given values of X k and w k. Observe that the (RMP) is a mixed 0-1 linear program with a single continuous variable. While such problems are not generally considered easy (they are, of course, NP-Hard), they are much more tractable than our original nonlinear mixed 0-1 program, and problems of reasonable size can be solved in acceptable time using CPLEX. The generalized Benders decomposition approach (and variants thereof) can be shown to be convergent when X is a finite discrete set. We next formalize our algorithm as follows: Step 1: Choose an initial vector X 0 that ensures a feasible solution for the subproblem and select an optimality tolerance ɛ. Solve subproblem (SP) at X 0, which is a multi-item constrained newsvendor problem, obtaining Y 0 and a corresponding optimal dual solution w 0 (Appendix A discusses the solution approach for the MINV problem). Set the upper bound UB = v(x 0 ) Q(X 0 ) and let (X, Y ) = (x 0, Y 0 ) denote the initial 28

29 solution. Generate the support function of v(.) at X 0 as the Benders cut using Theorem 2.1 and the results in Appendix B. Step 2: Solve the relaxed master problem with all previously generated cuts. Let (X, θ ) denote an optimal solution to (RMP) and set the lower bound LB = θ Q(X ). If (UB LB) < ɛ, stop; otherwise, go to Step 3. Step 3: Solve the subproblem at X = X, denoting Y as the optimal solution vector and v(x ) as the optimal solution value. If v(x ) Q(X ) < UB, set UB = v(x ) Q(X ) and update the incumbent solution, i.e., let (X, Y ) = (X, Y ). IF (UB LB) < ɛ, stop with (X, Y ) as an ɛ-optimal solution. Otherwise, recover the optimal dual multiplier w, add the corresponding Benders cut to the (RMP) formulation, and return to Step Heuristic Solution Approach for (P) The Master Problem in our Benders decomposition approach is a mixed 0-1 linear programming problem. As a result, solution via a commercial solver for extremely large problem instances will likely be impractical. We, therefore, propose a heuristic solution approach for very large problem sizes. To obtain a fast heuristic for solving this problem, we temporarily ignore the vehicle capacity constraint when identifying which customers to serve. Then, for a given vector X, the problem is a multi-item capacitated newsvendor problem, which can be solved efficiently. Thus, our solution approach identifies potential values for the vector X, and computes the corresponding optimal stock levels Y (X ). The resulting solution (X, Y (X )) is a feasible solution for problem (P). We begin by expressing problem (P) as a function of the vector X only, when the capacity constraint (2 2) is ignored. In the absence of this constraint, for a given vector X, we can solve the associated multi-item newsvendor problem and obtain the optimal stock value y i (X ) for each item i, given by y i (X ) = F 1 X,i (ρ i), where ρ i = e i c i e i, and F 1 X,i is the inverse cumulative distribution function (CDF) of the demand distribution for item i implied by the vector X. Since the demands are normally distributed, we can write the stock levels as y i (X ) = j µ ijx j + z(ρ i ) j σ2 ij x j, where z(ρ i ) = 1 (ρ i ) is the standard 29

30 normal variate value associated with the fractile ρ i, and 1 is the inverse CDF of the standard unit normal distribution. In addition, we define i (y i (X )) as the loss function for a given stock level y i (X ) and customer assignment vector X, i.e., i (y i (X )) = [ ( y i (X) (D i y i (X ))f i (D i )dd i = E j d ijx j y i (X ) ) ] +. Introducing the standard normal loss function L(z) = (u z)φ(u)du, where φ(u) is the probability density function z (pdf) of the standard normal distribution with CDF (u), we can write the loss function i (y i (X )) in terms of the standard normal loss function, i (y i (X )) = j σ2 ij x jl(z(ρ i )). Substituting these expressions for y i (X ) and the loss function into P(Y, X ) in Equation (2 1), with a slight abuse of notation, the expected cost can be written in the following form: P(X ) = P(Y (X ), X ) = j [π j i c i µ ij ]x j + i [c i z(ρ i ) + e i L(z(ρ i ))] σij 2x j. (2 9) j Let K i = c i z(ρ i ) + e i L(z(ρ i )) represent the i th coefficient value of the square root terms and denote r j = π j + i c iµ ij as the negative of the expected net revenue for serving customer j. We then need to solve the following problem in the case of normally distributed demands: (RP) min r j x j + i j K i σij 2x j j subject to: x j {0, 1}, j = 1,..., n. Note that we can, without loss of optimality, set x j = 0 for any j such that r j 0, i.e., any j such that c ij µ ij π j (which implies that the expected customer cost outweighs the customer s associated revenue). This relaxed problem RP, obtained by dropping the capacity constraint and expressing Y in terms of X has some special properties: Proposition 2.1. K i = c i z(ρ i ) + e i L(z(ρ i )) is a nonnegative constant under our assumption that e i > c i > 0 for all i. 30

31 Proof. For ease of exposition, we represent the standard normal variate value z(ρ i ) using z. To show K i 0 when assumption that e i > c i > 0, we only need to show z + L(z) 0, because the loss function L(z) is nonnegative for any z. Using L(z) = φ(z) z(1 (z)) we have z + L(z) = φ(z) + z(z) = φ( z) ( z)(1 ( z)) = L( z) 0. The second equality follows from the symmetry of the normal distribution, which completes the proof. Proposition 2.2. The objective function of RP is concave, and thus for the continuous relaxation of RP obtained by replacing each x j {0, 1} with 0 x j 1, the linear relaxation of RP is a concave minimization problem with an optimal solution at one of the integral extreme points of [0, 1] n. Proof. See Geunes, Shen, and Romeijn [41]. Proposition 2.3. The objective function of RP is a submodular function. Minimizing a rational submodular function is solvable by strongly polynomial combinatorial algorithms. Proof. Properties B.1 through B.4 and Lemma B.1 of Shen et al. [84] imply that that P(X ) is a submodular function. As discussed in Shen et al. [84], Grotschel et al. [48] showed that a submodular function can be minimized in polynomial time. Strongly polynomial time algorithms are available for this problem class as a result of work by Iwata et al. [54] and Schrijver [80]. We are interested in exploiting the particular special structure of our problem in order to obtain a fast heuristic approach for problem (P). We first discuss a solution method for a special case of our problem. This special case arises when either σ ij = σ j, for all j (the equal-item-variance case) or when only one type of item m = 1 is 31

32 considered. Because these special cases are mathematically equivalent, we illustrate the formulation for the former case below. (RPS) min r j x j + K i σj 2x j j i j subject to: x j {0, 1}, j = 1,..., n. We can solve this special case using a simple sorting scheme as in Taaffe et al. [91]. We first sort customers in nonincreasing order of the ratio of expected net revenue to the uncertainty in that customer s demand. This results in indexing customers such that r 1 σ 2 1 r 2 σ 2 2 r n σn 2. (2 10) Proposition 2.4. After indexing customers in nonincreasing order of the above ratio, an optimal solution to (RPS) exists such that if x k = 1 for some k {1,, n}, then x l = 1 for all l {1, 2,, k 1}. Proof. See Shen et al. [84]. For the general case with unequal item variances, we cannot show that a sorting scheme as in (2 10) is directly available. We can, however, utilize the insight from (2 10) to arrive at a heuristic ranking scheme for customers. Since we would like to capture the tradeoff between revenues and variance-related cost, we replace the denominator with a weighted-average variance across items for each customer. That is, we use a weighted-average variance value, where customer j s product i variance is weighted by a factor γ ij. We then define σ 2 j (γ j ) = i γ ij σ 2 ij i γ ij (where γ j is an m vector of γ ij values). Our heuristic approach is motivated by the fact that there exist weight values γ ij for all i, j such that, after indexing items in nonincreasing order of σ 2 j (γj ), an optimal selection of customers exists of the form defined in Proposition 2.4 (this follows because any one of the n! possible ordered vectors of customer indices can be obtained through an appropriate choice of γ ij values). One option is to set γ ij = µ ij, in which case r j 32

33 each variance value is weighted by the corresponding expected demand value. Another option is to set γ ij = K i, weighting the variance by the corresponding cost coefficient. To account for the influence of both of the above two factors, we might also set γ ij = µ ij K i. Alternatively, when γ ij = γ j for all i, we obtain the arithmetic average of variance, i.e., σ 2 j = i σ2 ij m. This approach, therefore, provides a family of heuristic solutions, with each member of the family defined by the choice of γ ij values. Our heuristic approach therefore sorts customers in nonincreasing order of for some choice of each of the n vectors γ j, and considers each solution containing r j σ 2 j (γj ) customers {1,..., k} for k = 1,..., n (in our computational tests, we consider the cases with γ ij = µ ij, γ ij = K i and γ ij = µ ij K i, as well as the simple arithmetic average case). We summarize our heuristic solution algorithm as follows. Step 1: Choose a value for each of the n values of m vectors γ j. Compute the ratio value R j = r j σ 2 j (γj ) for each customer j. Re-index all customers in nonincreasing order of R j values. Define each of the n assignment vectors {1,..., k} for k = 1,..., n; that is, assignment X j is such that x 1 = x 2 = = x j = 1, x j+1 = = x n = 0. Set p = 1. Step 2: Solve the multi-item constrained news vendor problem with the assignment vector X p to determine Y p (X p ) and record the optimal objective value v(x p ) Q(X p ). Step 3: Let p = p + 1; if v(x p ) Q(X p ) < v(x p 1 ) Q(X p 1 ) and p n, return to Step 2. Otherwise, let j = p 1; the heuristic solution consists of the assignment vector X j and stock levels Y (X j ) with objective function value v(x j ) Q(X j ). Note that the heuristic solution is integral and feasible. In addition, in Step 3, instead of enumerating the n assignment vectors {1,..., k} for k = 1,..., n, we stop once the objective value begins increasing in order to speed up the heuristic. We did this because, after testing instances with the data provided in the next section, we observed that the solutions obtained using this stopping criterion were the same as those obtained after enumerating and comparing all n assignments. 33

34 2.5 Computational Results In this section, we present computational results for our generalized Benders decomposition algorithm and heuristic approach for solving the stochastic resource allocation problem (P). We will demonstrate the benefits of our solution approach when compared to three commercial non-linear integer solvers. Another goal is to analyze the effect of different parameters on the results, such as the total number of served customers and the number of master and subproblem iterations (which also corresponds to the number of Benders cuts in the relaxed master problem). We implemented our algorithm in C++, with the 0-1 linear integer relaxed master problem solved using ILOG s CPLEX 12.1 solver with Concert Technology. We performed all tests on a computer with an Intel R Pentium 4 CPU, 3.4 GHz processor with 1.99 GB of RAM. In all of our experiments, we used a relative optimality tolerance of In order to avoid a divide by zero error, both in our algorithm and in GAMS, we added a constraint to ensure n j=1 x j 1. Then, if the optimal objective value for the minimization problem is larger than 0, we know it is better not to deliver items to any customers since X = 0 and Y = 0 are always feasible to the problem at a total cost of 0. Table 4-1 summarizes the common data used in our computational study. For each problem instance, the demand, revenue and all cost data were generated from uniform distributions. We let U(l, u) denote the continuous uniform distribution with lower bound l and upper bound u. To benchmark the performance of our algorithm against LINDOGlobal, SBB, and CoinBonmin, we tested our solution method for 16 problem sets and computed the running time and optimality performance (these 16 problem sizes use vehicle capacity level 3 and customer revenue level 2 shown in Table 2-2). Each of these problem sets is characterized by a unique combination of the number of customers and the number of items, (n, m), where we considered n {5, 10, 30, 50} and m {3, 10, 20, 50}. For each combination of (n, m) values, we tested 10 randomly generated problem 34

35 instances, for a total of 160 test cases. For comparison purposes, we also solved each problem instance using GAMS/LINDOGlobal, GAMS/SBB and GAMS/CoinBonmin, three commercial integer nonlinear solvers guaranteeing eventual convergence to globally optimal solutions for general nonlinear problems with continuous and/or discrete variables (assuming memory is not exhausted before convergence). We set the relative optimality tolerance to 10 5, the iteration limit to 200, 000, and the time limit to 5, 000 seconds in GAMS. In addition to the initial set of 160 problem instances described above, we also considered four levels of resource capacity and customer-specific unit revenue, respectively, (shown in Table 2-2) in order to study the impact of different values of these two important parameters. To this end, we will consider an additional 16 problem sets corresponding to each pair of distributions used for these two parameters, with the number of customers fixed at 10 and the number of items fixed at 3. We tested 10 randomly generated problem instances for each of these problem sets, which constitutes another 160 test cases with n = 10 and m = 3. For this latter set of 160 test problems, which were used to evaluate the impacts of capacity and revenue levels, we used our generalized Benders decomposition approach to obtain optimal solutions (although we did not compare the results of these cases with those using any of the GAMS solvers). We therefore tested a total of 320 problem instances, of which 160 cases were solved using both our algorithm and the three different GAMS solvers, and an additional 160 test problems were solved using our algorithm only. Table 2-1. Parameter distributions used in computational tests. Resource and customer data Unit Shortage Cost, ^e i Unit Salvage Value, g i Unit Handling Cost, ^c i Capacity, V U(2.5, 3.6) U(0.1, 0.5) U(1, 1.5) U(5mn, 80mn) Expected Demand, µ ij Standard Deviation, σ ij Unit item size, s i Fixed Revenue, ^π j U(50, 100) U(14, 50) U(0.5, 1) U(100m, 220m) 35

36 Table 2-2. Levels used for analysis of the impact of capacity level V and customer revenue value ^π j. Level Capacity distribution U(5mn, 10mn) U(10mn, 20mn) U(20mn, 40mn) U(40mn, 80mn) Revenue distribution U(100m, 140m) U(140m, 180m) U(180m, 220m) U(220m, 260m) GAMS Modeling In order to solve the full problem (P) using GAMS solvers, we needed to encode the equation for E [(D i y i ) + ] for each i, where D i N( ~U i, 2 i ). For ease of explanation, we suppress the index i in describing how we did this. We used the errorf (x) function (integral of the standard normal distribution from to x) in GAMS modeling, with the equation E [(D y) + ] = L(z), where z = y ~U, and ~U and are the mean and standard deviation of D, respectively (in terms of our model, ~U i = n j=1 µ ijx j and 2 i = n function, L(z), is given by j=1 σ2 ijxj 2 ). The standard normal loss L(z) = z (u z)φ(u)du = e z2 2 2π z(1 (z)), where φ(.) is the pdf of the standard normal distribution and (.) is the associated CDF. In GAMS, we can express (.) using (z) = errorf (z). Comparison with GAMS solvers To compare our algorithm with the commercial solver, we consider running time and optimality performance. As discussed previously, we first fixed the resource capacity and customer revenue levels to levels 3 and 2, respectively, and then tested the 16 different problem sets and studied the results. The reason for choosing capacity level 3 and revenue level 2 is that our initial computational testing showed that these settings led to a good balance between revenues and overflow costs, and the resulting problems were among the more computationally intensive problems we tested. The results of our tests, averaged over the 10 random problem instances for each given set of parameter 36

37 levels, are presented in Table 2-3. In the table, GBD represents our generalized Benders decomposition approach. The table shows that our approach is generally at least 30 times faster than GAMS/LINDOGlobal, and 2 times faster than GAMS/SBB and GAMS/CoinBonmin. All of the problems we tests were solved within 17 seconds using our algorithm. However, GAMS/LINDOGlobal takes around 30 seconds for smaller size problems, while the majority of problems were solved in 2 to 10 minutes, with the largest size problem requiring nearly 1 hour. To provide a benchmark for the performance of our algorithm, we use the Performance Ratio (PR) as an index, which corresponds to the average solution value as a percentage of the optimal solution value. Our algorithm, SBB and CoinBonmin can found an optimal solution for all problems, while LINDOGlobal works well only for smaller size problem (for larger size problems, LINDOGlobal does not perform quite as well). Much of the time it stops with only locally optimal solutions as a result of the iteration limit, which accounts for simplex iterations, barrier iterations, nonlinear iterations and box iterations. For the problem with (m, n) = (50, 50), for 10 randomly generated instances, three of the cases could not be solved within the time limit of 5, 000 seconds, and in one case it incorrectly concluded that the problem was infeasible. The average PR was as low as 62.14% for GAMS/LINDOGlobal. Based on the above comparison, we found that our algorithm significantly outperformed GAMS MINLP solvers LINDOGlobal, SBB and CoinBonmin across the 160 randomly generated problem instances under vehicle capacity level 3 and customer revenue level 2. Parameter Analysis In Table 2-4, we show how the different levels of two important parameters (capacity and revenues) affected the results for the case of (m, n) = (3, 10). The average value (across the 10 randomly generated instances) of expected profit is shown in column 6 of the table. The first column corresponds to the four levels of the resource capacity 37

38 Table 2-3. Computational test results for average running time (second) and performance Data Set GAMS MINLP Solvers Our Approach LINDOGlobal CoinBonmin SBB GBD m n Time PR Time* Time* Time* % % % % % % % % % % % % % % % % *PR=100.00% V and the second column corresponds to levels of customer-specific revenue ^π j. The average computing time is shown in the third column, and the average number of iterations is shown under the label Iterations in column 4 (representing the number of master/subproblem iterations), which we will discuss in more detail later in this section. The column labeled # Customers is the average number of customers served by the vehicle in the optimal solution, or equivalently n j=1 x j. Not surprisingly, the number of master/subproblem iterations is closely related to the required CPU time. The greater the number of iterations, the greater the number of Benders cuts added to the relaxed master problem. This implies a greater number of CPLEX solver calls, and a greater number of lower and upper bounds generated by the Lagrangian relaxation, both of which increase the required running time. The impact of different parameter levels on the running time is illustrated in Figure 2-1, again for the case of (m, n) = (3, 10). 38

39 Table 2-4. Computational test results for various levels of capacity and revenue. V ^π j Time Iterations # Customers Exp Profit In Figure 2-1, each bar in the chart corresponds to a customer-specific revenue level, and the results are grouped by vehicle capacity level. The vertical axis shows the average number of iterations required. We can see that for each vehicle capacity level, the heights of the four bars in a row are decreasing as the revenue level increases from level 1 to 4 (recall that the average customer revenue increases in the level number). When all customer revenues increase (all else being equal), we are more likely to assign a higher number of customers to a vehicle, and there will be relatively fewer attractive choices for demand allocation, thus resulting in fewer iterations. At the extreme, for example, when all customers have very high revenues relative to costs, an optimal solution would assign all customers to the vehicle and solve the corresponding multi-item constrained newsvendor problem. As the figure shows, we cannot establish a definite pattern as a function of the capacity level for a given revenue level (the variation in the average number of iterations is reasonably small, and can at least partially be attributed to random variation among problem instances). As we noted previously, the more computationally difficult problems occur when the revenues are closely matched to the overflow costs. As the capacity level increases (in particular at capacity level 4), the 39

40 expected overflow costs tend to decrease for a given number of customers, leading to less computationally intensive problem instances. Referring again to Table 2-4, under each capacity level, the number of customers served and the value of the expected profit both increase as the customer revenue levels increase; similarly, for each revenue level, when the vehicle capacity increases, the number of customers served increases and the expected profit increases. These effects are all quite intuitive and are in accordance with what we would expect in real-world practice. We note that most of the problems were solved with an active vehicle capacity constraint; in the last of the capacity settings (V at level 4), the vehicle capacity constraint was redundant for about half of the instances, which is responsible for the decrease in the number of required iterations at this capacity level. In order to illustrate the convergence of our generalized Benders decomposition approach for our problem, we illustrate an instance solved within five iterations with m = 3, n = 10, V at level 1 and ^π j at level 2, shown in Figure 2-2. We can see that the upper bound (UB) converges quickly at the beginning, which we found was the case in general. So the number iterations required to improve the lower bound (LB) is an important factor with respect to running time. To summarize, our numerical results showed that our generalized Benders decomposition algorithm can solve the multi-item, multi-customer resource allocation problem much faster than well-known benchmark commercial nonlinear solvers (LINDOGlobal, SBB and CoinBonmin), and typically solved the problems we tested within 17 seconds. The optimality performance comparison and parameter analysis provided further evidence of the efficiency and effectiveness of our algorithm. Heuristic Performance To test the performance of our heuristic approach described in the previous section, we consider four variants of our heuristic approach, one with the simple arithmetic average case, the second with γ ij = µ ij, which we refer to as the demand 40

41 Figure 2-1. Number of iterations required for solving problems with different parameter levels. Figure 2-2. GBD convergence illustration when m = 3, n = 10, V is at level 1, and ^π j is at level 2. 41

42 weighted-average case, the third with γ ij = K i which we refer to as the cost weighted-average case, and the fourth one using γ ij = µ ij K i. We denote these by HS, HD, HC and HDC, respectively, in Table 2-5. For comparison purposes, we use the same 160 test instances shown in Table 2-3. For these heuristics, we present the resulting optimality gap in Table 2-3, equal to one minus the Performance Ratio (PR). Table 2-5. Computational test results for average running time (second) and performance of the heuristic approach Data Set HS HD HC HDC m n Time GAP Time GAP Time GAP Time GAP % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % From Table 2-5, we observe that the computational requirements are substantially lower for our heuristic, resulting in computing times around times faster than LINDOGlobal and 4 16 times faster than our Benders decomposition method. The only time consuming element of the heuristic is the computation of the expected value term in the objective function v(x ) for multiple assignment vectors X k, k {1, 2,, n}, while in the Benders decomposition algorithm, each iteration requires the computation of an expected value term and the solution of a linear integer programming problem. The required computation time for HC and HDC is a little longer than for HS and HD as a result of the need to compute the cost parameter K i. The optimality gap on average is 2.61% for HS, 2.64% for HD, 2.33% for HC, and 2.31% for HDC. However, 42

43 the solutions obtained by HS, HD, HC, and HDC are typically different, and so we recommend running all of these methods and choosing the best solution among them, since the running time is very fast when compared to the exact algorithm. As the table shows, the performance of the heuristic approach improves as the number of items increases, as has often been shown to be the case for knapsack problem heuristics in the literature. Because the heuristic approach is intended for larger problem instances for which exact approaches are impractical, this is a promising result for handling problems of large size in reasonable computing times. 2.6 Concluding Remarks This study considered a stochastic resource allocation problem with normally distributed demands for multiple items and a resource capacity constraint. We simultaneously considered the demand allocation and vehicle stocking problem for multiple items. This problem has a broad set of applications in practice, although we focused on the sales visit context. Moreover, our stochastic resource allocation problem arises as a subproblem for solving a multiple resource allocation problem, which serves as one of our future research directions. In this study, we provided an exact solution method and a heuristic approach for solving this problem. The exact method we proposed uses Benders decomposition and transforms the nonlinear mixed integer problem into two smaller ones, one of which is a nonlinear continuous problem that can be solved using Lagrangian relaxation relatively easily, and the other of which is a 0-1 linear integer problem which can be handled effectively using the CPLEX solver. After solving these two problems iteratively, we converge to a globally optimal solution because of the problem s joint convexity properties. The heuristic approach expresses the stock level vector Y in terms of the binary X variables and, by sorting customers in nonincreasing order of a ratio of expected net revenue to variance (motivated by the solution for a single-item selective 43

44 newsvendor problem), we obtain near-optimal solutions (within 2.47% of optimality on average) typically within 1 second. Based on our numerical study, our exact algorithm has proven to be quite efficient when compared with three advanced commercial nonlinear solvers, GAMS/LINDOGlobal, GAMS/SBB, and GAMS/CoinBonmin. Our heuristic approach performed very well for larger problem sizes, which constitute the class of problems for which a heuristic approach would likely be most beneficial in practice. 44

45 CHAPTER 3 ALGORITHMS FOR MULTI-ITEM PROCUREMENT PLANNING WITH CASE PACKS 3.1 Motivation Consumer products such as apparel, home/office furnishings and supplies, and personal hygiene/ beauty products are often characterized by different styles, colors, features and sizes. While each style might sold in a few colors, the number of possible combinations of colors, features, and sizes is often very large. A mismatch between the product specifications sent to a store and those desired by customers can lead to lost sales in some items, while resulting in excess stocks and consequent markdowns in others. These supply and demand imbalances at the store level reduce inventory turnover and decrease gross margins. Retail chains have long recognized this as a store-level operations execution problem (Oracle, 2005). Examples of size and specification variety arise in numerous retail contexts. For example, men s shoes come in nineteen sizes (6 to 15, including half-sizes) and three widths (narrow, standard, and wide), while women s shoes range from 4 to 13 and come in extra-wide as well. The ASTM International subcommittee D13.55 on Body Measurement for Apparel Sizing has standards for, among others, adult female misses figure sizes 2-20, and men s sizes 34 to 60 regular. Most retail stores do not stock all possible sizes (although larger stores typically stock more sizes than smaller stores). Retailers have traditionally used simple solution approaches for the complex problem of deciding the assortment of sizes to hold in stock. In certain cases, firms have used a bell curve centered at the median size (Pessemier [74]) to determine how a store s total order quantity should be distributed among different sizes. Other firms use the previous year s sales history of different sizes at the chain level to make size decisions at the store level (Friend and Walker [37]; Oracle [69]). However, the true demand profile of a set of sizes (e.g., 21% Small, 34% Medium, 28% Large, 17% Extra-Large) can vary with merchandise type, store location, and merchandise attributes like vendor, season, 45

46 and/or fabric, among others. As an example, the retail chain Kohl s R, in conjunction with the software firm SAS R, found that size requirements differed for stores within a city and even among the different departments in a single store (Hajewski [51]). Retail chains sometimes receive bulk packages for each size and then break, sort, and repackage these to meet the demands of individual stores. However, many retail chains prefer to receive case packs (also called pre-packs) from vendors and then use combinations of these case packs to satisfy store demands. The term case pack refers to a set of individual Stock Keeping Units (SKUs) of merchandise packaged into bigger cases for easier handling in the supply chain. Case packs may be used in numerous industries including home/office products, personal hygiene/beauty products, and food products. For example, bulk case packs of food snacks may contain a mix of individual bags of different flavored potato chips, corn chips, and pretzels. Similarly, a department store who carries makeup may receive case packs containing various colors of lipstick or other makeup product. Another example may arise at theme parks (e.g., Walt Disney World), who carry souvenir items associated with multiple children s characters and receive deliveries of case packs containing a mix of different character-related items. The case pack serves as the lowest level of packaging hierarchy, and is designed to flow through from the vendor to the retail store. Handling of these larger case packs rather than individual SKUs proves to be cheaper and faster at all touch points in the vendor-to-retail-store supply chain. Breaking bulk typically involves a great deal of work at a distribution center and requires associated infrastructure and personnel, while distribution centers can process case packs quickly and at minimal cost, often through the use of cross-docking to transfer them to individual stores. A typical case pack for apparel might contain 12, 18, or 24 pieces, though packs of 6 may also be used for replenishment during the season. Each individual case pack typically contains many different sizes. For example, a wholesaler may sell boys solid pique polo shirts in case packs, each having 12 pieces of one color, with assorted sizes. For misses sizes 46

47 ranging from 2 to 16, vendors standard case packs contain more sizes in the mid-range and fewer of the extreme sizes (Reda [78]). An example of the contents of two case packs for casual shirts is shown in Table 3-1. Supply-demand mismatches at the store level result from decision processes involving (Oracle [69]; Parpia et al. [71]): (a) the size demand profile used to make purchases at different stores, (b) the contents of case packs, and (c) the allocation of case packs and sizes to individual stores from the distribution center. In this paper, we focus on (c) and assume that the composition of case packs has been pre-determined. Table 3-1. Example of composition of two case packs with four SKUs. SKUs Small Medium Large X-Large Case Pack Case Pack We consider a retailer s problem of procurement planning when a supplier of multiple items requires purchasing pre-defined case packs (defined by the assortment of included items). We use the term item in this paper to broadly refer to a distinct element of a case pack. Thus, the term items may refer to sizes of the same style-color in a multi-size case pack or multiple colors of the same style-size in a multi-color assortment. While the inclusion of multiple items in one case pack reduces the number of touches individual items experience in the distribution chain (thus reducing overall handling costs), the use of case packs can severely increase the complexity of the retailer s procurement planning. This complexity arises because different case packs (and the items within them) share certain (fixed) order costs and the composition of the case pack requires ordering in pre-defined combinations of multiple SKUs. Furthermore, since stores receive replenishments containing multiple SKUs from the distribution center on a common truck, a fixed order cost is incurred that is virtually independent of the number of different case packs contained in the order (any additional fixed cost for ordering each individual case pack is practically negligible). 47

48 The model and solution methods we develop address this complexity by providing algorithms for solving problems in this class in acceptable computing time. Retailers need to solve such problems routinely in pre-season planning when planning budgetary requirements. In doing so, they often use a demand profile (consisting of fixed forecasts) during planning to create a finite horizon plan (for a season). Clearly, this planning process provides estimates of future operational costs and requirements, and in-season replenishment decisions are made dynamically, based on observed demands and inventory levels (often in the presence of demand uncertainty). However, because little past research has addressed the need for planning in the presence of case packs, we focus on the planning problem using fixed forecasts. As we later discuss, a scenario-based approach can be used in the planning phase to more accurately account for the impacts of demand uncertainty. Large retail chains often have hundreds or thousands of stores and may, therefore, find it worthwhile to ask a vendor to create customized case packs, at an additional cost (Reda [78]). Since a vendors case packs are sold to many customers (while customized case packs are prepared specifically for a particular chain) it is understandable that a premium may be charged for customizing case packs. However, large retail chains can use their bargaining power with vendors to keep this premium low. Retailers can thus use our model to evaluate the impact of proposed customized case packs on potential savings when compared to a vendor s existing case packs. In this paper, we will study the use of vendor case packs and procurement planning for multiple case packs to meet store level demand. While the costs associated with a given set of proposed case packs may be accomplished using our model, the design problem for customized case packs will serve as a future research direction. The remainder of this paper is organized as follows. We discuss related literature in Section 3.2. Section 3.3 defines and formulates the case pack procurement planning problem. In Section 3.4, we study special cases containing one and two case packs, 48

49 and then generalize our methods to N case packs. We establish certain properties of optimal solutions to our case pack planning problem and use these properties to design a dynamic programming method for solving the problem in pseudopolynomial time for a fixed number of case packs. In order to solve larger problems more efficiently, we strengthen the problem formulation by adding specific cutting planes in Section 3.5, and provide a set of heuristic approaches in Section 3.6. We then provide a discussion of potential approaches for dealing with demand uncertainty in Section 3.7. In Section 3.8, we discuss the results of detailed computational results of the implementation of our heuristic approach. In particular, we compare the results for a variety of problem instances of different sizes with solution via the commercial solver CPLEX, and then investigate the sensitivity of the results to changes in several parameters used in the heuristic approach. Finally, concluding remarks are provided in Section Related Literature The procurement planning problem with multiple case packs that we consider is a variant of the class of multi-item dynamic lot sizing problems. Although we do not consider capacity limits, the interdependence among individual items contained in case packs (and associated shared fixed order costs) leads to increased complexity when compared to most uncapacitated lot sizing problems. (We show that this uncapacitated problem is strongly NP-hard in Section 3.3.) The uncapacitated, single-item dynamic lot sizing problem has received a significant amount of attention in the literature since it was first introduced in 1958 by Wagner and Whitin [100]. Their dynamic programming algorithm runs in O(T 2 ) under general cost parameters, where T is the planning horizon length. Subsequently, Federgruen and Tzur [32], Wagelmans et al. [99] and Aggarwal and Park [1] concurrently improved the worst-case complexity for obtaining an optimal solution to O(T log T ) under general cost parameter assumptions, and to O(T ) for problems with stationary costs. 49

50 Stowers and Palekar [90] considered a related problem arising in sheet/metal manufacturing, die-casting, and chemical processing, such that for each production setup, products must be produced in a fixed proportion to one another, and each product can be a part of multiple product families. They view the model as a generalization of the joint replenishment problem, discuss the complexity of the problem for special cases, and solve a model with continuous production quantities, applicable to chemical processing. They use a Lagrangian relaxation approach for solving the problem, while Bhatia and Palekar [17] propose a variable redefinition approach for the problem. In addition, the equivalent of a single case-pack model is transformed to the single-product uncapacitated lot sizing problem in Stowers and Palekar [90]). As we later discuss, the model we consider differs from theirs in the two (or, more generally, multiple) case-pack case, as we require that different case packs incur a shared (or joint) order cost in any period (whereas their model requires incurring a production setup cost for each product family). For multi-item problems with finite production capacities, efficient algorithms are not available, as such problems have been shown to be NP-hard (Florian et al. [35]). As a result, various heuristic approaches have been developed for this class of problems. Polyhedral methods have also led to improved model formulations in order to provide better lower bounds on optimal solution values. Barany et al. [12], Pochet and Wolsey [76], Loparic et al. [63], and Atamturk and Munoz [9] strengthened the original formulation by adding valid inequalities. Krarup and Bilde [59], Belvaux and Wolsey [14], and Wolsey [101] added new variables and/or reformulated the problems in order to strengthen their formulations. For heuristic approaches on generalizations of this problem class involving multiple levels (or machines) and multiple echelons (or facilities), see, for example, Bahl and Ritzman [10], Billington et al. [18], Blackburn and Millen [19], Roll and Karni [79], Parveen and Haque [72], and Akartunali and Miller [4]. 50

51 For multi-item problems with joint order costs, Federguen et al. [29] developed a class of progressive interval heuristics for a capacitated dynamic lot sizing problem with joint setup costs that are incurred once an order is placed in a given period. They also extended the model to allow for item-dependent setup costs in addition to the joint setup costs. Anily et al. [7] also considered a multi-item lot sizing problem with so-called joint setup costs. However, this setup cost is incurred for each batch of production, i.e., batch-dependent setup costs. Another variant of dynamic lot sizing that contains similarities to the case pack planning problem we consider is the class of batch ordering problems, in which items must be produced or ordered in multiples of a certain batch size or capacity Q. This requirement is applicable in situations when the product is produced in a batch process with a fixed capacity Q, or production is performed in order to fill containers or truckloads, each of size (capacity) Q. These batch-ordering problems usually consider a single item, and have been studied by Vander [96], Chand and Sethi [23], and Li et al. [62], among others. Anily et al. [7] provided a polynomial-time algorithm for a multi-item lot-sizing problem in which a batch may include any mix of items. The difference between case pack planning and batch ordering problems is that a batch does not have a specific assortment composition, but has a given total capacity or batch output size, while case pack assortment compositions are pre-determined. Thus, case pack planning problems have a reduced degree of flexibility, as both the batch size and the composition of the batch (number of units of each SKU) are fixed. To the best of our knowledge, the concept of retail case pack planning has received very little attention in the operations planning literature, although it is a widely accepted practice in industry. We have found only one working paper by Freimer et al. [36] that considers case pack ordering problems in a different setting. They assume stochastic and stationary demands, while minimizing expected holding and backorder costs over an infinite horizon. Their model considered only one type of case pack with a single 51

52 or multiple SKUs, and modeled the problem as a Markov decision process. They characterized the form of the optimal policy for both the single-sku and multiple-sku case. However, for the multiple-sku case, the form cannot be summarized quite as succinctly, and they found that the optimal policy may be difficult to implement, even for two SKUs. 3.3 Problem Definition and Formulation This section defines and formulates the case pack procurement planning problem with dynamic deterministic demand over a finite horizon. In order to formalize our model, we define the following notation: Parameters n: case pack index, n = 1,..., N. m: SKU (stock keeping unit) index, m = 1,..., M. s,t: time period indices, s, t = 1,..., T. E mn : nonnegative integer number of units of SKU m contained in case pack n. d mt : store-level demand for SKU m in period t. c n mt: unit production/procurement cost of SKU m in case pack n in period t. s m : clearance price of SKU m sold at the end of the planning horizon, s m < c n mt, t, n. K t : fixed setup cost for placing an order for any number of case packs of any type in period t. h mt : cost of holding a unit of SKU m at the retail store at the end of period t. Let ~c mt n = cmt n + T τ=t h mτ; we assume the values of ~c mt n are non-increasing in time for each item m and each case pack n, which is consistent with the commonly employed assumption of non-speculative motives, in which it costs at least as much to produce an item in period t and hold until period t + i as it does to produce in period t + i, for any nonnegative t, i with t + i T (this assumption is made because it frequently arises in practice and because it leads to properties of optimal solutions that enable solution via dynamic programming with a manageable state space; see, e.g., Van Hoesel and Wagelmans 2000). 52

53 We consider a product line containing M variants (or SKUs). A vendor offers N case packs, each containing a pre-defined assortment of (a subset of) these M SKUs. Case pack n = 1,..., N contains E mn units of item m = 1,..., M. The store may receive a replenishment from the distribution center at the beginning of every replenishment cycle (period) with the assumption of zero or constant lead time, and we assume there are T periods in the planning horizon. The time-varying demand is deterministic for each item at every store, and we assume this demand must be met without shortages. We assume deterministic demand, which is often appropriate for pre-season planning. (Later in Section 3.7 we will discuss methods to handle situations in which demands are uncertain.) We also assume that items within case packs cannot serve as substitutes for one another at the demand level (as a result of, for example, different sizes). We assume the unit cost for each item is known and also time-varying, and the unit clearance price is incurred only at the end of the planning horizon. Even though excess inventory may be marked down gradually during the season, during pre-season planning, a reasonable assumption is that at the end of the season, each unsold item is sold at a clearance (salvage) price that is less than its cost, and that all items can be disposed of at this price. For seasonal or fashion items this assumption is valid and, ideally, we would have no inventory at the end of the season so that no item is sold at clearance price. (For basic items sold throughout the year, e.g., men s white dress shirts, a planner might want to set the ending inventory in the final period to a positive value corresponding to a desired safety stock level.) We also consider a purchasing cost cmt n for SKU m that depends on the individual pack n, which permits modeling prices based on economies of scale and reduced handling requirements associated with different case-pack compositions. A generalization of our model would permit variations in how the items within case packs may be allocated to stores. For example, a distribution center may permit opening certain case packs and sending individual SKUs to stores at some premium. In such 53

54 cases, the individual unit purchase can be accommodated in our model by defining M additional case packs, each of which contains only one unit of item m. Then, since we permit different values of c n mt, this would allow charging a different amount when purchasing individual units. We next define our model s decision variables. Decision Variables x nt : number of case packs of type n ordered in period t. y t : binary variable equal to one if any case pack is ordered in period t, and zero otherwise. I mt : ending inventory of SKU m in period t, assuming no initial inventory at the beginning of period 1. Our model can be formulated as: (P ) min M m=1 [ T h mt I mt + T t=1 t=1 n=1 ] N cmte n mn x nt s m I mt + T K t y t t=1 subject to: N E mn x nt + I m(t 1) = d mt + I mt, m = 1,..., M, t = 1,..., T, (3 1) n=1 x nt M nt y t, n = 1,..., N, t = 1,..., T, (3 2) where M nt is a large number that we can set to M nt x nt Z +, n = 1,..., N, t = 1,..., T, (3 3) I mt 0, m = 1,..., M, t = 1,..., T, (3 4) y t {0, 1}, t = 1,..., T, (3 5) { T } τ=t = max d mτ m=1,...,m E mn, for each case pack n and in every period t, without loss of optimality. The symbol x represents the ceiling function, i.e., the smallest integer not less than x. We also assume the initial condition I m0 = 0 for m = 1,..., M. The first expression in the objective function expresses the holding, purchasing and clearance costs for all case packs, SKUs, and periods, while the second expression corresponds to the order setup costs incurred in all periods. The first constraint set (3 1) 54

55 ensures inventory balance for each item, while constraint set (3 2) ensures that we incur the setup cost in any period in which positive procurement occurs. Constraint set (3 3) requires that case pack order quantities take nonnegative integer values, while the final constraint sets (3 4) and (3 5) enforce nonnegativity of the inventory variables and binary values for the setup variables, respectively. Unlike standard dynamic lot sizing models, we must require that the number of case packs received (x nt ) takes integer values, since we cannot guarantee that this condition will hold in the linear programming (LP) relaxation, even when we require the setup variables to take integer values. We next consider a problem reformulation that permits reducing the number of variables required in the formulation. By rearranging terms in the above formulation, we know that: I mt = ( t N ) E mn x nk d mk, m = 1,..., M, t = 1,..., T. k=1 n=1 Substituting the inventory variables out of the formulation, dropping the constant term M m=1 s T m k=1 d mk, and defining ~C mnt = (cmt n s m + ) T k=t h mk E mn, we obtain: (P) min ( T N t=1 M n=1 m=1 ~C mnt x nt + K t y t ) (3 6) subject to: t N t E mn x nτ d mτ, m = 1,..., M, t = 1,..., T, (3 7) τ=1 n=1 τ=1 x nt M nt y t, n = 1,..., N, t = 1,..., T, (3 8) x nt Z +, n = 1,..., N, t = 1,..., T, (3 9) y t {0, 1}, t = 1,..., T. (3 10) Note that the objective function of (P) is a translation of the objective of (P ), where we have omitted the constant M m=1 T t=1 h mt t k=1 d mk from (3 6). We next characterize the complexity of problem (P). 55

56 Proposition 3.1. Problem (P) is strongly NP-Hard and cannot be approximated to within a factor better than ln M in polynomial time unless the class NP admits an M O(log log M) -time algorithm. Proof. Consider the special case of problem (P) when either the setup cost is zero or when the setup periods have been predetermined. The resulting special case is equivalent to the special case of the lot sizing problem with setup interactions (Stowers and Palekar, 1997) for a fixed setup sequence. Stowers and Palekar (1997) demonstrated the NP-Hardness of this special case by a reduction from set covering. Because the set covering problem is strongly NP-Hard (Garey and Johnson, 1979), this implies that problem (P) is strongly NP-Hard as well. Feige (1998) showed that set covering cannot be approximated to within a factor better than ln M unless the class N P of problems can be solved using slightly superpolynomial time algorithms. Stowers and Palekar [90] considered the case in which setup costs are case-pack dependent. Proposition 1 shows that the problem is strongly NP-Hard when multiple case packs share a joint setup cost in any period. Because of this negative complexity result, we will first consider practical special cases that permit efficient solution. We then generalize the results obtained for these special cases to provide a pseudopolynomial-time solution method for the general problem with a fixed number of case packs. As we have noted, we assume that case pack compositions are pre-defined. Thus, the problem we study falls within a hierarchy of decisions after case pack composition decisions have been made. As Proposition 3.1 indicates, the problem with pre-defined case pack compositions is strongly NP-Hard. We, therefore, begin by addressing this difficult problem class, and leave the integrated case pack composition and procurement planning problem as a promising area of future research. 56

57 3.4 Solution Methods for Special Cases Single Case Pack We first consider the case in which the supplier offers only a single case pack, which we refer to as problem (P 1 ). Suppose that a retail store plans to meet demands for small-, medium-, large-, and extra-large-sizes of a product over a time horizon, say the coming 13 weeks. The store must determine how many case packs to order at the beginning of each period. This model is an extension of the basic dynamic lot-sizing model to accommodate case packs. The basic Wagner-Whitin model contains only one item, whereas in our model, there are M different sizes. Furthermore, we can procure items only by ordering an integer number of case packs containing the items. For example, in Table 3-1, in order to obtain four small shirts, we must order two units of Case Pack 1. Stowers and Palekar [90] showed that the single product family version of the problem they consider is equivalent to a single-item lot sizing problem, after some pre-processing to account for multiple products. Because our single case pack problem is identical to the single product family version they consider, the same holds for problem (P 1 ). We next use problem (P 1 ) to illustrate a shortest path solution approach, which we later generalize to handle multiple case packs. We note first that the Wagner-Whitin zero-inventory-ordering property need not directly apply when multiple items must be purchased in case packs. For example, if an item can only be ordered in multiples of five and the demand in each period is 4 units, it is not necessary that inventory must equal zero before a new order arrives in an optimal solution. We show below that in an optimal solution, the inventory of all SKUs at the beginning of every order period (prior to receiving a replenishment) can be determined. For this new case pack procurement planning problem, the following modified minimum feasible inventory ordering (MFIO) property applies. In this subsection, E m and c mt correspond to E m1 and cmt, 1 respectively, as we omit the case-pack index for simplicity. 57

58 Property 1 (Minimum Feasible Inventory Ordering). An optimal solution exists for the single-case pack problem (P) such that if period t is an order period, then the ending inventory in period t 1 can be computed as I m,t 1 = E m max ~m=1,...,m { t 1 i=1 d ~mi E ~m } t 1 i=1 d mi for each SKU m. Proof. The minimum number of case packs required to meet demand without shortages { t 1 } i=1 in periods 1 through t 1 equals X 1,t 1 = max d ~mi ~m=1,...,m E ~m (ordering any fewer case packs will result in shortages). Because no shortages are allowed, this implies that I m,t 1 E m X 1,t 1 { t 1 i=1 d t 1 } i=1 mi = E m max d ~mi ~m=1,...,m E ~m t 1 i=1 d mi. Let {1 = t 1, t 2,..., t r = t} denote the first r order periods in an optimal solution, where the order in period t corresponds to the r th order placed (note that r 2 since we must order in period 1, assuming d m1 > 0 for at least one m). Suppose that an optimal solution exists such that the total number of case packs ordered in periods {t 1, t 2,..., t r 1 } equals some value ~X 1,tr 1 > X 1,tr 1. Next, consider a solution in which one fewer case pack k is ordered in period t r 1 and one additional case pack k is ordered in period t r, and observe that this new solution does not result in any shortages because ~X 1,tr 1 > X 1,tr 1. The cost of the original solution minus this new solution equals ( M m=1 E mk cm,t k r 1 cm,t k r + ) t r 1 τ=t r 1 h mτ = M m=1 E ( mk ~c k ) ~ck m,tr 1 m,t r 0 (because of our prior assumption of non-increasing values of ~c n mt in time for any item m and any n). Here for single case pack problems, k = 1. In addition, this new solution may also reduce the setup cost K tr 1 when the order amount in period t r 1 reduces to zero by ordering one fewer case pack, while the setup cost K t always exists since we have assumed t is an order period. Thus, the new solution has cost at least as low as the original solution, which implies that either an alternative optimal solution exists satisfying the property, or that the original solution is suboptimal, a contradiction. The MFIO Property (1) enables us to develop a dynamic programming (DP) solution approach for problem (P 1 ). In this DP, time periods serve as stages and end-of-period inventory levels correspond to states (note that the end-of-period t 1 inventory 58

59 equals the inventory at the beginning of period t before an order is placed). Suppose we wish to meet the demands in periods t through t + j, when I m,t 1 is the beginning inventory of size m in period t. Since a case pack contains E m units of size m, the number of case packs needed to meet the demand in periods t through t + j will equal X t,t+j = max ~m=1,...,m { t+j τ=t d ~mτ I ~m(t 1) E ~m }. Property 1 implies that the state vector (of beginning inventory levels before ordering) is independent of the path taken before arriving at a stage. Thus, the ending inventory in each period immediately prior to order placement can be calculated using the order sizes required to meet the demand through the end of the period, i.e., I mt = { ti=1 } d E m max ~mi ~m=1,...,m E ~m t i=1 d mi, t = 1, 2,... T. Since the end-of-period T inventory can be computed this way, the clearance cost is known and can be ignored when determining when and how much to order. Note that Property 1 does not imply that the beginning inventory (before receiving replenishment) in an order period must be insufficient to satisfy demand in the period for at least one item. To illustrate this counterintuitive property, suppose a case pack contains a single SKU in multiples of four, and that the demand in period one equals six. Suppose also that the demand for the SKU in period two equals one, and that positive demands occur in periods 3,..., T. If the holding cost in period one is sufficiently high, and the fixed order cost in period three is sufficiently large, it may be optimal to place an order in period two, despite the fact that sufficient inventory exists at the end of period one to cover demand in period two. Observe that if we know when orders will be placed, the order quantity in each period and the inventory at the end of each period can be determined. The only remaining decision is when to place orders. Thus, we can solve the DP model in a manner similar to the standard dynamic lot-sizing problem or the equipment replacement problem. In particular, we can formulate the single case pack problem 59

60 as a shortest-path problem (for example, Ahuja et al. [3], for a brief description of the shortest-path model for dynamic lot sizing with fixed ordering costs). We therefore create a shortest path network containing T + 1 nodes shown as Figure 3-1. An arc connects node t to each higher numbered node j = t + 1,..., T + 1. Thus, the arc from node t to j implies that consecutive orders occur in periods t and j, and the cost of this arc includes the minimum costs incurred while satisfying demands in periods t through j 1. This arc therefore has a cost equal to K t + M m=1 c mtx t,j 1 E m + H t,j 1, where H t,j 1 denotes the inventory holding cost from period t to period t + j 1, i.e., H t,j 1 = M j 1 h mτ [X t,j 1 E m + I m(t 1) m=1 τ=t ] τ d mi. The expressions in brackets denote receipts, starting inventory, and cumulative demand, respectively. The shortest path from node 1 to T + 1 corresponds to an optimal solution for the single case pack version of (P). i=t Order X 1,t-1 case packs for periods 1,2, 3,..t-1 Order X t,t case packs for periods t,t+1,..t 1 2 t T+1 I m,0 I m,1 I m,t-1 I m,t Figure 3-1. Single case pack problem network Clearly, the number of arcs is bounded by O(T 2 ), and the time required to compute an arc s cost is bounded by O(M(j t)) O(MT ). Thus, the construction of this network is bounded by O(MT 3 ). Using Dijkstra s algorithm (e.g., Ahuja et al. [3]) to solve the shortest path problem then requires O(T 2 ) time. Our discussion illustrates how the single case pack problem differs from the classic single item dynamic lot sizing problem. In particular, we have a known inventory determined by the MFIO property prior to receiving a new replenishment rather than 60

61 the classic zero inventory ordering property. As we have shown, both problems can be solved as shortest path problems Two Case Packs Many retailers have systems and processes to handle only one or two case packs for a particular class of merchandise. We next consider the special case involving two case packs, denoted as (P 2 ). Note that the effective joint order cost K t couples the decisions associated with the different case packs. If the fixed order cost is incurred separately for each case pack, then the nature of the problem changes significantly (this version of the problem is outside the scope of this paper, and we leave this for future research). Our solution approach relies on the insight that for a given number of case packs of the first type ordered up to any period, we can determine how many of the second type would be needed, and, given both of these quantities, we know the ending inventory of each size at the end of every period through a generalization of the MFIO Property. Let CP n denote case pack n (for n = 1, 2 for the two case pack problem). For this problem, we can create a shortest path graph in which each node contains two labels corresponding to the order period and the number of CP 1 cases ordered prior to the period, respectively. That is, the label (t, j) corresponds to order period t when j units of CP 1 have been ordered prior to period t. The associated network has a layered, acyclic structure, where each layer corresponds to a time period (an illustration of this layered network is displayed in Figure 3-2). Define V 1 (V 2 ) as the set of items contained in CP 1 (CP 2 ). Let N 1 (t) denote the minimum number of CP 1 cases required to cover { t demand for items in the set V 1 from period 1 to period t, i.e., N 1 (t) = max i=1 d mi m V E m1 }; { t 1 similarly, N 2 (t) = max i=1 d mi m V E m2 }. An arc from node (t, j) to (s, k) requires that we 2 order k j case packs CP 1 at the beginning of period t in order to cover demand in periods t, t + 1,..., s 1, and any remaining demand in these periods must be covered using a minimal number of case packs CP 2. The associated arc cost includes the setup 61

62 cost and variable production cost in period t, as well as the holding costs incurred from period t to s 1. The following property generalizes the MFIO Property (1). Property 2. An optimal solution exists for the two case pack problem (P 2 ) such that if period t is an order period and j units of case pack CP 1 have been ordered prior to period t, the ending inventory of item m in period t 1, denoted by Im,t 1 j, is given by } I j m,t 1 = E m1 j + E m2 max ~m V 2 { t 1 i=1 d ~mi E ~m1 j E ~m2 t 1 d mi. Proof. The proof follows the same logic as the proof of Property 1, using the condition that j units of CP 1 have been ordered prior to period t and our assumption that the c n mt values are non-increasing in t for each m and n (we thus omit the details). The following two propositions are implied by the above property and our description of the shortest path network structure. Proposition 3.2. The number of nodes required in layer (period) t in the shortest path network equals N 1 (t 1) + 1. Proposition 3.3. An arc exists from node (t, j) to node (s, k) in the shortest path network for with j k max m V 1 i=1 { s 1 i=t d } mi Im,t 1 j + j N 1 (s 1) E m1 max m V 1 [ s 1 ] d mi Im,t 1 j > 0. i=t Our construction of the network requires an additional dummy sink node (T +2) with only one label corresponding to the layer, and connecting all nodes in layer (T + 1) to this dummy sink node. These arcs account for the clearance cost; for example, the arc from node (T + 1, j) to the sink node has cost equal to A (T+1,j),(T+2) = M m=1 I j mt ( s m). 62

63 2, 0 3, 0 T, 0 T+1, 0 T+1, 1 1, 0 2, 1 3, 1 3, 2 T, 1 T, 2 T+1, 2 T+2 T+1, N 1 (T-1) 2, N 1 (1) 3, N 1 (2) T, N 1 (T-1) T+1, N 1 (T) Figure 3-2. Layered network for (P 2 ), the problem with two case packs. + All other regular arc costs can be computed using: s 1 M d mi I A (t,j),(s,k) = K t + c1 mte m1 (k j) + cmte 2 m,t 1 j E m1(k j) i=t m2 max m V 2 E m=1 m2 s 1 M s 1 d mi I h mτ I m,t 1 j m,t 1 j + E E m1(k j) i=t τ m1(k j) + E m2 max d m V τ=t 2 E m2 mi. i=t m=1 By first storing the demand data and taking advantage of previously computed arc costs, we can efficiently obtain an arc s cost within time O(M(s t)) O(MT ). In our network representation, the required number of nodes is O( T i=1 N 1(i)) and the worst-case number of arcs is no more than O(T N 1 (T ) T i=1 N 1(i)) O(N 2 1 (T )T 2 ). Combining this with the fact that computing the cost of an arc requires O(MT ) time, the construction of the network requires O(MT 3 N 2 1 (T )) time. This leads to a pseudopolynomial time approach, because the value of N 1 (T ) depends on the demand quantities (d mt ) and the case pack quantities (E mn ). 63

64 Solving (P 2 ) requires finding a shortest path in this network. Dijkstra s algorithm can guarantee finding the optimal solution with a worst-case running time of O(( T i=1 N 1(i)) 2 ) O(T 2 N 2 1 (T )) N Case Packs In practice, most retailers do not use a large number of case packs for a particular type of merchandise and its various sizes. The benefits gained by more precisely supplying demand are often offset by increased logistics costs. Most vendors do not support a large number of case packs for a particular style color either. However, it is worthwhile to develop solution approaches to solve the more general N case pack problem for any fixed value of N. As mentioned in the introduction, the N case pack problem and its extensions (in future research) could be used to determine the optimal number of case packs to use by iterating on the number of case packs. If there are N case pack types, the above network-based approach for two-case packs can be generalized, although the associated network turns out to be very large. We briefly describe the associated network as follows. Each node contains N labels corresponding to the order period and the number of case packs of type CP n, n = 1,..., N 1, ordered prior to the order period. At layer (period) t in the network, the number of required nodes increases to N 1 n=1 N n(t 1). An arc emanates from node (t, j 1, j 2,..., j N 1 ) to node (s, k 1, k 2,..., k N 1 ) when j n k n s 1 i=t max d mi I m(t,j1,j 2,...,j N 1 ) m Vn E mn + j n N n (s 1), n = 1, 2,..., N 1. Then, generalizing our discussion in the two-case pack case, the worst-case running time of this approach equals O(MNT 3 N 1 n=1 N 2 n (T )), which implies pseudopolynomial time solvability for any fixed value of N. 3.5 Strengthening the Formulation Since the linear programming relaxation of (P) is rather weak, we consider the use of strong valid inequalities to improve the lower bound provided by its LP relaxation. A class of inequalities that has been successful for strengthening the formulation of the 64

65 economic lot sizing problem is known as the set of (l, s) inequalities (Barany et al. [12], who show that these inequalities are facets for the single-item uncapacitated lot sizing problem that are sufficient to describe the convex hull of solutions for this problem). For problem (P), we define associated (l, s) cp inequalities for case pack problems as follows. Given a dividing period l {1,..., T 1}, a nonempty subset S belonging to the period set from period 1 through the dividing period l, S {1,..., l} and its complement S, = S = {1,..., l}\s, we define a case pack based (l, s) cp inequality as: N i S n=1 E mn x ni + i S d m il y i d m 1l, m (3 11) where d m rt = t τ=r d mτ. Proposition 3.4. The (l, s) cp inequalities (3 11) are valid for (P). Proof. For a feasible solution (X, Y ), if y i = 0 for all i S, then x ni = 0 for all i S because x nt M n y t, so N i S n=1 E mn x ni + i S d m il y i = i S N E mn x ni + 0 = n=1 l N E mn x ni d1l m, m i=1 n=1 where the last inequality follows from constraint set (3 7) in (P). Otherwise, let k = min{i S : y i = 1}. Then x ni = 0 for all i S, i < k and i S,i<k n=1 N E mn x ni = i S,i<k n=1 N E mn x ni + i S Combining the above inequalities, we obtain i S,i<k n=1 N E mn x ni = d m il y i d m kl y k = d m kl. k 1 i=1 N E mn x ni d1,k 1, m n=1 N i S n=1 E mn x ni + i S d m il y i i S,i<k N n=1 E mn x ni + i S d m il y i d m 1,k 1 + d m kl = d m 1l, m. This completes the proof. 65

66 Our separation approach for (l, s) cp cuts will be based on the same approach used in the usual separation algorithms for (l, s) inequalities for lot sizing as described by Pochet and Wolsey [77]. We next consider an additional set of inequalities. Letting E max m = max n=1,...,n {E mn}, then we have E max m t τ=1 n=1 N x nτ t N E mn x nτ τ=1 n=1 t d mτ, m = 1,..., M, t = 1,..., T, τ=1 where the second inequality above corresponds to constraint set (3 7). The above inequality, along with the integrality of the x nt variables, implies that the following inequalities are valid for (P): t τ=1 n=1 Then, defining D t = the form N t τ=1 x nτ d mτ max m=1,...,m t E max m, m = 1,..., M, t = 1,..., T. { t } τ=1 d mτ E, we only need to add T such inequalities of m max τ=1 n=1 N x nτ D t, t = 1,..., T. (3 12) Adding inequalities (3 11) and (3 12) helps in improving the quality of the LP relaxation-based lower bound. In order to improve the solution upper bound (UB), we next turn to a heuristic solution approach for providing good feasible solutions, and, hence, upper bounds. 3.6 Heuristic Solution Approach When dealing with large-size, multi-period problems with numerous integer variables, exact algorithms such as conventional branch-and-bound methods can prove to be far too computationally expensive. Heuristic approaches, on the other hand, can often find good solutions relatively early in the search. We thus provide a heuristic framework for problem (P) that guarantees finding a feasible solution. The computational results provided later in Section 3.8 show that the resulting solution 66

67 is often competitive with the best known methods for such large-scale mixed integer programming problems. Relax-and-fix is a constructive heuristic method for solving multistage integer problems. Its general procedure divides the integer variables into several stages. Initially, only the first stage variables are required to be integer, while the remaining variable restrictions are relaxed to be continuous, and this relaxed subproblem is solved to optimality. At successive stages, the integer variables from earlier stages are fixed at the (integer) values they took in the corresponding relaxation, the current stage variables are declared integer, and the remaining variables are relaxed to be continuous. After continuing this process for all stage variables, we may arrive at a feasible solution, or an infeasible solution may result due to, for example, capacity constraints. Such problems sometimes have a special structure in which the earlier stage variables are more critical than later ones, in the hope that trying to maintain the integrality of early stage variables might result in better solutions. By iteratively solving subproblems with fewer integer decision variables, the relax-and-fix heuristic will require much less time than solving the entire mixed integer problem directly. For production planning or lot-sizing problems, the stage usually corresponds to a time interval that includes consecutive periods, called time windows. For a general review of mixed integer programming (MIP) heuristics, including LP-and-fix and relax-and-fix, please see Pochet and Wolsey [77]. The first production planning tool to use relax-and-fix was proposed by Belvaux and Wolsey [13]. Under their approach, the time windows are predefined in length without any overlap. Stadtler [89] extended the basic idea of successive time windows to include overlap between consecutive windows for higher quality results. After solving the subproblem for each time window, only the variables associated with the time periods that do not overlap with the next time window are fixed. The disadvantage of Stadtler s heuristic lies in its more complex implementation. The recent paper by Akartunali and Miller [4] modified prior relax-and-fix approaches by using overlapping time windows and 67

68 time limits for the heuristic. In addition, they added an LP-and-fix subroutine for updating upper bounds, and generated multiple solutions throughout the algorithm. Our heuristics are based on the relax-and-fix scheme modified to account for the case pack planning model and setting. We will allocate a time limit to each of our heuristic approaches. Because the relax-and-fix approach requires repeatedly solving a set of MIPs, the way in which we allocate this time to each time-window subproblem can substantially affect overall heuristic performance. Because the problems associated with earlier time windows are larger (with a greater number of free variables) and hence harder to solve, it is preferable to allocate more time to earlier time-window subproblems in order to employ upper bounding methods and generate good solutions. In order to allocate time to individual time-window subproblems, we first compute the average time per time-window subproblem based on the total time limit set for the heuristic. Then, we divide the time-window subproblems into four sets of equal-size groups; after initial tests we found it advisable to allocate a multiple of 1.75, 1.25, 0.75 and 0.25 of the average time per time-window subproblem to the four groups, respectively, to determine the maximum time allocated to each time-window subproblem. Note that these values add to 4. For example, if T denotes the overall time limit for the heuristic, and W time-window subproblems must be solved, then we allocate 1.75T /W time units to each of the first set of time-window subproblems, 1.5T /W to the second set, 0.75T /W to the third set, and 0.25T /W to the final set. Thus, in addition to a time limit for the overall heuristic approach, a time limit is set for each individual time-window subproblem. Recall that our model differs from traditional lot-sizing problems in that we require an additional set of integer variables corresponding to the number of case packs ordered in each period x nt, for all n and t, in addition to the binary setup variables y t, for all t. We wish to capture the step-wise relaxation approach of the relax-and-fix method and apply it to this case pack planning problem. Our heuristic approach considers two different 68

69 relax-and-fix approaches, along with two approaches for obtaining upper bounds at each step. We first describe the relax-and-fix schemes we used. Scheme A. One application of (classic) relax-and-fix: our time windows (stages) refer to time intervals in which all integer variables, x nt and y t, are forced to be integer. The later periods variables are relaxed to be continuous, while earlier periods variables have been fixed to the optimal values obtained in the earlier stages. We solve this MIP subproblem and fix the variables in the current time window using the solution obtained. We then repeat the process for the following time window, and so on, until the end of horizon is reached. Scheme B. Successive relax-and-fix: in each time window (stage), two MIPs are solved before moving to the next time window. We first require only the binary variables in the current time window to take integer values, while the non-binary integer variables in the time window are relaxed. After solving the first MIP in this stage to obtain binary setup variable values within the time window, we then force the non-binary variables in the time window to take integer values, and solve a second MIP. We then fix the remaining variables in the time window to the resulting integer values, and repeat this process for the next time window, continuing until reaching the end of the horizon. In order to provide multiple solutions (upper bounds) and cutoff values for later windows, we apply two methods, each of which is applied within every time window, described as follows. Method 1. LP-and-fix: this technique first requires solving the LP relaxation of an MIP, fixing the variables that take integral values, and re-solving the restricted MIP with these variables fixed. In our heuristic framework, during the consideration of each time-window stage, if a sufficient amount of allocated time is available, we update the upper bound by fixing the later periods variables that happen to take integer values in the relax-and-fix procedure, and then require all variables in later windows to be integer and solve this restricted MIP. 69

70 Method 2. Rounding-up: since our lot-sizing problems do not have capacity constraints, we can always round up the solution values resulting from the relax-and-fix procedure for each time window stage and obtain a feasible integer solution. This is a simple but quite efficient method, especially for solving large-size problems within a fixed time limit. Depending on which relax-and-fix variant is chosen (A or B) and which upper bounding method is applied (1 or 2), we denote our four heuristic variants as A1, A2, B1 and B2, respectively. The general framework for our heuristic approach is as follows. After generating and adding violated (l, s) cp cuts, the heuristic begins with the first time window, solving (P) with all variables assumed continuous for any period beyond the first time window. (If the successive relax-and-fix variant is chosen, we need to relax the non-binary variables x nt in this window as well. We solve the first MIP and fix the binary variable values y t in this first time window, then require x nt within the window to be integer, solve a second MIP, and then fix these x nt values.) If a sufficient amount of allocated time is available for the time-window subproblem, an upper bound (UB) generating method (1 or 2) is applied to generate a feasible solution for the original problem. When compared with Akartunali and Miller [4], our heuristics do not apply an initialization phase to provide an initial solution, and we only begin generating upper bounds during the second time window, in order to reduce the required running time. One key parameter that affects our heuristic performance is the length of the time window, denoted by α. When using overlapping time windows, the length of the first part of the window that does not overlap with the following time window is called the fixing interval, since the integer variables within this time interval will be fixed once the subproblem is solved. The length of the fixing interval is denoted by β. We report the results of tests on the effects of these parameters in the next section. 70

71 3.7 Addressing Demand Uncertainty This paper primarily focuses on case-pack procurement problems with deterministic demands. Such an assumption may be appropriate or may serve as a close approximation within application areas such durable goods, basic apparel, and sportswear, for example, when demand forecasts are relatively accurate. Accounting for products with unpredictable and/or volatile demand, such as fashion goods, requires adapting our solution approaches to account for the need for safety stock and for the inability to precisely state optimal time-phased order quantities for the entire planning horizon at the beginning of the first period (to illustrate our proposed strategies for handling demand uncertainty, we consider the single-case-pack multi-item special case). We first observe that by virtue of ordering in case packs, we implicitly have a kind of natural or built-in safety stock level for many items, because in the deterministic version of the problem we have I mt > 0 for any item m and period t, except for those item and period pairs (t, m) such that t + 1 is an optimal order period and m = arg max m=1,...,m { ti=1 d mi E m }, for which I m t = 0. In other words, even if we were to treat forecasted demands as if they corresponded to deterministic demands, the use of case packs would, for most items, provide some degree of safety stock buffer, with the degree of this safety stock depending on the case-pack composition, the time between case-pack order replenishments, and the deviation of demands from forecasted values. However, because of significant demand uncertainty in some contexts, it is still not unlikely to encounter supply shortages when using fixed forecasts by substituting expected demands for the model s deterministic demand values. When item demand distributions are bounded (e.g., the demand d mt is a random variable contained in the interval [lb mt, ub mt ]), then if shortages are extremely expensive and/or the decision maker is risk averse, a simple strategy would use the upper bound (ub mt ) values as deterministic demand values, which would enable meeting a 100% service level (subsequent order quantities after the first period could be adjusted to net out remaining 71

72 unplanned item inventory when demand values fall below their upper bounds). In many practical contexts, however, this approach may prove too costly, and we therefore discuss two additional useful approaches below for adapting our model and solution methods to handle demand uncertainty. Strategy 1: Probabilistic Model with Service Level Constraints Under this approach, we state a stochastic version of our case-pack planning problem and transform this to a deterministic model by using the static uncertainty strategy for the probabilistic single-item lot-sizing problem studied by Bookbinder and Tan (1988), under which an order plan must be set at the beginning of the horizon. We assume that the demands d mt are not known with certainty, but their probability density functions are known and all random variables are mutually independent. We then apply service level constraints on the probability of a stockout, i.e., Pr{I mt 0} = Pr{I m0 + t E m x i i=1 t d mi } α, i=1 where α is a desired probability of not stocking out. Since α incorporates management s perception of the cost of backorders, a precise shortage cost is not required. If a stockout occurs, demands are backordered and filled as soon as adequate supply arrives. Letting F 1 (m, t)(α) be the inverse function for the cumulative distribution D function (CDF) of D(m, t) = d m1 + d m2 + + d mt, then we can write our demand satisfaction constraints as t i=1 E m x i F 1 D(m,t) (α) I m0, m, t. We now have an uncertain demand model with an objective of minimizing expected total costs, subject to inventory service-level constraints. We can transform this uncertain demand model to a deterministic one by setting d mt = FD(m,t) 1 (α) FD(m,t 1) (α) and t i=1 d mi = FD(m,t) 1 (α) in the deterministic model, respectively, for all values of m and t (Bookbinder and Tan, 1988). The resulting deterministic problem can be solved 72

73 using exactly the same methods as in subsection 3.4.1, since the optimal solution properties still hold. Moreover, we can adjust order quantities before ordering in an order period based on observed demands (as we are more concerned with the amount on the shelf after receiving an order than we are with adhering to the original order quantity prescribed by the model at the beginning of the planning horizon). We can alternatively modify the Silver-Meal (1973) heuristic (as well as other similar heuristic approaches), which can enable quicker solution as well as an ability to adjust successive order periods and quantities in response to observed demands. Using this approach, we would begin with period 1 and compute the expected cost per F 1 D(m,t) period associated with stocking max (α) m=1,...,m case packs at the beginning of period 1 to satisfy demand in periods 1 through t, for t 1. Let τ denote the smallest value of t such that the expected cost per period associated with satisfying demand in periods 1 through τ + 1 (using the setup in period 1) exceeds the expected cost per period associated with satisfying demand in periods 1 through τ. We then stock F 1 D(m,τ) max (α) m=1,...,m case packs at the beginning of period 1 in order to satisfy E m demand in periods 1 through τ. At the beginning of period τ + 1, we reset the time to period 1, subtract any existing inventory from the FD(m,t) 1 (α) values (or add any item shortages to these values), and repeat the procedure. We can thus apply this approach in a rolling fashion in response to observed demand values by netting out leftover inventory from the stock levels (or adding any existing shortages to the stock levels) required to meet the desired service levels. The approach we have described assumes negligible lead times, and target stock levels (or order-up-to) levels may alternatively be used in the case of positive lead times, although the details of such an approach are beyond the scope of this paper. Strategy 2: Deterministic Model with Lower Bounds on Stock Under this approach, we treat fixed forecasts as deterministic demands, but we incorporate a lower bound (L mt ) on inventory (safety stock) in each period and for every E m 73

74 item, in order to reduce the risk of stockouts as in Loparic et al. (2001). That is, instead of nonnegativity restrictions on inventory levels, we use the following constraints: I mt L mt, m = 1,..., M, t = 1,..., T, (3 13) I m0 = L m0, m = 1,..., M. (3 14) After adding the above constraints to the problem formulation (P ), we can then obtain an equivalent formulation with nonnegative inventory variables by substituting I mt = I mt L mt throughout the formulation, and requiring I mt 0 for all m and t. In order to ensure equivalence, we then define e mt = L mt L m,t 1 and add e mt to the forecasted demand value d mt for all values of m and t (observe that the e mt values may be negative, unless L mt L m,t 1 for all values of m and t; this implies that the modified demand values may be negative as well, although for the majority of practical settings it is likely safe to assume that d mt e mt ). Clearly this serves as a heuristic approach to setting safety stock levels (even in cases in which a target service level is specified), as the lower bound (L mt ) levels must be set prior to having knowledge of the time between replenishment orders, and the resulting model sets all order quantities in advance of the planning horizon. 3.8 Computational Results This section presents the results of a set of computational results intended to characterize the performance of our heuristic approaches for solving the economic lot sizing problem with multiple case packs. All test instances were run on a personal computer with an Intel R Pentium 4 CPU, 3.4 GHz processor with 1.99 GB of RAM. We implemented our heuristic algorithm in C++, with the embedded MIP subproblems solved using ILOG s CPLEX 11.2 solver with Concert Technology. Table 4-1 illustrates the general parameter settings we used in our computational study. Except for the first row, which provides specific values of the time horizon T, the number of case packs N, and the number of items M, all other data shown in the 74

75 table are generated from uniform distributions using the lower and upper limits shown in the table. In all of our computational tests, we used time-invariant cost parameters and case-pack-independent purchasing costs. In such cases, the total purchasing cost over the whole horizon is constant and can thus be ignored. The three choices for the parameters T and M, and the four choices for N, form a total of ten problem sets, where each set is characterized by a unique combination of these three parameter values. Table 3-2. Parameter distributions used in computational tests. Parameter data Period, T Case pack, N SKU, M 10, 30, 50 3, 5, 7, 10 3, 6, 10 Unit clearance loss, c m s m Unit holding cost, h m Fixed ordering cost, K U[0.1, 0.5] U[1, 1.5] U[10, 25] Demand, d mt Composition of case packs, E mn U[40, 100] U[3, 20] Heuristic performance and comparison with CPLEX In this subsection, we consider our heuristic approach with a time window length of α = 3 and a fixing interval β = 1, with an overlap of two periods between successive windows. As discussed in the previous section, we consider four variants of our heuristic framework by considering each combination of the relax-and-fix variants (A and B) and the upper bound methods (1 and 2), denoted as A1, A2, B1 and B2. For heuristics A1 and B1, we also set an absolute MIP gap parameter for LP-and-fix, so that if we obtain an upper bound with an integer gap less than this preset amount (here we set it to be 1%), the LP-and-fix upper bounding method stops for this window and moves on to the next window. We compare these four approaches to determine which provides superior performance. For each problem set, we ran 10 randomly generated problem instances, for a total of 100 test instances. Then we compared the average running time and optimality performance of our heuristic approach with those obtained by solution of the problem directly via the CPLEX 11.1 solver. 75

76 Table 3-3 indicates the number of windows processed and the number of (l, s) cp cuts and the additional set of inequalities, denoted as Extra Cuts, added to the problem formulation. Generally speaking, the (l, s) cp inequalities help to improve the lower bounds obtained at the root node during the application of CPLEX s branch and bound method. The larger the problem s size, the more cuts are added to the initial problem formulation. For the Extra Cuts, although the number of added such cuts is smaller, they still play a role toward getting better lower bounds. Additionally, problems with a longer planning horizon require processing a greater number of time-window subproblems. Table 3-3. The number of time windows and cuts used within relax-and-fix heuristics. T N M # Windows #(l, s) Cuts # Extra cuts Tables 3-4 and 3-5 summarize the average running time, optimality gap, and solution information resulting from the application of our heuristics and CPLEX (assuming a 600s time limit for the heuristics and CPLEX). Our heuristic approach runs faster when compared to CPLEX. For most cases, it stops within 1 minute and generates a good solution with a smaller optimality gap. CPLEX, on the other hand, frequently reached the time limit of 600s, and stopped with a suboptimal solution, even for medium-size problems. For the largest problem set, the solutions obtained by our heuristics have gaps as small as half that of CPLEX, obtained in substantially less time. Table 3-5 reports the number of times that the solution method corresponding to the column provided the best solution values, among 10 randomly generated instances 76

77 Table 3-4. Computational test results for average running time (seconds) and optimality gap for solution approaches. Data set CPLEX A1 B1 A2 B2 T N M Time %Gap Time %Gap Time %Gap Time %Gap Time %Gap Gap is calculated as (Sol. Best LB)/Best LB 100%. within each problem set. Note that in some cases, each of the methods generated the same bounds. Except for the smallest two problem sets, most of the best solutions were generated by our heuristics. Comparison between our heuristic variants shows that the A series (one application of relax-and-fix) is faster than B (successive relax-and-fix). The reason for this can be attributed to the fact that solving two MIPs within each time window for the B series consumes more time than solving one MIP. However, the B series generally provides better solutions than A. Method 2 (rounding) is faster than 1 (LP-and-fix) because of its simpler implementation, while method 1 generally provides a tighter UB (cutoff value) and thus better solutions than 2, except for the largest problems, where the time limit for each window and the stopping criterion for LP-and-fix restrict its performance. Overall, Heuristic A2 is the fastest, while B1 provides the smallest optimality gaps Sensitivity analysis of parameters We next consider the effects of two parameters that have a strong effect on heuristic performance: the time-window length α, and the length of the fixing interval β. For these tests, in the interest of speed, we chose to focus on heuristics A1 and A2. First we compare the results (shown in Table 3-6 and Figure 3-3) under different time-window lengths (i.e., α = 2, 3, 4, 5) for our medium-size problem set with T = 30, 77

78 Table 3-5. Number of times (out of ten) each solution method attained the lowest-cost solution. Data set # Best solutions T N M CPLEX A1 B1 A2 B N = 5, and M = 6. As expected, as the time-window length increased, the average running time increased, because more time is needed to solve the MIPs with fewer variables relaxed beyond the time window; the number of windows processed is reduced accordingly. From the distribution of best and average solution values, we see that there is a relative best choice for the length of the time window. For this problem size, α = 4 appears to be the best choice. As the α value increases, the performance tends to improve initially and then become worse. This phenomenon can be explained as follows. The shorter the time-window length, the easier it is to solve the related subproblem and thus solution quality tends to deteriorate because the decisions become more myopic. Moreover, the number of time windows grows and the time allocated to each time-window subproblem decreases. On the other hand, if the time window tends to be very big, finding a time-window subproblem solution may take quite long. The best choice of time window α thus depends on the size of the problem. We next perform sensitivity analysis for the length of the fixing interval, β (shown in Table 3-7 and Figure 3-3). It indicates that β = 1 is the best choice for this medium-size problem set. As the β value increases, the running time decreases for each heuristic, because more variables become fixed and the MIP subproblems are easier to solve; the 78

79 Table 3-6. Effects of the time-window length, α, on solution value for the medium-size problem set. T = 30, N = 5, M = 6 Instance A1 A2 α value: Avg. time(s): Avg. win #: Avg. solu: number of time windows processed decreases as the overlap between windows gets smaller. Table 3-7. Effects of the length of fixing interval, β, on solution value for the medium-size problem set. T = 30, N = 5, M = 6 Instance A1 A2 β value: Avg. value: Avg. time(s): Avg. win #: Cost of requiring a case pack In addition to gauging the algorithmic performance of our solution approaches, an interesting question for management concerns the impact that requiring case 79

80 Figure 3-3. Effects of time window parameters on solution value. (T=30, N=5, M=6) packs has on the buyer s cost performance. In order to estimate this cost, we would like to be able to determine an optimal case-pack composition for the buyer. Finding an optimal case-pack design and procurement plan, however, requires solving an extremely challenging nonlinear optimization problem with a bilinear objective and a set of bilinear constraints (in which both the objective and constraint set (3 1) of formulation (P ) would now contain a product of E mn and x nt variables). For example, for even small problem sizes (with N = 3 or more case packs), our computational experience 80

81 indicates that the nonlinear integer commercial solver GAMS/BARON may often require more than 5,000 seconds to find an optimal solution (a more detailed analysis of the complexity of the integrated case-pack design and procurement planning problem can be found in Chen and Geunes [24]). Thus, we are only able to obtain an optimal case-pack composition for relatively small problem sizes using GAMS/BARON. In order to gain some initial insight on this problem, we considered a small set of test problems involving M = 2 items and T = 8 planning periods. Clearly if M = 2, then we need only consider problems such that the number of case packs, N, is equal to either one or two (with N = 1 we must necessarily include both SKUs in a single case pack, while the case of N = 2 permits each item to have its own case pack). Thus, this setting enables us to consider the retailer s incremental cost that results from requiring a case pack. Using the cost and demand generation parameters in Table 3-2, we generated a baseline set of 12 problem instances in order to assess the value of permitting N = 2 case packs (relative to the cost of requiring N = 1 case pack). For this baseline set of instances, the average percentage savings of permitting N = 2 case packs was equal to 3.72%. Thus, the holding cost penalty for requiring one case pack is nontrivial, and these costs must be balanced against the cost savings that can be gained in warehousing operations from using the case pack. In addition to considering the baseline set of 12 problem instances, we also considered three additional test sets that altered the original 12 instances by (a) doubling the variable purchase and holding costs of one of the items; (b) doubling the demand values of one of the items; and (c) multiplied each fixed order cost by ten (each of these changes was made with respect to the original baseline problems, i.e., these changes were not applied successively to the data). Our goal in running these additional 36 test problems was to determine how each of these factors influences the relative cost of requiring a case pack. These changes resulted in cost penalties of (a) 3.48%, (b) 2.58%, and (c) 1.6%, for each of the respective cases. Our results showed that these percentage decreases (with respect 81

82 to the baseline case) were largely a result of the fact that the magnitude of savings changed very little, while the overall problem cost increased as a result of the problem data increasing. We also observed that when we doubled the demand of one item (to be roughly twice that of the other item), then the optimal single-case-pack composition contained a 2-to-1 ratio of the higher-demand item to the lower-demand item. We must interpret these results with some caution, however, because these results are highly dependent on our parameter choices (which we selected to reflect settings involving a mix of relatively low cost items). 3.9 Concluding Remarks This chapter considered a procurement planning problem with multiple case packs. Although such pre-packing can lead to significant savings in handling and delivery costs compared to breaking bulk, it may reduce flexibility. The resulting model provides a previously unexplored, but practical variant of the dynamic lot size problem. We showed that if items are available only in case packs, i.e., they cannot be obtained in units, the classic Wagner-Whitin zero inventory ordering property does not hold. Exploiting special properties of optimal solutions permitted solution via dynamic program within pseudopolynomial time for a fixed number of case packs. In particular, we showed that although the zero-inventory ordering property does not directly apply, we can determine the ending inventory prior to order replenishment in an optimal solution. This leads to a dynamic programming approach for solving the single case-pack problem. In order to solve multiple case pack problems more efficiently, we strengthened the formulation by adding case-pack specific (l, s) cutting planes, and provided a set of heuristic approaches. These heuristics were based on a framework that uses a relax-and-fix procedure and embedded upper bounding subroutines in order to provide high quality feasible solutions in fast computing time. To make the chapter more applicable to practice, we also provide several efficient strategies and guidelines to cope with demand uncertainty. 82

83 We compared the performance of our heuristic approaches with the benchmark solver CPLEX, and showed that these heuristics are able to quickly provide very good solutions. We also discussed the results of detailed sensitivity analysis for three heuristic control parameters (the length of the time window, and the length of the fixing interval). For the choice of the time-window length and the fixing interval, we demonstrated the trade off between running time and solution quality. An interesting extension of this problem is the allowance for the possibility of customized case packs. In such cases, a retail chain must determine the composition of the customized case packs, as well as the associated procurement plan, which is the subject of Chapter 4. Another interesting area for further research would consider problems that assign available case packs at warehouses to downstream retail stores in order to maximize system-wide profit. 83

84 CHAPTER 4 INTEGRATED CASE-PACK CONFIGURATION AND PROCUREMENT PLANNING 4.1 Motivation In retail supply chains, merchandise is often packed in cases containing a mix of multiple stock keeping units (SKUs). These case packs reduce the number of individual touches an individual product receives in a distribution chain, which streamlines handling costs. Case packs sometimes, therefore, form the lowest level of packaging in a distribution system, and are designed to move from the vendor to a retail store without being opened. More and more retail chains are moving to the use of case packs to satisfy store demands instead of receiving bulk packages for each SKU at distribution centers (DCs), and then breaking, sorting, and repackaging individual units for delivery to stores. This is because breaking bulk typically involves a great deal of coordination and work at the DC, including associated infrastructure and personnel, while DCs can more quickly process case packs at lower cost, often by using cross-docking to transfer items to individual stores. The use of case packs is prevalent in the shoe and apparel industry, where a typical case pack might contain 12, 18, or 24 items, though packs of 6 may also be used for replenishment during a selling season. A case pack may contain similar items with small differences in only one feature or dimension, such as size. For example, a wholesaler may sell sport shirts in case packs, each having 12 pieces of one style, with assorted sizes. The primary reason for this practice is that there is a large variety of SKUs in each style, while the demand for each individual SKU might be low. For example, for men s shoe sizes ranging from 3 to 14, vendors standard case packs contain more sizes in the mid-range and fewer of the extreme sizes. While the inclusion of multiple items in a case pack reduces the number of touches individual items experience in the distribution chain, determining the composition of a case pack can be a major challenge, and the use of case packs can substantially 84

85 increase the complexity involved in retailer procurement planning. This complexity arises because different case packs (and the items within them) share certain (fixed) order costs, and the composition of the case pack requires ordering in defined combinations of multiple SKUs. Fully exploiting the advantages of using case packs in distribution requires understanding how case packs reduce handling costs as well as how case-pack composition affects replenishment and inventory holding costs. Corresponding relevant research questions include: 1. How many case pack patterns should be used in a given system? 2. Which SKUs and what SKU quantities should be contained in each case pack? 3. How should inventory replenishment be planned to ensure that a retailer meets demand with as little leftover inventory as possible, with a goal of minimizing the total procurement-related cost? Designing larger case-pack configurations containing a greater number of SKUs implies handling cost advantages, but also reduces ordering flexibility and consequently increases the probability of overstocking and/or the need for breaking bulk at the DC. Thus, it is necessary to balance these tradeoffs in order to take full advantage of the reduced DC handling costs associated with case packs. Case-pack designs are normally determined by the retailer in consultation with a vendor (Chettri and Sharma [26]). In practice, the research questions above are usually addressed sequentially (by first determining case-pack configurations and then making replenishment decisions), which leads to sub-optimal decision making. Clearly, the replenishment problem is heavily influenced (and constrained) by the number and composition of the different case packs available. This motivates our modeling and solution approaches for solving integrated joint case-pack configuration and procurement planning decision problems. In this chapter, we assume that individual items must be assigned to a limited number of case packs (defined by an assortment of items included within a larger case 85

86 or container). Each item has an item- and period-dependent, per-unit ordering and holding cost. Furthermore, since stores receive replenishments containing multiple SKUs from a DC on a common truck, a fixed order cost is incurred that is essentially independent of the number of different case packs contained in the order (any additional fixed cost for ordering each individual case pack is practically negligible). The goal is to jointly determine the case-pack composition and the case-pack replenishment quantities such that the demand for all items is satisfied and the total cost is minimized. We develop a model that addresses this complex and practical problem class and provide algorithms for solving it in acceptable computing time. The remainder of this chapter is organized as follows. We begin by discussing related literature in Section 4.2. Section 4.3 describes and formulates the joint case-pack design and procurement problem, which is an integer nonlinear (nonconvex) optimization problem. We first propose an exact linearization method for small problem instances using a reformulation in Section Then, in Section 4.4.2, we describe a basic iterative heuristic approach that alternatively fixes case-pack composition and procurement decision variables and solves the remaining restricted problem optimally. In Section 4.4.3, we then modify the iterative heuristic to improve performance by changing the initial bound on case-pack capacity. To capture the benefits of integrated decision making, Section further studies another iterative approach that alteratively considers a geometric programming problem and a dynamic lot-sizing problem, while optimizing the case-pack composition and order quantities simultaneously. In section 4.5 we discuss the details of an implementation of our approaches. Finally, concluding remarks are presented in Section Related Literature Our combined case-pack design and procurement planning problem is related to several streams of past literature. The first of these streams is the multi-item lot-sizing problem with joint setup costs and capacity. Federguen et al. [29] developed a class 86

87 of progressive interval heuristics for a capacitated, multi-item dynamic lot-sizing problem with joint setup costs. Anily et al. [7] also considered a multi-item problem with joint setup costs. However, in this case, the setup cost is incurred for each batch of production, where each batch (or truckload) has a constant capacity consisting of any mix of the items. After assuming the storage costs obey certain dominance conditions, Anily et al. [7] reformulate the model by introducing surrogate products and adding inequalities. They then describe the convex hull of solutions and propose a linear programming approach. In general, dynamic lot-sizing problems with capacity constraints are notoriously difficult problems. When only one batch is allowed in each period and capacity limits are time-varying, even the single-item case is NP-Hard (Florian et al. [35]). No efficient solution methods are known for such multi-item problems, with the exception of Anily and Tzur s [8] dynamic programming algorithm for the case involving constant capacities and cost parameters with deterministic demands, which they show is polynomially solvable when the number of items is fixed (but is exponential otherwise). The lot-sizing literature considers hard capacities (e.g., Florian et al. [35]) as well as soft capacities in the form of batch quantities (e.g., Pochet and Wolsey [75]). In the latter case, nq units of capacity may be made available for any positive integer value n (where Q denotes the batch capacity) at a cost of Fn, where F denotes the fixed batch capacity cost. Case-pack procurement problems differ from problems with soft capacities because the latter class of problems does not require a fixed number of each product in each batch, but allows the batch size Q to contain any mix of items, while case-pack compositions are fixed. Thus, case-pack procurement problems have a reduced degree of flexibility, as both the batch size and the composition of the batch (number of units of each SKU) are fixed. Stowers and Palekar [90] and Bhatia and Palekar [17] considered a similar class of lot-sizing problems with so-called strong set-up interactions, where multiple products can be produced only in fixed 87

88 proportions to each other. They studied the continuous production variant (i.e., when order quantities are not necessarily integer), when setup costs exist for each product group. Stowers and Palekar [90] showed that the problem is NP-hard and presented a Lagrangian relaxation approach for solving the linear programming (LP) relaxation, while Bhatia and Palekar [17] used a variable redefinition approach. Chen et al. [25] proposed a polynomial-time shortest path algorithm for the special case involving a single case pack, and a pseudo-polynomial time algorithm for the general multiple-case-pack problem when the number of case packs is fixed. For larger problems, they applied valid inequalities to strengthen the problem s LP relaxation, and proposed heuristics for solving the mixed integer problem. In each of these cases, the assortment and number of items in a case pack (or the proportions in which different items must be produced) are exogenous to the model. In contrast, we consider the problem in which the decision variables include not only the order quantities (of the individual case packs) as addressed in aforementioned literature, but the composition (or design) of each case pack as well. If we restrict our problem to a single period setting, then this special case is similar to the cutting stock and modular design problems. The cutting stock problem is a well studied class of integer linear programming problems first introduced by Gilmore and Gomory [42]. A few past works have considered combined cutting stock and lot-sizing problems. Nonas and Thorstenson ([67], [68]) considered the combined problem under static and deterministic conditions. In the absence of joint setup costs and integer lot sizes, they suggested several specialized heuristics. Gramani and Franca [46] considered a dynamic lot-sizing problem where the integrality constraints on lot sizes were relaxed. They solved this problem heuristically, using an analogous shortest path algorithm in which each arc corresponds to a cutting stock problem parameterized by the cutting pattern variables. This approach did not apply any limit on the number of plate (case-pack) types, and assumed that pattern changes are costless. As a result, a large number of different 88

89 patterns may be used to cover the exact demand, and the chosen patterns can also change over time. Gramani et al. [47] then developed a slightly different model that ignored setup costs and relaxed integrality restrictions on lot sizes, and solved this problem using a column generation technique. However, this model also applied no restrictions on the number of patterns, and thus the number of different patterns in the solution is usually very large. To reduce the number of cutting pattern changes, heuristic methods have been studied for cutting stock problems, including a sequential heuristic procedure where the pattern change cost is incorporated in the objective (Haessler, [50], Lefrancois and Gascon [61]), as well as a pattern combination heuristic (Johnston [55], Goulimis [45]). For pattern minimization problems in which the objective is only to minimize the number of different cutting patterns, exact algorithms exist (e.g., Vanderbeck [97], Umetani et al. [94]). In addition to these linear formulations of cutting stock problems that parameterize on cutting pattern variables, some past works formulate this problem as a non-convex, integer, non-linear trim-loss problem, and then linearize and solve the problem using existing linearization methods (Harjunkoski and Westerlund [52]), an approach that is consistent with the linearization approach we later discuss. If there is no physical restriction on the sheet size in the cutting stock problem, the resulting problem becomes a nonlinear Modular Design Problem (MDP). This is a single-period problem first proposed by Evans [28], where both the module size and its composition, in terms of parts, must be determined. Most papers in the literature on MDPs target the single module and continuous versions of the problem. Passy [73], Smeers [87], and Shaftel and Thompson [82] successively developed and refined similar efficient simplex-like algorithms by exploiting the special geometric programming structure, or by using the Karush-Kuhn-Tucker (KKT) conditions. Only the paper by Shaftel [81] investigated the problem in which an integer solution is required, using an enumeration method when the problem is small. Al-Khayyal [6] modeled the MDP as 89

90 a generalized bilinear program, and derived an LP relaxation as a subproblem that is solved based on outer approximation and branch and bound methods. A recent paper on the MDP by Benson [16] presented duality results and provided two approaches (generalized Benders decomposition and the separable programming approach) for solving the MDP. Although illustrative examples were provided for these approaches, no computational results for problems of practical size were presented. Silverman [86], Shaftel and Thompson [83], and Al-Lhayyal [6] have studied the multiple module design problem, although the approaches they suggested are only guaranteed to lead to good local solutions for the problem. In the problem we consider, we not only need to design the case-pack composition, as in the MDP problem, but we must also solve an embedded multi-period, multi-item dynamic lot-sizing problem for case-pack replenishment. For practical purposes, an upper limit on the number of case-pack types used over time should be enforced, since most retailers do not wish to handle a large number of case packs, as this imposes too high a degree of complexity on the distribution system and reduces the benefits of using case packs. To the best of our knowledge, the combined dynamic case-pack procurement and design problem with pattern limitations has not been addressed previously in the literature. Our contributions include analyzing the problem s special structure and providing efficient linear and nonlinear solution methods for solving this class of problems. 4.3 Problem Definition and Formulation This section defines and formulates the case-pack design and procurement planning problem with dynamic deterministic demand over a finite horizon. In order to formalize our model, we define the following notation: 90

91 Parameters n: case-pack index, n = 1,..., N, where N is the total number of all possible case packs. m: SKU (stock keeping unit) index, m = 1,..., M. s,t: time period indices, s, t = 1,..., T. π: maximum number of case packs allowed over all time periods. R: case-pack capacity in terms of the number of items (we assume that each individual item consumes one unit of case-pack capacity, and R M). d mt : store-level demand for SKU m in period t. c mt : unit production/procurement cost of item m in period t. O t : cost for placing an order containing any number of case packs of any type in period t. h mt : cost of holding a unit of SKU m at the retail store at the end of period t. Let ~c mt = c mt + T τ=t h mτ; we assume the values of ~c mt are non-increasing in time for each item m and each case pack n. We next define our model s decision variables. Decision Variables x nt : integer number of case packs of type n ordered in period t. z t : binary variable equal to one if any case pack is ordered in period t, and zero otherwise. E mn : nonnegative integer variable representing the number of units of SKU m contained in case pack n. I mt : ending inventory of SKU m in period t, assuming no initial inventory at the beginning of period 1. We formulate the case-pack design and procurement planning problem as follows. (P ) min T M h mt I mt + T t=1 m=1 π M c mt E mn x nt + T O t z t t=1 n=1 m=1 t=1 91

92 subject to: π E mn x nt + I m(t 1) = d mt + I mt, m = 1,..., M, t = 1,..., T, (4 1) n=1 M E mn R, n = 1,..., π, (4 2) m=1 x nt M t z t, n = 1,..., π, t = 1,..., T, (4 3) I mt 0, m = 1,..., M, t = 1,..., T, (4 4) x nt, E mn Z +, n = 1,..., π, t = 1,..., T, m = 1,...,(4 5) M, z t {0, 1}, n = 1,..., π, t = 1,..., T, (4 6) where M t is a large number that we can set to M t = without loss of optimality. { T max m=1,...,m j=t mt} d in period t, The objective function expresses, respectively, the the inventory costs for all SKUs, the purchasing cost for all case packs, and the order setup costs over all periods. The first constraint set (4 1) ensures inventory balance for each item, while the case-pack capacity limit is given by (4 2). The constraint set (4 3) ensures that we incur the setup cost in any period in which positive procurement occurs, while the final constraint sets (4 4), (4 5) and (4 6) enforce nonnegativity of the inventory variables, nonnegative integer values for the case-pack order quantities x nt and composition variables E mn, and binary values for the setup variables, respectively. In the case-pack capacity constraints (4 2), the parameter R provides an upper bound on the largest feasible case-pack size. In practice, the case-pack size may also be a decision variable itself. Thus, the quantity R may serve as an upper limit, and an optimal solution may result containing fewer than R items. That is, if an optimal solution results in which constraint (4 2) is not tight for some case-pack n, we can design a case pack whose size is determined by the optimal value of the left-hand side of this constraint. As we later discuss, the results of our iterative heuristic methods may be strongly influenced by the value of R used in the problem formulation, and our heuristic approach will therefore parameterize on R in order to improve solution quality. 92

93 Observe that if the maximum number of case packs, π, is greater than or equal to M (the number of SKUs), an optimal solution will create M case packs, each containing one unit of one SKU. In practice, however, the goal is to design a small number of case packs in order to reduce DC handling costs. Thus, we typically observe practical problem instances in which π is much smaller than M. By rearranging terms in the above formulation, we know that: ( t π ) I mt = E mn x nk d mk, m = 1,..., M, t = 1,..., T. k=1 n=1 Substituting the inventory variables out of the formulation (P ) and defining ~c mt = c mt + T τ=t h mτ and D mt = t τ=1 d mτ, we obtain the following more compact expression of the model: (P) min ( T π ) M ~c mt E mn x nt + O t z t t=1 n=1 m=1 (4 7) subject to: t τ=1 n=1 π E mn x nτ D mt, m = 1,..., M, t = 1,..., T, (4 8) M E mn R, n = 1,..., π, (4 9) m=1 x nt M t z t, n = 1,..., π, t = 1,..., T, (4 10) x nt, E mn Z +, n = 1,..., π, t = 1,..., T, m = 1,..., M,(4 11) z t {0, 1}, t = 1,..., T. (4 12) We might also consider formulating problem (P) by enumerating all possible case-pack patterns N and choosing some subset of these, using a set partitioning formulation. In this approach, the set of E mn values play the role of a set of parameters of dimension M N instead of a set of decision variables of dimension M π. However, limiting the number of case packs would then require defining a new binary variable y n, for n = 1,..., N, indicating whether pattern n is selected. We would then need to add the constraint N n=1 y n π to an appropriate set-partitioning formulation of 93

94 the problem. Such large-scale problems are often effectively solved using column generation approaches. However, in addition to the set partitioning constraints, we will also need forcing constraints of the form T t=1 N n=1 x nt My n, where M is a large positive number. This will effectively destroy the special structure of the set partitioning formulation that enables effective solution via column generation (because we cannot determine appropriate dual prices for such constraints before a new pattern is generated). We will therefore focus on the problem as formulated in (P) and develop effective algorithms for solving this problem. We first characterize the complexity of problem (P). First note that the nonlinear terms E mn x nt exist both in the objective and constraints, which implies that it is a quadratically constrained quadratic problem (QCQP). To be more specific, the quadratic terms are all bilinear terms, so it is a bilinear programming problem, which belongs to the class of nonconvex optimization problems (Pardalos and Rosen [70]). The following observation further addresses its complexity. Observation 1. Problem (P) is Strongly NP-Hard. Proof. If we fix the values of all E mn variables in (P), the problem turns out to be a case-pack procurement problem which has been shown to be strongly NP-Hard by reduction from set covering in Chen et al. (2010). It implies that if we have an instance of the strongly NP-Hard case-pack procurement problem with parameters ^E mn, then we can form an instance of our (P) problem by setting E nt = ^E mn in polynomial time. This completes our proof. 4.4 Solution Methods This section proposes four different solution approaches. In Section we discuss a reformulation and linearization approach that permits solving the problem using a commercial mixed integer linear programming (MILP) solver, such as CPLEX. Section then considers an iterative heuristic approach that alternates between fixed values of the case-pack design variables and production planning variables. 94

95 Section modifies this heuristic by applying it using each level of possible case-pack size. We then present an integrated geometric programming based approach for the single-case-pack special case in Section Reformulation and linearization method This section proposes a reformulation and linearization approach that will permit solution via an MILP solver. The nonlinear terms E mn x nt in both the objective function and constraints of (P) make the problem more difficult than a typical MILP problem. Moreover, these two sets of variables are both general integers, and so the first step we take in linearizing the problem is to change each of these general integer bilinear terms to the product of a single binary variable and a general integer variable. We first express each E mn as a weighted sum of binary variables, β mnk, as follows: E mn = K 2 k 1 β mnk, K = log 2 R, m = 1,..., M, n = 1,..., N, k=1 and introduce a set of continuous slack variables, s mntk, in place of each corresponding bilinear term β mnk x nt. Then, using an outer approximation, it is well known that the convex envelope of the two dimensional bilinear function f (x, y) = xy on the hyperrectangle {(x, y) R 2 : x l x x u, y l y y u } is given by ϕ(x, y) = max{x l y + y l x x l y l, x u y + y u x x u y u }, while its concave envelope is given by ψ(x, y) = min{x l y + y u x x l y u, x u y + y l x x u y l } (Al-Khayyal [6]). In our problem, since 0 β mnk 1, 0 x nt M t, we can add the following four sets of inequalities to the original problem to eliminate the nonlinear term E mn x nt in both the objective function and the constraints, while maintaining equivalence with our original formulation. s mntk x nt + M nt β mnk M nt, m, n, k, t, (4 13) s mntk 0, m, n, k, t, (4 14) s mntk M nt β mnk, m, n, k, t, (4 15) s mntk x nt, m, n, k, t. (4 16) 95

96 The linearized reformulation of problem (P) is then written as follows. (PL) min T π M K ~c mt 2 k 1 s mntk + T t=1 n=1 m=1 k=1 Ot=1z T t t=1 subject to: t N τ=1 n=1 k=1 M m=1 k=1 K 2 k 1 s mntk D mt, t, m, K 2 k 1 β mnk R, n, x nt M t z t, n, t, s mntk x nt, m, n, t, k, s mntk x nt M nt (1 β mnk ), m, n, t, k, s mntk M nt β mnk, m, n, t, k, x nt Z +, n, t, z t, β mnk {0, 1}, m, n, t, k. This formulation clearly has a high dimension and therefore may only be practical for small-size problems, as our computational tests later illustrate An iterative heuristic Because the problem s difficulty stems from the bilinear terms, we consider an iterative procedure that successively fixes one set of variables in the bilinear terms and uses the results as input for the next iteration. While this kind of algorithm often performs well in practice, it tends to converge to a suboptimal solution (Mangasarian [64]). This approach begins with a case-pack design and determines an optimal procurement plan for this case-pack composition. Then, we determine an optimal case-pack design for the given procurement plan. For ease of exposition, we present the approach in detail for the single-case-pack special case. At the end of the section we then discuss the implications for the multiple case-pack problem. As we show below, our algorithm quickly converges within at most two iterations, due to certain special properties in the 96

97 single-case-pack special case. Later in Section we further modify the iterative heuristic in an attempt to improve upon locally optimal solutions. In the single-case-pack problem (π = 1, and thus the index n is omitted), if the case-pack design variables (E m, m = 1,..., M) are fixed to values E m, m = 1,..., M, the remaining problem can be formulated as a dynamic lot-sizing problem with case packs, denoted as PE, as follows: (PE ) min T M ~c mt E m x t + T O t z t (4 17) t=1 m=1 t=1 subject to: t τ=1 Dmt x τ max, m = 1,..., M, t = 1,..., T, (4 18) m E m x t M t z t, t = 1,..., T, (4 19) x t Z +, t = 1,..., T (4 20) z t {0, 1}, t = 1,..., T. (4 21) We can solve PE as a shortest path problem in order to determine the optimal order periods and case-pack order quantities (Chen et al. [25]). On the other hand, if we fix the x t and z t variables to fixed values x t, z t, t = 1,..., T, and let C m = T t=1 ~c mtx t, m = 1,..., M, and w t = t τ=1 x τ, t = 1,..., T, the resulting problem P x,z becomes: (P x,z ) min C m E m (4 22) subject to: w t E m D mt, m = 1,..., M, t = 1,..., T, (4 23) M E m R, (4 24) m=1 E m Z +, m = 1,..., M. (4 25) We can solve this problem by inspection, because feasibility of the first set of constraints and the integrality restrictions requires E m max t=1,...,t D mt /w t. Because 97

98 C m 0 for m = 1,..., M, and R > 0, this implies that E m = max t=1,...,t D mt /w t is an optimal solution, since any solution with E m > max t=1,...,t D mt /w t can not improve the objective function value. The quality of the locally optimal solution obtained using the iterative heuristic we have described will depend on the initial case-pack composition used to set the E m values, m = 1,..., M. In order to gain some insight for constructing a good initial solution, we first consider the single-period version of the problem. Eliminating the index t, the customized case-pack design problem can be written as: { M } min E m R; E m, x Z +. m=1 ~c m E m x E m x D m, m; m Note that this is a special case of the module design problem, i.e., a one-product module design problem with a side constraint. Evans [28] showed that an infinite number of optimal solutions exist for such problems when the variable integrality restrictions are relaxed, since for any optimal solution (E, x), (E/θ, θx) is also an optimal solution for any θ > 0. Goldberg [43] observed that for continuous module design problems with side constraints, an optimal solution exists in which at least one of the side constraints holds at equality. In our single-period problem, there is only one side constraint ( M m=1 E m R), and it is easy to show that an optimal solution exists when the integrality restrictions are relaxed such that this inequality is tight. In addition, in any optimal solution to the continuous version of this problem, each of the demand constraints will be tight, and thus an optimal solution exists that satisfies M m=1 E m = R, E p E q = D p D q, p, q = 1,, M, (4 26) assuming positive demand values. The unique solution to this system is then given by E m = RD m / M m=1 D m, m = 1,..., M. Thus, an optimal solution to the continuous relaxation of the single-period problem fills the capacity R with SKUs in proportion to each SKU s percentage of the total demand in the period. This suggests a potential 98

99 heuristic approach for the T period version of the problem, described by setting E m = RD mt / M p=1 D pt, for m = 1,..., M. That is, we fill the capacity R in proportion to each SKU s percentage of total demand over the T period horizon (note that this is equivalent to filling the capacity in proportion to each SKU s average demand per period). This will result in a non-integer solution for the E m values, which we then adjust by applying a rounding heuristic that sets E m equal to the rounded value of (R 0.5M)D mt / M p=1 D pt, i.e., (R 0.5M)E m / M m=1 E m, for m = 1,..., M, in order to produce a feasible case-pack design 1. We next summarize the steps involved in implementing the iterative heuristic. Step 1. Choose an initial case-pack composition by setting E 0 according to (4 26) and setting D m to the average demand per period for SKU m (and applying a rounding heuristic in order to satisfy the case-pack capacity constraint). Set i = 0. Step 2. Solve the production planning problem for the fixed E i and let x i, z i denote the resulting optimal order quantities and setup variables for this problem (P E i ). Step 3. Compute w i, C i for the given x i, z i and solve the corresponding problem (P x i,zi ) by setting let E i+1 m = max Dmt /wt i, m = 1,..., M. t=1,...,t Step 4. If E i+1 = E i, then terminate. Otherwise, set i = i + 1 and return to Step 2. Proposition 4.1. In the iterative algorithm described above, Em i+1 Em, i m = 1,..., M. Proof. From the shortest path solution to problem (P E i ), wt i = t τ=1 x i τ = max Dmt /E i m m=1,...,m if period t + 1 is an optimal order period, and wt i max Dmt /E i m otherwise. We m=1,...,m solve (P x i,zi ) by setting E i+1 m = max Dmt /w i t ; let t m = arg max Dmt /w i t. We then t=1,...,t t=1,...,t have E i+1 m = Dmt m w i t m Dmt m D mt m /Em i = Em, i (4 27) 1 If (R 0.5M)D mt / M p=1 D pt < 0.5, then this quantity rounds to zero. If this occurs, we will need to round up and subtract one from E j for some other SKU j whose demand is the maximal; this is possible because of our assumption that M R. 99

100 where the inequality holds because wt i max Dmt /E i m max {D mt/em}. i m=1,...,m m=1,...,m Proposition 4.2. The objective function value is nonincreasing at each iteration of the iterative heuristic. Proof. Let f (x i, z i, E i ) denote the objective function after iteration i. Because (x i+1, z i+1 ) = arg minf (x, z, E i+1 ) by definition, and E i+1 = arg minf (x i, z i, E ), this implies f (x i+1, z i+1, E i+1 ) x,z E f (x i, z i, E i+1 ) f (x i, z i, E i ), and therefore that the objective function value is nonincreasing. Proposition 4.3. If tm(i) = arg max Dmt /w i t, then (t m (i) + 1) Q i, where Q i is the set t=1,...,t of optimal order periods set obtained by solving (P E i ). Proof. By contradiction, suppose E i+1 m = max t=1,...,t D mt/w i t is achieved at t / Q i with t 1 < t < t 2 where t 1, t 2 Q i and t 1 is the most recent order period prior to period t and t 2 is the next order period after t 1. Since x i t = 0 for each t such that t t < t 2, this implies w i t = t τ=1 x i τ = t 2 1 τ=1 x i τ = w i t 2 1. This combined with the fact that D m(t2 1) > D m t gives D m(t2 1)/w i t 2 1 > D m t/w i t, which contradicts the premise that t is the maximizer. Proposition 4.4. For the iterative heuristic method, if E 1 E 0, then E 2 = E 1, and the algorithm will stop within two iterations. Proof. For any t such that (t + 1) Q 1, w 1 t m = arg max D mt /Em, 1 we have m=1,...,m w 1 t = Dm t E 1 m = t τ=1 x 1 τ = max m=1,...,m D mt/e 1 m. Letting Dm t D m t/wt 0 = wt 0, (4 28) where the second inequality holds since E 1 m D m t/w 0 t for all t = 1,..., T. Then, Proposition 4.3 implies that: E 1 m = max Dmt /w 0 t (t+1) Q 0 t max Dmt /w 1 t (t+1) Q 0 t Combining this with the result of Proposition 4.1 implies E 2 = E 1. max Dmt /w 1 t = E 2 m. t=1,...,t 100

101 For the multiple-case-pack problem, our iterative heuristic framework can still be applied in principle, although the corresponding subproblems (P E ) and (P x,z ) are both strongly NP-hard (Chen et al. [25], Akçay et al. [5]). For the multiple case-pack procurement planning subproblem for a given set of case-pack configurations (P E ), a pseudopolynomial time approach exists for any fixed value of π, in addition to heuristic solution methods (Chen et al. [25]). The associated case-pack design subproblem (P x,z ) for a given set of procurement decisions x, z will be: min C mn E mn n m subject to: w nt E mn D mt, m = 1,..., M, t = 1,..., T, n E mn R, n = 1,..., π, m = 1,..., M, m E mn Z +, n = 1,..., π, m = 1,..., M, where C mn = T t=1 ~c mtx nt and w nt = t τ=1 x nτ. This problem can be shown to be strongly NP-Hard by a reduction from the multi-dimensional knapsack problem (Akçay et al. [5]) An iterative heuristic with alternative case-pack sizes The iterative heuristic we described in the previous section runs very quickly, although it tends to quickly become trapped in a locally optimal solution. In addition, the starting solution described by (4 26) strongly influences the final solution quality. The value of R used in (4 26) is, therefore, a critical determinant of solution quality. Recall that the case-pack size may in practice be a decision variable itself, and we use R as an upper limit on case-pack capacity. By considering various starting values of R, we are therefore able to evaluate numerous locally optimal solutions using the iterative heuristic approach. While the iterative approach improves the solutions at each iteration, these improvements only occur over two iterations. Therefore, while we do make some progress in determining a better case-pack size (from the upper limit R to m E m, which 101

102 is typically less than R) this progress does not continue. To overcome the restrictions implied by the brief number of iterations, we embed the iterative heuristic we have described as a subroutine within a larger framework that enumerates case-pack sizes starting with the upper bound R and progressing to some reasonable lower bound. That is, for each case-pack size between the lower and upper bound, we apply the basic iterative heuristic using this case-pack size as the value of R and storing its corresponding locally optimal solution. The best among these solutions is then taken as the heuristic solution value. We will refer to this method as the Iterative+Mod heuristic, because it is a modification of the basic iterative algorithm with a goal of jumping out of locally optimal (but globally suboptimal) solutions by setting the initial case-pack capacity at numerous starting points. The results based on this method are particularly encouraging, and are discussed in Section Geometric programming based iterative heuristic In this section, we describe an additional heuristic method for solving the nonlinear model (P) when only a single case-pack type is permitted (π = 1, and thus the index n is omitted). We also briefly address the implications of extending our method to handle additional case packs (π 2). This method uses a geometric programming based approach to jointly set the case-pack design variables and the case-pack order quantities for a given set of order periods. We then alternate between solving the production planning problem (PE) for a given set of case-pack design variables, and a geometric program that determines optimal values of case-pack design and case-pack order quantity variables for a given set of order periods. We formulate this geometric program by first defining new cumulative production decision variables w t = t τ=1 x τ, t = 1,..., T (with w 0 = 0), and directly using these decision variables instead of the case-pack order quantity variables x t, t = 1,..., T. Clearly then for a given set of w t variable values, we have x t = w t w t 1. Based on this substitution, the objective 102

103 function of (P) can be expressed as: which is equivalent to where T M O t z t, t=1 ~c mt E m (w t w t 1 ) + m=1 t p mt E m w t + O t z t, t m t p mt = ~c mt ~c m,t+1, t = 1, 2,, T 1, and p mt = ~c mt. Let Q = {Q 1, Q 2,, Q q } denote the set of q T order periods (that is, z t = 1 if t Q and z t = 0 otherwise), where Q 1 = 1 and we use the convention that Q 0 = 0 and Q q+1 = T + 1. As discussed in the previous section, since we can always scale any solution to ensure that the side constraint ( M m=1 E m R) is tight, we temporarily ignore this side constraint, and formulate the problem (P Q ) as follows. (P Q ) min M p mt E m w t (4 29) t Q m=1 s.t. E m w Qi D m(qi+1 1), i = 1, 2,..., q, m = 1,..., M, (4 30) E m, w t Z +, m = 1,..., M, t Q {1,..., T }. (4 31) Using this formulation, although (P Q ) is still a quadratically constrained quadratic problem, we can transform its continuous relaxation to a convex optimization problem using the following proposition. Proposition 4.5. The continuous relaxation of (P Q ) can be formulated as a geometric programming problem, which can be transformed to a nonlinear but convex optimization problem. Proof. First notice that in problem (P Q ), there is an implicit requirement that the variables are all positive. From Boyd et al. [21], we know that a geometric program 103

104 (GP) is an optimization problem that takes the following standard form: min f 0 (x) s.t. f i (x) 1, i = 1,, m, g i (x) = 1, i = 1,, p, where the f i are posynomial functions, g i are monomials and x i are positive decision variables. To briefly describe the basic form of a GP, a monomial in the variable x is a real valued function g(x) of the form: g(x) = cx a 1 1 x a 2 2 x a n n where c > 0, a i R. A sum of one or more monomials with positive coefficients is called a posynomial, i.e., a function of the form f (x) = k c kx a 1k 1 x a 2k 2 x a nk n. Now we can write (P Q ) with the integrality restrictions relaxed in standard geometric program form as: (P R Q) min s.t. p mt E m w t t Q m D mt E 1 m w 1 t 1, m = 1,..., M, t Q. To convert this GP to a convex optimization problem, we make the substitutions u m = log E m (so E m = e um ) and v t = log w t (so w t = e vt ). We also apply a logarithmic transformation to the objective function. This results in the following convex optimization problem: ( ) (P R Q) min log p mt e (u m+v t ) t Q m s.t. log(d mt ) (u m + v t ) 0, m = 1,..., M, t Q. This transformed problem is a convex program, and thus can be solved very efficiently using existing algorithms, which enables solving a GP efficiently. According to Boyd and Vandenberghe [22], standard interior-point algorithms can solve a GP with 1,000 variables and 10,000 constraints in under a minute on a small desktop computer. With recently developed solution methods, we can solve even 104

105 large-scale practical GP problems very efficiently and reliably. In our discussion of our computational tests in the following section, we use a software package developed specifically for solving GPs. Hence, to solve subproblem (P Q ), our heuristic method considers its continuous relaxation, and solves this using a geometric programming solver. We then round the continuous solution ~E m in a way similar to that discussed in Section 4.4.2, except that the value of R used in the rounding scheme changes at successive iterations (in order to capture the benefit of evaluating various values of R as previously discussed in section 4.4.3). That is, at each iteration, we update R to be equal to the summation of E m values (over all m) obtained in the previous iteration, and use this value of R for the rounding scheme discussed in Section Given an integer feasible solution E, we then set Dm(Qi+1 1) w Qi = max m=1,...,m E m, i = 1, 2, q, and x Qi = w Qi w Qi 1 for Q i Q and x t = 0 otherwise. Given the ability to solve (P Q ), the remaining problem requires determining the set Q, i.e., the set of z t values that will be set to one. We use an initial solution z such that z = {1, 0, 0,, 0}, so that in the first iteration the number of variables in the geometric program is small. Then, for the given z (and the implied set Q), we solve (P Q ) to obtain E and the associated x as described above; for given values of E obtained by solving (P Q ), we then solve the restricted problem (PE ) described in Section This provides a new value of the vector z, and thus a new definition of the set Q. We thus apply an iterative approach as described in the previous section, where the geometric program is used to determine the best case-pack composition for a fixed set Q. This iterative approach is applied until the z variables do not change, or until reaching an iteration number limit, which we set to 5 in our computational tests. The advantage of this iterative approach over the one described in the previous section lies in the fact that the geometric program (P Q ) permits integrated optimization of the case-pack composition and the case-pack order quantities, even though the order periods are fixed. 105

106 Thus, we expect it to lead to superior performance in some problem instances when compared to the basic iterative procedure discussed in the subsection In this section, we described a geometric programming based heuristic as it applies to the single-case-pack design and procurement planning problem. Generalizing this approach to the multiple case-pack setting introduces several difficulties. In particular, the geometric programming approach requires assuming that all variables must take positive values. However, in the multiple case-pack setting, we cannot generally require that every case pack contains at least one unit of each SKU, or that every order contains a positive order quantity for each case pack. For the special case with stationary costs, the multiple-case-pack version of subproblem (P Q ) will correspond to a multiple module design problem. Although Silverman [86], Shaftel and Thompson [83] and Al-Khayyal [6] have developed some heuristic schemes for such problems, these approaches are often difficult to implement, and optimal solutions are not guaranteed. 4.5 Computational Results This section discusses the setup and results for a set of computational tests of our exact linearization method and heuristic approaches. We implemented our linearization method, iterative heuristic and its modification in C++, with the embedded MIP problems in the linearization method solved using ILOG s CPLEX 11.0 solver with Concert Technology. Our geometric programming based algorithm was executed in MATLAB, with the embedded geometric program (GP) solved using the GGPLAB toolbox, which uses a primal-dual interior-point solver, GPCVX. To evaluate the effectiveness of our methods, we also modeled the nonlinear integer optimization problem (P) in GAMS, and solved it using the BARON solver, which is a commercial solver for general mixed integer nonlinear programming problems. We performed all tests on a Unix machine with two pentium 4, 3.2 Ghz processors and 6 GB of RAM. We designed seven sets of single case-pack problems, with each characterized by a unique combination of the number of items (SKUs) and periods (M, T ) equal to (3,5), (3,12), (6,12), (6,20), 106

107 (10,12), (10,20) and (10,30). Ten randomly generated replications were done for each problem setting. Our computational tests focused solely on single-case-pack problems due to the high complexity of this problem class. Table 4-1 summarizes the common data used in our computational study. For each problem instance, the data were generated from uniform distributions. We let U[l, u] denote the continuous uniform distribution with lower bound l and upper bound u. The capacity R here indicates the upper bound for an allowable case-pack size, which is set to twenty times the number of items, i.e., R = 20M. The data were chosen so that trivial cases would be avoided (e.g., problems with only one replenishment at optimality or problems in which a replenishment occurs in every period). Table 4-1. Parameter distributions used in computational tests. Size data (integer) and cost data (continuous) Period, T SKU, M Capacity, R Demand, d mt 5, 12, 20, 30 3, 6, 10 20M U[50m, m] Purchasing, c m Holding, h m Fixed Ordering, O t U[10, 60] U[0.05, 1.00] U[100, 180] The results of our tests, averaged over the 10 random problem instances for each problem set are presented in Table 4-2. We compared the average running time and optimality gap of our approaches with those obtained by GAMS/BARON. We set the time limit for each test to 5000 seconds. In the first row, CPLEX corresponds to our linearization method, which uses CPLEX to solve the transformed MILP model, which is an exact method. The Iterative and Integrated columns correspond to our basic and geometric programming based iterative heuristic introduced in Sections and 4.4.4, respectively. The last bold column is our modified Iterative+Mod approach. The gap is the difference between the value of the solution obtained by each approach and the lower bound (LB) achieved by CPLEX, taken as a percentage of the LB value. From Table 4-2, we see that although BARON is a computational system for solving 107

108 non-convex optimization problems to global optimality, unfortunately our joint case-pack design and procurement problem requires a great deal of computational effort, and thus the gap it achieves within 5000 seconds is very large, even for medium size problems. The gap is as large as %, which means that the solution obtained by the BARON solver after 5000 seconds is more than 38 times as large as the lower bound obtained by our linearization method using CPLEX. In addition, BARON reached the time limit of 5000 seconds without obtaining an optimal solution for the second smallest problem set we tested (M = 3, T = 12). We also performed a set of tests in which only the binary order variable vector z was forced to be integer (while the order size x and case-pack composition variables E were both relaxed to be continuous), and BARON still could not solve this problem set to optimality within 5000 seconds. Our linearization method (labeled CPLEX ) has an obvious advantage in terms of the Gap, especially for the small and medium size problems, although it is also a time-intensive approach. For the largest problem set, this linearization method fails to solve the problem, with a large optimality gap value of 22.16%. Between the two exact approaches (using BARON for the nonlinear model and CPLEX for the linearized model), we can conclude that the linearization method is superior, although neither method is efficient, since very long execution times are required to find a solution of reasonable quality. The results for our heuristic approaches are shown in Table 4-2 under the column headings Iterative, Integrated and Iterative+Mod. As the table shows, the iterative approach is extremely quick, with running times of less than 0.01 seconds even for the large problem sizes. However, the integrated approach provides much lower gaps, on average, at the expense of longer computing times. Its running time is also acceptable, with solution times within one minute for small and medium problems, and 2-20 minutes for the larger instances. We observe that our heuristics quickly provide better solutions than the exact approaches for the larger size problems (with 108

109 Table 4-2. Computational test results for average running time (seconds) and optimality gap Data Set BARON CPLEX Iterative Integrated Iter+Mod M T Time %Gap Time %Gap Time %Gap Time %Gap Time %Gap < < < < < < < < < Gap is calculated as (Sol. CPLEX LB)/CPLEX LB 100%. M = 10). Table 4-3 provides the number of iterations used by each heuristic approach. Recall that the Iterative heuristic terminates within two iterations. The Integrated heuristic, on the other hand, usually requires 2-10 iterations, depending on the size and difficulty of jumping out of the local optimums associated with each problem. While the shortest-path based Iterative heuristic guarantees fast solution, the limited number of iterations restricts its performance. On the contrary, by alternating between the geometric program P Q and the shortest-path solution as discussed in Section 4.4.4, the Integrated approach enables moving away from a local minimum, although the geometric programming subproblem requires a greater amount of time. Figure 4-1 illustrates the progress of the Integrated approach for an instance with M = 6 and T = 20. This problem instance required 10 iterations before termination. As the figure shows, at the final iteration, both the red dot (corresponding to the objective value for the subproblem (P Q ) with fixed z variables) and the blue dot (the objective value for the dynamic program with fixed E variables) reach the same level, meaning 109

110 Table 4-3. The iteration number for iterative and integrated heuristics M T Iterative Integrated that no improvement was seen in successive iterations. At previous iterations, the solution corresponding to the blue dot always slightly improves upon its immediate preceding solution indicated by the red dot (by decreasing the objective value). At subsequent iterations, the red dot following each blue dot either jumps out of the previous local area (as in Iterations 4, 7, and 10), or further improves upon the objective (as in Iterations 2, 3, 5, 6, 8, and 9). If there is no improvement over the best solution found after 5 iterations, the algorithm terminates. Figure 4-1. Objective value vs iterations for the integrated approach (2nd instance M = 6, T = 20) 110