Integrated Capacity and Inventory Management with Capacity Acquisition Lead Times

Transcription

1 Integrated Capacity and Inventory Management with Capacity Acquisition Lead Times Gergely Mincsovics a, Tarkan Tan a,, Osman Alp b a Department of Technology Management, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands b Industrial Engineering Department, Bilkent University, Ankara, Turkey June 2006 Abstract: In this paper we consider a make-to-stock production system whose permanent production capacity can be increased temporarily by making use of some contingent resources, while this capacity acquisition has a constant lead time associated with it. We present a model where the capacity management problem is handled in an integrated manner with the inventory management, under non-stationary and stochastic demand. We analyze the characteristics of the optimal policies for the integrated problem. Finally, we evaluate the value of utilizing flexible capacity under different capacity acquisition lead times and other parameter settings, which enable us to develop managerial insights. Keywords: Inventory; Production; Stochastic Models; Capacity Management; Flexible Capacity; Capacity Acquisition Lead Time 1 Introduction and Related Literature In a make-to-stock production system that faces volatile demand, system costs may be decreased by managing the capacity as well as the inventory in a joint fashion, in case there is some flexibility in the production capacity. In many cases it is possible to increase the production capacity temporarily while it may take some time to do so, which we refer to as capacity acquisition lead time. In this paper we consider such a make-to-stock production system subject to periodic review in a finite horizon under non-stationary stochastic demand, where our focus is on the effects of capacity acquisition lead time. The production capacity of a system can be permanent or contingent. We define permanent capacity as the maximum amount of production possible in regular work time by utilizing internal resources of the company such as existing workforce level on the steady payroll or the machinery owned or leased by the company. Total capacity can be increased Corresponding author addresses: G.Z.Mincsovics@tm.tue.nl (G.Z. Mincsovics), t.tan@tm.tue.nl (T. Tan), osmanalp@bilkent.edu.tr (O. Alp) 1

2 temporarily by acquiring contingent resources, which can be internal or external, such as hiring temporary workers from external labor supply agencies, subcontracting, overtime production, renting work stations, and so on. We refer to this additional capacity acquired temporarily as the contingent capacity. Such increases in capacity may be subject to a certain time lag associated with the operations required for the increase. For example, an external labor supply agency may not be able to send immediately the temporary workers asked by a company. According to our experience in the Netherlands, it takes 1 or 2 days to acquire temporary workers from the external labor supply agencies for the jobs that do not require high skill levels. This time may be longer for the jobs that require certain skills. Changing the level of permanent capacity as a means of coping with demand fluctuations, such as hiring and/or firing permanent workers frequently, is not only very costly in general, but it may also have many negative impacts on the company. Utilizing flexible capacity is a possible remedy to this problem and we consider it as one of the two main operational tools of coping with fluctuating demand, along with holding inventory. Flexible capacity management refers to adjusting the total production capacity with the option of utilizing contingent resources on top of the permanent ones. The integrated inventory and flexible capacity management that we deal with in this paper refers to determining the contingent capacity to be ordered which will be available in a future period as well as determining the optimal production quantity in a certain period given the available capacity which is determined in an earlier period. Since the long-term changes in the state of the world can make permanent capacity changes unavoidable, we consider the determination of the permanent capacity level as a tactical decision that needs to be made only at the beginning of the planning horizon and not changed until the end of the horizon. For the ease of exposition, we refer to the workforce capacity setting in general. More specifically, we use the temporary (contingent) labor jargon to refer to capacity flexibility. Consequently, the problem we consider can be viewed as one where the production is mostly determined by the workforce size, permanent and contingent. There exists a significant usage of flexible workforce in many countries. For example, 6.6% of the active labor force of the Netherlands was composed of flexible workers (temporary, standby, replacement, and such other workers) in 2003 (Beckers [2005]), and 32% of the employees in the manufacturing industry were regularly working overtime in 2004 (Beckers and Siermann [2005]). Temporary workers can be hired in any period (provided that they are ordered from the external labor supply agencies timely) and they are paid only for the 2

3 periods that they are hired. Clearly, not all the temporary workers that are hired need to be utilized in the optimal solution. The productivity of contingent workers are allowed to be different than that of permanent workers in our model. The size of the permanent workforce is determined only at the beginning of the planning horizon and considered as fixed during the whole planning horizon. In this paper, we first develop a model to represent this relatively complicated problem. Making use of our model, we characterize the optimal solution and show some properties. Finally, we develop several useful managerial insights on the effects of capacity acquisition lead time. In specific, how the value of employing contingent workforce changes with the lead time, how demand uncertainty influence this change, what implications to expect on the permanent capacity level. Holt et al. [1955] present a modelthat exploit the trade-offs in coping with fluctuations in demand. In specific, they describe the possible alternatives as (1) adjusting in-house capacity using a large pool of capacity outside the company, (2) changing productivity employing the same workforce or set of machines, and (3) allowing inventory and backlog of orders. Indeed, this very idea of trading off between capacity and inventory that is the basis of aggregate production planning constitutes the essence of our problem too, nevertheless we consider non-stationary stochastic demand and a positive lead time for acquiring capacity, which reflects the reality closer in most cases. Our work builds on Tan and Alp [2005], which consider the same problem except for capacity acquisition lead time. Therefore, we review only the literature on capacity acquisition lead time issue here, while referring the reader to Tan and Alp [2005] for the literature on the remaining aspects of the problem that we consider. In a particularly relevant paper, Yang et al. [2005] model subcontracting via a discrete dynamic programming formulation. Subcontracting takes a positive lead time, which is assumed to be one period greater than or equal to the production lead time. In-house capacity is modeled as a Markov process independent from the demand. Production and subcontracting decisions alternate throughout the planning horizon, where the use of subcontracting incur an additional setup cost. The optimal policy on subcontracting is shown to be of capacity-dependent (s,s)-type. The authors also show a complementarity condition between production and subcontracting, namely, if subcontracting is more costly than production, no subcontracting will take place unless production capacity is fully utilized. The lead time involved in expanding the capacity is first modeled and presented by 3

4 Maccini [1973], to the best of our knowledge. Maccini used a continuous time net present value model with capacity adjustment penalties to determine stationary modularity relations among the rate of capacity investment, the investment lead time, the delivery price, and the capital tied up in capacity. In concern of lead time, Maccini showed that optimal level of capital goods decrease (increase) when investment lead time increases, which decreases (increases) the delivery price. If investment lead time does not affect the delivery price, then the rate of investment decreases (increases) for an increasing lead time when the capital goods are expanding (contracting). For further literature on capacity expansion in general, we refer the reader to the extensive introduction in Ryan [2003]. Within the capacity expansion literature, investment lead time is sometimes modeled with the necessary learning involved (see for example, Gaimon and Burgess [2003]). Inventory management is rarely coupled with capacity management if lead time is involved in expanding the capacity, whatever means is used for capacity expansion. Our study aims to contribute filling in this gap in literature. The rest of the paper is organized as follows. We present our dynamic programming model in Section 2. The optimal policy and some properties of it are discussed in Section 3 and our computations that result in managerial insights are presented in Section 4. We summarize our conclusions and suggest some possible extensions in Section 5. 2 Model Formulation In this section, we present a finite horizon dynamic programming model to formulate the problem under consideration. Unmet demand is assumed to be fully backlogged. The relevant costs in our environment are inventory holding and backorder costs, and the unit cost of permanent and contingent capacity, all of which are non-negative. There is an infinite supply of contingent workers, and any number of contingent workers ordered become available with a given time lag. We assume that the raw material is always available and the lead time of production can be neglected. The notation is introduced as need arises, but we summarize our major notation in Table 1 for the ease of reference. We consider a production cost component which is a linear function of permanent capacity in order to represent the costs that do not depend on the production quantity (even when there is no production), such as the salaries of permanent workers. That is, each unit of permanent capacity costs c p per period, and the total cost of permanent capacity per period is Uc p, for a permanent capacity of size U, independent of the production quantity. We do not 4

5 Table 1: Summary of Notation T : Number of periods in the planning horizon L : Lead time for contingent workforce acquisition c p : Unit cost of permanent capacity per period c c : Unit cost of contingent capacity per period h : Inventory holding cost per unit per period b : Penalty cost per unit of backorder per period α : Discounting factor (0 < α 1) W t : Random variable denoting the demand in period t G t (w) : Distribution function of W t U : Size of the permanent capacity x t : Inventory position at the beginning of period t before ordering y t : Inventory position in period t after ordering θ t : Contingent workforce available in period t (that is ordered in period t L) θ t : { (θt, θ t+1,..., θ T 1, θ T ) if T L + 1 t T (θ t, θ t+1,..., θ t+l 2, θ t+l 1 ) if 0 < t < T L + 1 f t (x t, θ t, U) : Minimum total expected cost of operating the system in periods t, t + 1,..., T, given the system state (x t, θ t, U) : optimal solution ˆ : unconstrained optimum consider material-related costs in our analysis, but it can easily be extended to accommodate this component. In order to synchronize the production quantity with the number of workers, we redefine the unit production as the number of actual units that an average permanent worker can produce; that is, the production capacity due to U permanent workers is U unit s per period. We also define the cost of production by temporary workers in the same unit basis, where the cost for flexible workers is related to their productivity. In particular, let c c be the hiring cost of a temporary labor per period, and let c c denote all other relevant variable costs associated with production by temporary workers per period. It is possible that the productivity rates of permanent and temporary workers are different. Let γ be the average productivity rate of temporary workers, relative to the productivity of permanent workers; that is, each temporary worker produces γ units per period. Assuming that this rate remains approximately the same in time, the unit production cost by temporary workers, c c, can be written as c c = (c c + c c)/γ. It is likely that 0 < γ < 1, but the model holds for any γ > 0. In every period, a decision is made to determine the number of contingent workers to 5

6 be available in exactly L periods after the current period. If θ t contingent workers are ordered in period t L then that many workers become available in period t with a total cost of c c θ t which is charged whenever they are available. In any period t, we keep a vector θ t = (θ t, θ t+1,..., θ t+l 1 ) which consists of the number of contingent workers that are ordered in periods t L, t L + 1,..., t 1. In the next period, the vector θ t+1 consists of the information on the hired contingent workers for periods t + 1, t + 2,..., t + L 1, carried from the vector θ t, as well as the decision made for period t + L, θ t+l, in period t. The order of events is as follows. At the beginning of each period t, the initial inventory level, x t is observed, and the number of previously ordered contingent workers, θ t, become available. The total amount of productive resources of period t becomes U + θ t, that is the upper limit on the production quantity of this period. Then, the operational decisions are made. That is, the production decision given the available capacity and the decision on the number of contingent workers to be available in period t + L. According to the production decision, the inventory level is raised to y t x t + U + θ t by utilizing the necessary capacity means; that is, if y t x t + U then only permanent capacity is utilized, otherwise contingent capacity is utilized as well as full utilization of permanent capacity. We note that the optimal production quantity may result in partial utilization of the available capacity. At the end of period t, the realized demand w t is met/backlogged, resulting in x t+1 = y t w t and the vector θ t+1 is constructed as explained above. We denote the random variable corresponding to the demand in period t as W t and its distribution function as G t. Consequently, denoting the minimum cost of operating the system from the beginning of period t until the end of the planning horizon as f t (x t, θ t, U), we use the following dynamic programming formulation to solve the integrated Capacity and Inventory Management Problem with Lead Times (CIMPL): f t (x t, θ t, U) = Uc p + θ t c c + min {L t (y t ) + αe[f t+1 (y t W t, θ t+1, U)]} if T L + 1 t T y t [x t ;x t +θ t +U] if 1 t T L min {L t (y t ) + αe [f t+1 (y t W t, θ t+1, U)]} 0 θ t+l,y t [x t ;x t +θ t +U] f 0 (x 1 ) = min 0 U, 0 θ 1 f 1 ( x1, θ 1, U ) z where f T +1 ( ) 0 and L t (z) = h 0 z ω dg t (ω) + b ω z dg t (ω). z 6

7 3 Analysis of the Optimal Policies In this section we first characterize the optimal solution to the problem that is modeled in Section 2. Then we introduce an important property regarding whether all of the available capacity should be utilized or not and we discuss some settings where this is or is not the case. Finally we consider a special case of the problem to gain some additional insights. 3.1 Optimal Policy Characterization Let J t be the cost-to-go of period t, as a function of the actual decision so that we exclude the period s capacity related costs; J t (y t, θ t+l, U) = L t (y t )+αe [f t+1 (y t W t, θ t+1, U)]. For T L + 1 t T we denote the same function by J t (y t, U), as θ t+l no longer exists in θ t+l. f t (x t, θ t, U) can be rewritten as f t (x t, θ t, U) = Uc p + θ t c c + min J t (y t, U) y t [x t ;x t +θ t +U] min J t(y t, θ t+l, U) 0 θ t+l, y t [x t ;x t +θ t +U] if T L + 1 t T if 1 t T L In every period t, the optimal decisions (inventory level after production, y t and number of contingent workers hired, θ t+l ) are made by minimizing the function J t over the feasible region. First we show in Theorem 1 that J t is convex in the decision variables. This result allows us to find the structure of the optimal policy in Theorem 2. Theorem 1 also shows that the recursive minimum cost function of the dynamic programming formulation, f t (x t, θ t, U) is convex. This convexity aids in the optimization of the permanent capacity level (U) which is defined as a decision variable to be optimized at the starting period of the planning horizon. Theorem 1 f t (x t, θ t, U) is convex, and minimizing J t is a convex optimization problem for arbitrary capacity acquisition lead time, L N. Proof: See Appendix. Let (ŷ t, ˆθ ) t+l be the global optimizers of the function J t ( ). We need the following definition in our further discussion: ȳ t := arg min J t (y t, 0), θ ta := arg minj t (x t, θ L+t ), x t y t x t +θ t +U 0 θ t+l and θ tb := arg minj t (x t + θ t + U, θ L+t ). 0 θ Let ( yt, θt+l) be the aggregate optimal production and contingent capacity hiring decision in period t given that the state variables are x t, θ t and U. The following theorem provides the form of the optimal decisions.. 7

8 Theorem 2 The optimal production and contingent workers hiring policy is given by (ŷ, ˆθ) ifx t ŷ t x t + θ t + U and 0 ˆθ t+l (ȳ, 0) ifx t ŷ t x t + θ t + U and 0 > ˆθ t+l (yt, θt+l) (x = t, θ ta ) ifŷ t x t and 0 ˆθ t+l (ȳ t, θ ta ) ifŷ t x t and 0 > ˆθ. t+l (xt + θ t + U, θ ) tb ifŷt > x t + θ t + U and 0 ˆθ t+l (ȳ t, θ tb ) ifŷ t > x t + θ t + U and 0 > ˆθ t+l Proof: The convexity of the function J t imply this policy. Next, we present a series of submodularity and supermodularity results for the functions in our model. The implications of each result are explained thereafter. The reader is referred to Porteus [2002] and Topkis [1998] for further details on submodular and supermodular functions. Theorem 3 f t (x t, θ t ) and J t (y t, θ t+1 ) are supermodular for all t for general capacity acquisition lead time of L N. Proof : See Appendix. Supermodularity of function J t (y t, θ t+1 ) indicate that y t and θ t+1 are economic substitutes. That is to say, for a larger value of y t, the optimal θ t+1 decreases. A similar interpretation also holds for the function f t (x t, θ t ). At the start of a period t, if we had the chance of determining the contingent capacity to be available in this period, then higher starting inventory levels would lead to lower contingent capacity levels. Note that we can denote the function J t by J t (W t, y t, θ t+1, U) as W t can also be considered as an argument of the function. We use the notation J t (W t, θ t+1 ) to denote the same function when y t is fixed to an arbitrary value. Similarly, J t (W t, y t ) denotes the function J t for an arbitrary value of θ t+1. Theorem 4 J t (W t, θ t+1 ) is submodular for all t for general capacity acquisition lead time of L N where W t D and D is the poset of random variables with the first order stochastic dominance as partial order. Proof : See Appendix. Theorem 5 J t (W t, y t ) is submodular for all t for general capacity acquisition lead time of L N where W t D and D is the poset of random variables with the first order stochastic dominance as partial order. 8

9 Proof : See Appendix. Submodularity of the function J t (W t, θ t+1 ) indicates that W t and θ t+1 are economic complements. That is to say, stochastically higher demand distributions lead to hiring more contingent labor. A similar relation also exists between W t and y t by Theorem Complementary Slackness Property In our model, we have two decision variables to be determined in every period: the inventory level after production and the contingent capacity hired for L periods ahead. The former decision variable is bounded from above by the maximum amount of capacity we have (the permanent capacity level plus the contingent capacity that was ordered L periods ago) whereas the latter decision variable has only the nonnegativity constraint. Let s t denote the slack capacity in period t after the optimal course of action is implemented. In notation, s t = x t +U +θ t yt. We define the complementary slackness property for an optimal solution as follows: Definition 1 The optimal solution of period t, (y t, θ t ) is said to have the complementary slackness property if s t θ t = 0. Figure 1: Graphical illustration for the analysis of the complementary slackness property In an optimal solution where the complementary slackness property does not hold, one can observe a situation where a positive contingent capacity is ordered for future use, while not fully utilizing the current capacity which has already been paid for. Such a solution prefers to defer possible production activities that could have taken place with the slack capacity and prefers to acquire contingent capacity to be available in L periods time, rather than producing in the current period with the slack capacity and carrying inventory. To illustrate this situation, assume that we have a deterministic demand stream, yt < x t + θ t + U, and θt+l > 0. Consider increasing y t and decreasing θt+l by one unit. In such a case, one 9

10 unit less production in period t + L (since the capacity is decreased by one unit) is to be compensated by the additional unit of production in period t. Therefore, the additional unit produced in period t will be carried as an inventory up to period t + L but a saving will occur by hiring one unit less contingent capacity. Referring to Figure 1, note that this additional cost of carrying inventory is given by L 1 k=0 αk h whereas the savings by decreasing θ t+l by one unit is given by αl c c. Therefore, if L 1 k=0 αk h < α L c c then it is trivial to observe that the optimal solution would have the complementary slackness property for a deterministic demand stream. Otherwise, one may easily find cases where this property does not hold. For a stochastic demand stream, we prove that similar conditions exist that secure the complementary slackness property to hold in optimality in two extreme cases: in a stationary demand, infinite horizon problem with L = 1 (Theorem 6) and in a non-stationary demand, two period problem (Theorem 8). However, we do not have a corresponding result for the general finite planning horizon problems. We present a parameter setting, which allows illustration of complementary slackness property for stochastic, time-dependent demand pattern in a finite horizon setting. Let the parameters be L = 1, h = 1, b = 5, c p = 2.4, c c = 3.2, α = 1, and U = 10. We use a starting state x 1 = 0, and θ 1 30, arbitrarily chosen. The demand stream starts with the stochastic random variable W 1, such that P (W 1 = 0) = 1 p and P (W 1 = 30) = p for a given probability parameter, p. The demand in the following periods is deterministic, W 2 W 3 W 10 W 11 0 and W 12 W 13 W 14 W Interpretation of this demand pattern is that a notified holiday close down of the procurer is coming forth throughout 10 periods, from the second period on. We vary the p probability parameter of first period s demand. In the range, p [0; 0.39], the optimal decision is not producing anything and not ordering contingent workers. Thus, this solution satisfies complementary slackness. However, when p is in [0.40; 0.51], a noncomplementary decision is optimal. Namely, the best is to order 10 unit of contingent capacity while no production takes place. If p 0.52, then the optimal solution is again complementary: production of either 10, 20 or 30 units, depending on p while ordering no contingent workers. This example shows, that non-complementary decision can be optimal for realistic settings. For a slightly different setting, when we omit the last four stages (T = 11), the same solutions and p intervals can be gained. This shortened setting with e.g. p = 0.45 is an example for stochastically decreasing demand, for which non-complementary decision is 10

11 optimal. As a final remark to wrap up this discussion, we note that the cases where the complementary slackness property does not hold in the optimal policy structure presented in Theorem 2 are the cases where the unconstrained optimum of the decision variables fall strictly within the feasible region and where yt = x t and θ ta > 0. In the rest of the cases, the optimal solution has the complementary slackness property. The following theorem characterizes the condition where the complementary slackness property hold for a stationary infinite horizon problem with a capacity acquisition lead time of 1 period. Theorem 6 For the infinite horizon stationary demand case (T, W t W ) with L = 1, the optimal solution has the complementary slackness property if αc c > h for 0 < W. Proof : See Appendix. 3.3 A Special Case: Two Period Problem In this section we analyze the two period problem with non-stationary demand. Assume that the lead time for capacity acquisition is one period. In this special case, a one-time contingent capacity acquisition decision is made in the first period to be effective in the second period whereas the production decisions are made for two times, one for each period. Consider a firm producing items that sell only in a specific period, like the Christmas season. Possible examples of such items include Christmas gifts, winter garments, etc. As the duration of the time that these items sell is short, especially when compared to possible lead times of production, such a firm may have to plan its production operations in only two stages. In the first stage, a production decision is made by considering any possible starting inventory (e.g. the leftovers from the previous years) and existing total productive capacity (may include permanent and available temporary resources) before observing any demand. In cases where the permanent resources are not sufficient to satisfy the future demand, the firm also makes a decision to acquire contingent capacity to be available in the second stage. Such a capacity can be in the form of temporary labor or outsourced resources, which may take some time to be available. In the second stage, the firm only makes a production decision based on their starting inventory levels and the updated demand information, by using some or all of the available capacity. The model under consideration in this section fits to such environments. 11

12 We first present a set of equations of which the solution in terms of the unknown variables provide the optimal values of the decision variables of the first period. Lemma 1 Optimal decision of stage 1 is the solution of the following equalities. c c b + bg 1 (y 1 ŷ 2 + θ 2 + U) + (h + b) y 1 ŷ 2 +θ 2 +U G 2 (y 1 ω + θ 2 + U) g 1 (ω) dω = 0 (1) L 1 (y 1 ) + α( b(g 1 (y 1 ŷ 2 + θ 2 + U) G 1 (y 1 ŷ 2 ) 1 y 1 ŷ 2 + (h + b) G 2 (y 1 ω) g 1 (ω) dω + (h + b) y 1 ŷ 2 +θ 2 +U G 2 (y 1 ω + θ 2 + U) g 1 (ω) dω ) = 0 (2) ( where ŷ 2 = G 1 b 2 h+b). Proof : See Appendix. Our next result provides an interval for the optimal value of the decision variable denoting the inventory level after production in an unconstrained situation. When h = 0, this interval becomes a single point, and the target inventory level is definite. The result aids in finding the optimal solution by restricting the search space. Theorem 7 L 1 1 (α (c c h)) ŷ 1 L 1 1 (αc c ) holds. Proof : See Appendix. Our final result in this section provides a condition which warrants the complementary slackness property to hold in the optimal solution. Note that this result is valid for any non-stationary demand stream in a two period problem. Theorem 8 If h (1 + α) < αc c, then the complementary slackness property holds in the optimal solution. 4 Numerical Results and Discussion The main goal of this section is to gain insight on how the value of flexible capacity and the optimal permanent capacity levels change as the following system parameters change: capacity acquisition lead time, unit cost of contingent capacity, backorder cost, and the 12

13 variability of the demand. For this purpose, we conduct some numerical experiments by solving CIMPL. We use the following set of input parameters, unless otherwise noted: T = 12, b = 5, h = 1, c c = 3.5, c p = 2.5, α = 0.99, and x 1 = 0. We consider Normal demand with a coefficient of variation of CV = 0.2 that follows a seasonal pattern with a cycle of 4 periods, where the expected demand is 10, 15, 10, and 5, respectively. For the results containing different lead times to be comparable, we also select the values of the pipeline contingent capacity at the beginning of the first period that minimizes the system costs. In the results that we present, we use the term increasing ( decreasing ) in the weak sense to mean non-decreasing ( non-increasing ). We provide intuitive explanations to all of our results below and our findings are verified through several numerical studies. However, one should be careful in generalizing them, as for any experimental result, especially for extreme values of problem parameters. 4.1 Value of flexible capacity The option of utilizing contingent capacity provides additional flexibility to the system and leads to reduction of the total costs, even though there is a certain lead time associated with it. We measure the magnitude of cost reduction in order to gain insight on the value of flexible capacity. We compare a flexible capacity (FC) system with an inflexible one (IC), where the contingent capacity can be utilized in the former but cannot in the latter. We define the absolute value of flexible capacity, V F C, as the difference between the optimal expected total cost of operating the IC system, ET C IC, and that of the FC system, ET C F C. That is, V F C = ET C IC ET C F C. We also define the (relative) value of flexible capacity as the relative potential cost savings due to utilizing the flexible capacity. That is, V F C% = V F C/ET C IC. We note that both V F C and V F C% are always non-negative. We first test the value of flexibility with respect to the backorder and contingent capacity costs under different capacity acquisition lead times, by varying the value of one of the parameters while keeping the rest fixed. We present the results in Table 2, which verifies intuition in the sense that V F C% is higher when 1. capacity acquisition lead time is shorter, 2. contingent capacity cost is lower, and 3. backorder is more costly, or equivalently when a higher service level is targeted. 13

14 L c c V F C% b V F C% Table 2: V F C% as a function of the lead time, L, for varied c c, and b We note that, while the value of flexibility decreases with an increasing capacity acquisition lead time, the value persists to be significant. Consequently, while there is a value in trying to decrease capacity acquisition lead time in the system through means such as negotiating with the external labor supply agency or forming a contingent labor pool perhaps within different organizations of the same company, the existence of a lead time in acquiring contingent capacity should not discourage the production company to make use of capacity flexibility, even if this lead time is relatively long. The effect of the cost of contingent capacity on the value of flexibility can also be seen by reverting to the definition of V F C and %V F C: As c c decreases while the other parameters are kept constant, ET C IC remains the same, since flexible capacity is not utilized in this case, but ET C F C decreases due to decreased costs. Consequently, both V F C and %V F C increase. Nevertheless, Table 2 gives an indication on the magnitude of this increase and verifies that this result holds for any given capacity acquisition lead time. Finally, Table 2 also indicates that integrated inventory and capacity management outperforms isolated inventory management in an increasing manner as the backordering costs increase, for any given capacity acquisition lead time. We note that the permanent capacity level is optimized in the inflexible case (as well as in the flexible one), so the difference is not necessarily caused by the insufficiency of permanent capacity in the inflexible case. Demand uncertainty has a remarkable effect on the value of flexible capacity. The results presented by Tan and Alp [2005] indicate that the value of flexibility is higher when the demand is more variable for sufficient permanent capacity levels, under zero lead time assumption. Nevertheless, under the existence of positive capacity acquisition lead times, 14

15 6 Relative value of flexible capacity (VFC%) Normal, CV=0.1 Normal, CV=0.2 Normal, CV= Lead time (L) Figure 2: V F C% as a function of the lead time, L, for different demand distribution streams we observe that this may cease to hold. Indeed, for the parameter settings that we present, we observe that the value of flexibility increases when the demand variability decreases. In Figure 2, we depict V F C% as a function of the capacity acquisition lead time, for different demand streams. Namely, we employ normal distributions with coefficient of variation 0.1, 0.2 or 0.3 throughout the horizon. A longer capacity acquisition lead time deteriorates the effectiveness of capacity flexibility, as argued before. This effect is amplified in case of higher demand variability. In other words, the demand prediction is more reliable for lower demand variability and this makes the use of contingent capacity -which has to be ordered one lead time before the realization of the prediction- more effective compared to the high variability case. This also explains why the decrease in the value of flexibility as lead time increases is steeper when the variability is higher. 4.2 Optimal level of permanent capacity In this section we investigate how the optimal level of permanent capacity changes as the problem parameters change in the same way as we discussed in Section 4.1. We present our results in Table 3. We first note that the optimal permanent capacity decreases as the contingent capacity acquisition lead time decreases, in all of the cases that we consider. That is, since the This is not the case for all parameter settings that we have conducted tests on. Therefore we do not mean to generalize this finding and our discussion is based only on the parameter setting that is presented. 15

16 L c c U b U demand U Poisson Normal, CV= Normal, CV= Normal, CV= Table 3: U as a function of the lead time, L, for varied c c, b and demand distribution streams decreased lead time makes the capacity flexibility a more powerful weapon, it decreases the required level of permanent capacity. It is also interesting to note that the optimal level of permanent capacity is less than the average demand of 10 when the contingent capacity acquisition lead time is 1 period, except for one case where contingent capacity is very expensive to use. Combining these results together, one can deduce that less permanent capacity is needed if contingent capacity is more easily available. Finally, we observe that more permanent capacity is required when the demand is more variable. In other words, flexibility is more reliable when the demand variation is lower, which is in line with our previous observation as to the value of flexibility decreasing in demand variability. 5 Conclusions and Future Research In this paper the integrated problem of inventory and flexible capacity management under non-stationary stochastic demand is considered, when lead time is present for flexible capacity acquisition. In our model, permanent productive resources may be increased temporarily by hiring contingent capacity in every period, where this capacity acquisition decision becomes effective with a given time lag. Other than the operational level decisions (related to 16

17 the production and capacity acquisition levels), we also keep the permanent capacity level as a tactical decision variable which is to be determined at the beginning of a finite planning horizon. Our main major contribution in this paper is to provide insights into the role of capacity acquisition lead time in the integrated inventory and flexible capacity management problem. We first prove that all of the decision making functions under consideration are convex in all decision variables. This result helps us to provide an optimal policy for the operational decisions. Moreover, we prove that the decisions related to the inventory level after production and to the amount of contingent capacity hired in any given period are economic substitutes in the sense that for higher inventory levels obtained after production, the corresponding optimal levels of contingent capacity acquisition decreases. A similar result also holds for the starting inventory levels and the available contingent capacity in the current period. We also show that the stochastic demand variable and the optimal contingent capacity acquisition decisions are economic complements, i.e. for stochastically higher demand streams, we observe higher contingent capacity levels in optimality. A similar interpretation is also true stochastically higher demand streams and the optimal inventory levels obtained after production. We introduced the complementary slackness property of the aggregate production-contingent capacity ordering decision, as being satisfied if contingent capacity ordering can take place only when all productive resources are already fully utilized. We declared conditions, which ensure that the optimal decision has the complementary slackness property in two extreme cases, in an infinite horizon problem and in a two period problem. For a finite planning horizon problem instance, an example shows that optimal decision may not hold the complementary slackness property for a non-stationary demand stream. Our numerical results show that (i) an increasing contingent capacity acquisition lead time reduces the value of employing contingent workforce, (ii) the higher the demand uncertainty the steeper the value drops, and (iii) the reduced value of contingent capacity entails an increased use of permanent capacity. This research may be extended in several ways. Introducing an uncertainty on the permanent and contingent capacity levels would enrich the model. For example, the supply of contingent capacity may be certain for larger lead times whereas it may be subject to an uncertainty for shorter lead times. Some other extension possibilities include consideration of the fixed costs for production and/or acquiring contingent capacity, inclusion of expansion 17

18 and contraction decisions also for the permanent capacity, and developing efficient heuristic methods for the problem. Appendix Preliminaries for Theorem 1 Lemma 2 Let x R n, b R m, A R m n and e R s. Assume that f(x, e) is convex in x and e. Then the function g(b, e) = min f(x, e) is also convex in b and e. Ax b Proof : Let λ be such that 0 λ 1. Then λg(b, e) + (1 λ)g(b, e ) = λ min f(x, e) + (1 λ) min Ax b Ax b f(x, e ) = λf(x 0, e) + (1 λ)f(x 1, e ) f(λ(x 0, e) + (1 λ)(x 1, e ) (3) min {f(x, λe + (1 Ax λb+(1 λ)b λ)e } (4) = g(λ(b, e) + (1 λ)(b, e )) (3) is due to convexity of f and (4) is because Ax 0 b and Ax 1 b implies that A(λx 0 + (1 λ)x 1) λb + (1 λ)b. This completes the proof. Lemma 3 If J(y, e) is convex then g(x, θ, e) = min {J(y, e)} is also convex. y [x,x+u+θ] [ ] [ 1 Proof : Let h(b, e) = min {J(y, e)} where A = and b = Ay b 1 that h(b, e) is convex. Since h(b, e) = g(x, θ, e), g is also convex. Lemma 4 If J(y, δ, e) is convex then g(x, θ, e) = Proof : Let h(b, e) = min {J(y, e)} where A = Au b x x + U + θ min {J(y, δ, e)} is also convex. y [x,x+u+θ],0 δ [ ] y, u = and b = δ Lemma 2, we conclude that h(b, e) is convex. Since h(b, e) = g(x, θ, e), g is also convex. ]. By Lemma 2, we conclude x x + U + θ 0. By Proof of Theorem 1: Lemmas 3 and 4. The theorem can be proven by regular inductive arguments and the results of Preliminaries for Theorem 3 Definition 2 Let x x + and θ θ +. Then the function H(x, θ) is said to be a supermodular function if H(x, θ ) + H(x +, θ + ) H(x, θ + ) + H(x +, θ ). Definition 3 A function f is submodular if f is supermodular. Here is a list of properties of submodular and supermodular functions. The reader is referred to Porteus (2002) and Topkis (1998) for proofs of these properties. Property 1 1. H + (x, y) := g (x + y) is supermodular iff g (x) is convex. H + (x, y) := g (x y) is submodular iff g (x) is convex. 18

19 2. If g (w, θ) is submodular, then E [g (W, θ)] is submodular with W D, where D is the poset of random variables with the first order stochastic dominance as partial order. 3. If f and g are submodular (supermodular) then, for each positive scalar a, af and f +g are submodular (supermodular). 4. If f(x, y) is submodular (supermodular) then g(x, y) = f(x+a, y+b) is also submodular (supermodular) for any (a, b) R For any cost parameters, the newsboy function (loss function) y L (W, y) = h (y w) df W (w) + b y (w y) df W (w) is submodular with W D, where D is the poset of random variables with the first order stochastic dominance as partial order. Lemma 5 If g is convex then H (x, θ) := min g (y) is supermodular (0 θ, 0 U). y [x;x+θ+u] Proof : We define the global optimum point as ŷ := ming (y) R {, + }. For supermodularity, we y R aim to show x x +, θ θ + : H (x +, θ ) + H (x, θ + ) H (x, θ ) + H (x +, θ + ) The domain of H (x, θ) can be divided into three parts. g (x) increasing in x, if ŷ x H (x, θ) = g (ŷ) constant, if ŷ U x + θ and x ŷ g (x + θ + U) decreasing in x + θ, if x + θ ŷ U Step 1) H (x, θ) is convex (see the proof of Theorem 1). Step 2) H (x, θ + ) H (x, θ ) holds for all x, θ, θ +, when θ θ + Step 3 / Case a) x + ŷ U : H (x +, θ ) = H (x +, θ + ) which means that the supermodularity equation holds. Step 3 / Case b) x + ŷ U : We can define a function with a single variable h (x + θ) := H (x, θ) which is convex by step 1. Applying Property 1-1 completes the proof. Lemma 6 If g (y, θ ) is convex then H (x, θ) := min y [x;x+θ+u] 0 θ g (y, θ ) is supermodular (0 θ, 0 U). Proof : We can take the steps of the previous lemma and the result follows. Proof of Theorem 3: f T +1 (x T +1, θ T +1 ) 0 is supermodular. ( J T (y T, θ T +1 ) = L T (y T ) + f T +1 is supermodular being invariant on θ T +1. Next, we use convexity of J t yt, θ t+l) ( or min J t yt, θ t+l), and 0 θ t+l apply Lemma 5 or Lemma 6 depending on the value of t and observe that ( H t xt, θ t) min J ( t yt, θ t+1), if T L + 1 t T y := t [x t;x t+θ t+u] ( min min J t yt, θ t+1), if 0 < t < T L θ t+l y t [x t ;x t +θ t +U] is supermodular in (x t, θ t ). By Properties 1-2, 1-3, and 1-4, we can conclude that the function f t (x t, θ t ) = Uc p +θ t c c +H t (x t, θ t ) is supermodular in (x t, θ t ) and the functions J t (y t ) = L t (y t )+αe[f t+1 (y t W t, θ t+1 ), and J t (y t, θ t+l ) = L t (y t ) + αe [f t+1 (y t W t, θ t+1 )] are supermodular. Proof of Theorem 4: 19

20 ( ) We prove the theorem for only discrete random variables in D. By Theorem 3 that f t+1 xt+1, θ t+1, θt+1 rest is supermodular in (x t+1, θ t+1 ) for all t where θ rest t = { (θt+1,..., θ T 1, θ T ), if T L + 1 t T (θ t+1,..., θ t+l, θ t+l 1 ), if 0 < t < T L + 1. By definition ( of supermodularity, ) ( for x x ) +, θ ( θ + we have ) ( ) f t+1 x +, θ, θt+1 rest + ft+1 x, θ +, θt+1 rest ft+1 x, θ, θt+1 rest + ft+1 x +, θ +, θt+1 rest. We introduce new variables w := y x +, w + := y x with an arbitrary y. For w w +, θ θ + we have ( ) ( ) ( ) ( ) f t+1 y w, θ, θt+1 rest +ft+1 y ( w +, θ +, θt+1 ) rest ft+1 ( y w +, θ, ) θt+1 rest +ft+1 y w, θ +, θt+1 rest for all y and θt+1 rest. This means that H t w, θ, θ rest t+1 := ft+1 y w, θ, θ rest t+1 is submodular in (w, θ) for all t. By Properties 1-2 and 1-3, we have E [ ( H t Wt, θ t+1)] submodular in (W t, θ t+1 ) and hence J t (W t, θ t ) := L t (y t ) + αh t (W t, θ t+1 ) is also submodular. Proof of Theorem 5: We show only among discrete random variables in D. Since f t+1 ( xt+1, θ t+l) is convex for all t, we have H t (x) := f t+1 ( x, θ t+1 ) convex for all t. By the Properties 1-2, 1-3, and 1-5, H t (w, y) := H t (y w) submodular in (w, y) for all t, E [ H t (W t, y t ) ] submodular in (W t, y t ), L (W t, y t ) is submodular, and finally J t (W t, y t ) := L (W t, y t ) + αh (W t, y t ) is also submodular. Proof of Theorem 6: We prove the statement indirectly. Let W min be the smallest possible realization of W. Assume, that the optimal strategy does not satisfy complementarity; that is (y1, θ2) [x 1 ; x 1 + θ 1 + U) (0; ). We can define another feasible strategy (y 1, θ 2 ) such that y 1 := y1 + ε and θ 2 := θ2 ε with ε := min { W min 2, x 1 + θ 1 + U y1, θ2} > 0. We study the cost difference J1 between the two strategies. J 1 := J 1 (y 1, θ 2) J 1 (y 1, θ 2 ) = P (ŷ 1 [x 2 + ε; )) (L (y 1) L (y 1 ) + αc c ε) + P (ŷ 1 [x 2 ; x 2 + ε)) C 1 +P (ŷ 1 (; x 2 )) C 2 with C 1 and C 2 expected costs for the remaining stages. As 0 < ε < W min, we have P (ŷ 1 [x 2 ; x 2 + ε)) = P (ŷ 1 (; x 2 )) = 0 and P (ŷ 1 [x 2 + ε; )) = 1. Therefore, J 1 = L (y 1) L (y 1 + ε) + αc c ε. Since L is convex and L < h, we have L (y 1 + ε) L (y 1) < hε which leads to J 1 > hε + αc c ε > 0 by the second condition. Positive J 1 value contradicts with (y 1, θ 2) being the optimum. Proof of Lemma 1: We introduce two functions H (y T 1, θ T ) := E [f T (y T 1 W T 1, θ T, U)] J T 1 (y T 1, θ T ) := L T 1 (y T 1 ) + αh (y T 1, θ T ) The unconstrained and constrained optima of the last stage are ( ) ŷ T, if ŷ T [x T ; x T + θ T + U] ŷ T = G 1 b T h+b and yt (x T, θ T, U) = x T, if ŷ T < x T x T + θ T + U, if x T + θ T + U < ŷ T Step 1) We write H (y T 1, θ T ) eliminating the minimization of the last stage. H (y T 1, θ T, U) = E [f T (y T 1 W T 1, θ T, U)] = = f T (y T 1 ω, θ T, U) dg T 1 (ω) = Uc p + θ T c c + min {L T (y T ) + f T +1 } dg T 1 (ω) = y T [y T 1 ω;y T 1 ω+θ T +U] = Uc p + θ T c c + L T (y T (y T 1 W T 1, θ T, U)) dg T 1 (ω) = 20

21 = Uc p + θ T c c + = Uc p + θ T c c + L T L T ŷ T, if ŷ T [y T 1 ω; y T 1 ω + θ T + U] y T 1 ω, if ŷ T < y T 1 ω dg T 1 (ω) y T 1 ω + θ T + U, if y T 1 ω + θ T + U < ŷ T ŷ T, if ω [y T 1 ŷ T ; y T 1 ŷ T + θ T + U] y T 1 ω, if ω < y T 1 ŷ T dg T 1 (ω) y T 1 ω + θ T + U, if y T 1 ŷ T + θ T + U < ω = Uc p + θ T c c + L T (ŷ T ) [G T 1 (ω)] y T 1 ŷ T +θ T +U + + y T 1 ŷ T y T 1 ŷ T +θ T +U y T 1 ŷ T L T (y T 1 ω) g T 1 (ω) dω L T (y T 1 ω + θ T + U) g T 1 (ω) dω Step 2) We take derivatives θt J T 1 and yt 1 J T 1 to find the optimal decision of stage T 1, ( yt 1, ) θ T. The derivatives turn out to be c c b + bg T 1 (y T 1 ŷ T + θ T + U) θt J T 1 (y T 1, θ T, U) = α + (h + b) G T (y T 1 ω + θ T + U) G T 1 (ω) dω y T 1 ŷ T +θ T +U and bg T 1 (y T 1 ŷ T + θ T + U) G T 1 (y T 1 ŷ T ) 1 y T 1 ŷ T yt 1 J T 1 (y T 1, θ T, U) = L T 1 (y + (h + b) G T (y T 1 ω) g T 1 (ω) dω T 1)+α + (h + b) G T (y T 1 ω + θ T + U) g T 1 (ω) dω) y T 1 ŷ T +θ T +U When we take the roots of the derivatives we find the stated equations. Proof of Theorem 7: We introduce an equivalent form of equation (1) (h + b) G T (y T 1 ω + θ T + U) G T 1 (ω) dω = b bg T 1 (y T 1 ŷ T + θ T + U) c c (5) y T 1 ŷ T +θ T +U and two new optimality equations based on the substitution of (5) into (2). bg T 1 (y T 1 ŷ T ) c c L T 1 (y T 1 ) + α y T 1 ŷ T + (h + b) G T (y T 1 ω) g T 1 (ω) dω = 0 (6) (h + b) y T 1 ŷ T G T (y T 1 ω) G T 1 (ω) dω = bg T 1 (y T 1 ŷ T ) + c c L T 1 (y T 1) α (7) Case (A) ŷ T < ŷ T 1 Step 1) y T 1 ŷ T y G T (y T 1 ω) G T 1 (ω) dω = T 1 ŷ T 0 0 (1 G T (y T 1 ω)) G T 1 (ω) dω + G T 1 (y T 1 ŷ T ) = We apply an intermediate value theorem for (1 G T (y T 1 ω)) monotone (increasing), bounded, Riemann-integrable, non-negative, and G T 1 bounded, Riemann-integrable. There exist ζ [0; y T 1 ŷ T ] so that = (1 G T (ŷ T )) y T 1 ŷ T ζ G T 1 (ω) dω + G T 1 (y T 1 ŷ T ) = 21

22 = G T 1 (ζ) + G T (ŷ T ) G T 1 (y T 1 ŷ T ) G T (ŷ T ) G T 1 (ζ) Step 2) From equation (7) and Step 1): there exist ζ [0; y T 1 ŷ T ] so that (h + b) (G T 1 (ζ) + G T (ŷ T ) G T 1 (y T 1 ŷ T ) G T (ŷ T ) G T 1 (ζ)) = bg T 1 (y T 1 ŷ T )+c c L T 1 (y T 1) α This equation is equivalent to L T 1 (ŷ T 1) = α (c c hg T 1 (ζ)) from which we can deduce two inequalities: L T 1 (ŷ T 1) α (c c h), and L T 1 (ŷ T 1) αc c As L T 1 is convex, we have L T 1 (y T 1) increasing, so from the inequalities above, we have L 1 T 1 (α (c c h)) ŷ T 1 L 1 T 1 (αc c) Case (B) ŷ T 1 ŷ T For y T 1 ŷ T, we have G T 1 (y T 1 ŷ T ) = 0, y T 1 ŷ T G T (y T 1 ω) G T 1 (ω) dω = 0 and from (6) we get L T 1 (y T 1) αc c = 0, which results ŷ T 1 = L 1 T 1 (αc c) Proof of Theorem 8: The curve of intersection of J T 1 with the plane y T 1 + θ T = 0 defines a new function, JT 1, which we parameterize with variable y T 1. J T 1 (y T 1 ) = L T 1 (y T 1 ) + αuc p αy T 1 c c + αl T (ŷ T ) [G T 1 (ω)] U ŷ T y T 1 ŷ T y T 1 ŷ T +α L T (y T 1 ω) g T 1 (ω) dω +α U ŷ T L T (U ω) g T 1 (ω) dω We take the derivative of function J T 1 (y T 1 ) and look for negative values. 0 > yt 1 JT 1 (y T 1 ) = +L T 1 (y T 1) αc c αl T (ŷ T ) g T 1 (y T 1 ŷ T ) y T 1 ŷ T L T (y T 1 ω) g T 1 (ω) dω +α yt 1 As a result, we have the inequality, L T 1 (y T 1 ) + α (h + b) y T 1 ŷ T G T (y T 1 ω) g T 1 (ω) dω < αc c + αbg T 1 (y T 1 ŷ T ) (8) By increasing its LHS, we create a sufficient condition for inequality (8) to hold. y L T 1 (y T 1 ŷ T T 1) + α (h + b) G T (y T 1 ω) g T 1 (ω) dω h + α (h + b) = h + α (h + b) G T 1 (y T 1 ŷ T ) h (1 + α) + αbg T 1 (y T 1 ŷ T ) y T 1 ŷ T 1 g T 1 (ω) dω = When we check in (8) if the increased LHS is still below its RHS, we find h (1 + α) + αbg T 1 (y T 1 ŷ T ) < αc c + αbg T 1 (y T 1 ŷ T ) which is equivalent to h (1 + α) < αc c. Consequently, {y T 1 + θ T = 0, y T 1 + } is an always decreasing ray for J T 1 (y T 1, θ T ) when h (1 + α) < αc c. In this case, the constrained optimum of the T 1th stage is on the border, either in B 1 = {y T 1 = x T 1 + θ T 1 + U, θ T R + } or in B 2 = {y T 1 [x T 1, x T 1 + θ T 1 + U], θ T = 0}, and complementariness holds between x T 1 +θ T 1 +U y T 1 and θ T. 22