A LAGRANGEAN DECOMPOSITION ALGORITHM FOR MULTI-ITEM LOT-SIZING PROBLEMS WITH JOINT PURCHASING AND TRANSPORTATION DISCOUNTS 1. Alain Martel.

Transcription

1 A LAGRANGEAN DECOMPOSITION ALGORITHM FOR MULTI-ITEM LOT-SIZING PROBLEMS WITH JOINT PURCHASING AND TRANSPORTATION DISCOUNTS 1 Alain Martel Nafee Rizk Amar Ramudhin Network Enterprise Technology Research Center, Faculté des sciences de l'administration, Université Laval, Québec, Canada. Septembre This research was supported by the Natural Sciences and Engineering Council of Canada and CAPS LOGISTICS, inc.

2 1. INTRODUCTION In the consumer goods wholesaling and retailing industry, a large number of stock keeping units must be managed on a regular basis. The items are typically purchased in families, each family being associated to a specific external vendor, and usually they are shipped together. The transportation costs thus incurred account for a significant part of the items price, and their structure may incorporate economies of scale, as well as constraints on the total load which can be carried. In some cases, the transportation costs are included in the item prices and they are not paid directly by the buyer. In such cases, however, the vendor often offers discounts on the total amount of the order and imposes constraints on its size. In either cases, this give rise to the class of dynamic multi-item lot sizing problems with oint shipping and/or purchasing cost discounts studied in this paper. This problem is a generalization of several of the dynamic lot-sizing problems that have been treated in the literature. Much work has been done to find exact and heuristic methods to solve the numerous versions of the dynamic lot-sizing problem. The complexity of the problem depends largely on whether it deals with a single item or several items, on the resource constraints imposed on the lot sizes (unconstrained or oint lower bounds and/or oint upper bounds) and on the type of cost structure which applies. A dynamic programming algorithm to solve the basic single-item case was published by Wagner and Whitin [1958]. Comparisons of the numerous heuristics that have been developed to solve the single-item problem can be found in Zoller and Robrade [1988]. Bitran and Yanasse [1982] provide a review and analysis of the problem under capacity constraints. When several items are involved, additional complexity is introduced by cost interactions and/or by resource interactions. The simplest multi-item case is the one where the interaction is due only to a oint setup (or order) cost. Dynamic programming algorithms to solve this problem were proposed by

3 2 Zangwill [1966], Veinott [1969], Kao [1979], Silver [1979] and Haseborg [1982]. Atkins and Iyogun [1988] developed a heuristic procedure. Maes and Van Wassenhove [1986] reviewed the case where the interaction is due only to capacity constraints and where costs are stationary. They also provide a comparison of three "simple" heuristics. An integer programming cutting plane algorithm was proposed by Barany, Van Roy and Wolsey [1984] and Lagrangean relaxation procedures were developed by Thizy and Van Wassenhove [1985], Gelder, Maes and VanWassenhove [1986], Trigeiro, Thomas and McClain [1989] and Diaby, Bahl, Karwan and Zionts [1992]. A number of set partitioning and column generation heuristics are also proposed by Cattrysse, Maes and VanWassenhove [1990]. To the best of our knowledge, little work was done on the case where the interaction is caused by oint order cost functions as well as by restrictions on the amount of resources used/available. Such situations are very common in the consumer goods wholesaling and retailing industry and occur when shipment costs depend on the total amount of transportation resources used by all the items, and/or when all unit or incremental discounts are offered by vendors on the total invoice value, instead of on the quantity ordered for individual items, as is usually assumed in the literature. In practice, shipping costs depend on several factors. Pricing mechanisms are not the same for private fleets and for public carriers. In the former case, for our purposes, only variable costs related to specific shipments are relevant. In the latter case, freight-rate structures depend on the mode used, the shipping distance and weight, the type of shipment (TL or LTL), and the commodity class of the item shipped. For this reason, few studies have attempted to model freight-rate structures explicitly. Buffa and Munn [1989] provide a review of the work done in this area. Most of the currently available models that incorporate a realistic representation of freight-rate structures deal only with the single-item, constant-demand case.

4 3 Little work has been done on the modeling of oint discounts based on total invoice value. Some aspects of the problem are studied by Chakravarty [1984] and by Silver and Peterson [1985] for the unconstrained constant-demand case. To the best of our knowledge, however, no previous work is available on oint discounts for the multi-item dynamic-demand problem treated in this paper. The paper is organized as follows. In section 2 the problem is defined and formulated as a mixed-integer program. In section 3 Lagrangean relaxation is used to decompose it in a set of simple sub-problems. A heuristic method based on subgradient optimization is then proposed, and a branch-and-bound algorithm is designed to solve the original model. Finally, section 4 presents the experiments performed to assess solution methods and it discusses computational results. 2. PROBLEM DEFINITION AND FORMULATION For most practical situations, oint shipping cost structures, or oint price discount structures, can be represented by a general (i.e., not necessarily concave, nor continuous) piecewise-linear function z(s) of the amount S [0, S max ] of a critical resource (measured in cwt, cube, $,...) associated to an order. S max is an upper bound on the availability of this resource in a given time period. This complex structure for the shipment and purchasing costs is required to model the various types of discounts offered by public carriers in practice. It also accounts for the situation where one switches from one mode of transportation to another based on volumes, and for the constraints imposed by carriers to qualify for certain types of discounts. It can also adequately model the costs incurred when a private fleet is used, as well as incremental or all units cost discount structures. The situations that are encountered the more often in practice are illustrated in Figure 1.

5 4 z(s) A fixed cost available capacity z(s) large artificial cost A 2 = A 1 fixed cost fixed cost - A 1 Smax a) Fixed cost with capacity constraint S min b) Fixed cost with bounds on load S max z(s) A = cost of a truck trip z(s) A A truck capacity S 2S c) Private fleet TL costing Smax =3S S 1 S 2 S 3 S 4 d) Public carrier / all units discount Smax z(s) z 3 (s 3, λ 3 ) v 3 A 3 s 3 A 1 v 1 =0 v 2 minimum order quantity technological constraint S b 1 b 2 b 3 b 4 e) General case (not continuous nor concave) b 5 S max Figure 1: Shipping/Purchasing Costs Discount Functions

6 Let b, = 1,...,γ, denote the length of the th interval defined on the S-axis by the linear segments of z(s) and define new linear functions, 5 z (s, λ ) = A λ + v s, λ {0, 1}, s [0, b ], = 1,..., γ, over each of these intervals as illustrated in Figure 1e). Then, assuming that when there is a discontinuity the lowest point is preferred, the shipment size is given by S = γ = 1 the amount procured at rate v, and the shipment/purchasing cost by s, where s is γ z(s) = =1 z (s, ) λ = ( A λ + v s ), provided that, γ = 1 i) if A 0 a) λ = 1 and s = b if S k = 1 b k b) λ = 1 and 0 < s b if 1 k= 1b < S k = 1 b k k c) λ = 0 and s = 0 if S 1 k= 1b k ii) if A < 0 a) λ = 1 and s = b if S k = 1 b k b) λ = 1 and 0 s b if 1 k= 1b S k = 1 b k k c) λ = 0 and s = 0 if S 1 k= 1b k Since our obective is to minimize costs, condition i-b) will be satisfied even if the < is replaced by a and hence the required conditions can be stated as b λ +1 s b λ, λ {0, 1}, = 1,..., γ, λ γ+1 = 0,

7 To simplify the notation, we assume without loss of generality that the intervals on the S-axis of the shipment/purchasing costs function do not change from period to period. 6 i : an item type, i F, F = {1,, m} t : a planning period, t H, H = {τ+1,..., T} τ d it : procurement lead-time (in planning periods) for the family of items : effective demand for item i during period t r it : resource absorption rate for item i in period t (in cwt, cube, $,...) A t v t : fixed cost/saving associated to the th interval on the oint shipment/purchasing cost function in period t : th shipment/purchasing cost per unit of resource for period t b : maximum amount which can be procured at a rate v t, for all t (in cwt, cube, $,...) c it h it R it I it : procurement cost of an item i received at the beginning of period t : inventory holding cost for item i in period t : quantity of item i received for the beginning of period t : inventory of item i on hand at the end of period t S t : resources absorbed by the items procured for period t (in cwt, cube, $,...) s t : amount procured at rate v t for period t (in cwt, cube, $,...) λ t : binary variable associated with the th interval on the shipment/purchasing cost function for period t Figure 2: Basic Notation Using the notation defined in Figure 2, the dynamic lot-sizing problem with purchasing and/or transportation cost discounts can be formulated as follows: (P) min T t = τ + 1 { γ m = 1 ( A λ + v s ) + i = 1 [c it R it + h it I it ] } t t t t subect to (1) R it + I i,t-1 - I it = d it, i, t = τ+1,..., T γ (2) = 1 s it - = m i 1 r it R it = 0, t = τ+1,..., T (3) s t - b λ t 0,, t = τ+1,..., T

8 7 (4) -s t + b λ t,+1 0,, t = τ+1,..., T (λ t,γ+1 = 0) (5) R it, I it, s t 0, λ t {0, 1} i,, t = τ+1,..., T (6) I iτ = I it = 0 i The obective function of (P) simply says that we want to minimize total shipment, purchasing and inventory costs. Constraints (1) are the inventory balance equations. They insure that all demands are met. Constraints (2) establish the relationship between the receptions R it and the variables s t defined over the intervals of the shipment/purchasing cost functions. Constraint sets (3) and (4) enforce the piecewise-linear nature of the shipment/purchasing costs. Finally, constraints (6) reflect our assumption that the inventory levels at the beginning and at the end of the planning horizon are zero. 3. SOLUTION METHOD 3.1. Lagrangean Decomposition If we multiply the constraints (2) in (P) by Lagrange multipliers υ t, and add the resulting expressions to the obective function of (P), the following two primal Lagrangean sub-problems are obtained: T m (SP1) min t= τ + 1 i= 1 [c it R it + υ t r it R it + h it I it ] subect to R it + I i,t-1 - I it = d it, i, t = τ+1,..., T R it, I it 0 i, t = τ+1,..., T I iτ = I iτ = 0 i.

9 8 and (SP2) min T = + γ t τ 1 = 1 [A t λ t + (v t -υ t )s t ] subect to s t - b λ t 0,, t = τ+1,..., T -s t + b λ t,+1 0,, t = τ+1,..., T (λ t,γ+1 = 0) s t 0, λ t {0, 1}, t = τ+1,..., T. problems: Clearly, (SP1) decomposes into the following m unconstrained single-item lot-sizing (SP1-i) L1 i (υ) = min T t = τ + 1 [(c it + υ t r it )R t + h it I t ] subect to (7) R t + I t-1 - I t = d it, t = τ+1,..., T (8) R t, I t 0 t = τ+1,..., T (9) I τ = I T = 0. and (SP2) decomposes into the (T-τ) independent sub-problems: γ (SP2-t) L2 t (υ) = min = [A t λ + (v t -υ t )s ] 1 subect to (10) s - b λ 0, (11) -s + b λ +1 0, ( λ γ+1 = 0)

10 9 (12) s 0, λ {0, 1}. Problem (SP1-i) is a linear network flow model and it can be solved efficiently with the transportation algorithm or with Wagner-Whitin's dynamic programming algorithm (Wagner and Whitin [1958]). Problem (SP2-t) can be solved efficiently with the O(γ) time algorithm proposed by Diaby and Martel [1993]. From Lagrangean decomposition theory (Guignard [1987]), we know that m (13) L D (υ) = i= 1 L1 i (υ) + T t = τ + 1 L2 t (υ) is a lower bound to the cost of the optimal solution of (P). We also know that the greatest lower bound attainable with our Lagrangean decomposition is provided by the multipliers obtained by solving the Lagrangean dual problem (LD) L D = max υ L D (υ) Problem (LD) is solved efficiently by subgradient optimization. If, for a given υ, the solution provided by the primal sub-problems is a feasible solution of (P) with value V*, then we know that it is within ([V* - L D (υ)]/v*)100% of the optimum. This provides the basis for the elaboration of a good heuristic procedure, as well as a branch-and-bound algorithm, for the solution of (P). 3.2 Branch-and-Bound Algorithm The branch-and-bound (B&B) algorithm developed involves a branching strategy based on feasibility considerations, a bounding scheme based on Lagrangian relaxation, and a heuristic procedure to generate upper bounds at each node examined. We discuss these three issues in the following sections.

11 Branching Strategy A depth-first strategy is used in our B&B procedure. Each level of the branching tree corresponds to a period t T, and each node of this level generates γ+1 new candidate subproblems with each candidate corresponding to one of the following vectors: (14) (λ t1 = λ t2 =...= λ tγ = 0) (λ t1 =1, λ t2 =...= λ tγ = 0) (λ t1 = λ t2 = 1, λ t2 =...= λ tγ = 0)... (λ t1 = λ t2 =...= λ tγ = 1). The general B&B scheme is as follows. A candidate sub-problem is examined only if the set of all feasible solutions to (P), given the restrictions imposed at its parent nodes and at its node in the branching tree, is not empty and its lower bound is less than the best upper bound. When a node cannot be eliminated it generates (γ+1) new candidate sub-problems as mentioned above. Recall that levels of the tree correspond to periods. The period to branch on is the one that has the greatest violation of constraint (2) according to the Lagrangean relaxation solution of the candidate sub-problem associated to the node examined Lower Bounds Lower bounds are obtained by optimizing L D using a sub-gradient optimization approach (see Bazarra and Goode 1979 or Fisher 1981). Initial values for the multipliers are obtained heuristically by setting υ 0 = max t ( v ) t t = τ+1,, T At iteration k, those multipliers are changed according to the formula: υ k t = υ k-1 m t + β k-1 [( 1 r it R k-1* it - i= γ = 1 s t k-1* )]

12 where (R k-1*, s k-1* ) is the solution to the Lagrangean sub-problem at iteration k-1, and β k-1 is a scalar given by: 11 β k-1 = k-1 ( L - D * k D -1 T m L ) / [ τ 1 ( 1 r it R k-1* it - t = + i = γ = 1 s t k-1* ) 2 ], where k-1 (0, 2]; L D is an upper bound on the value of the Lagrangean dual problem, and * k D -1 L is the value of the Lagrangean sub-problem at iteration k-1. We start with 0 = 1 and multiply k-1 by 0.45 if the improvement in the Lagrangean bound is not more than 0.25% in two consecutive trials. We stop the procedure if the improvement in the Lagrangean bound is not more than 0.25% in 35 consecutive trials, or if the total number of iterations reaches Upper Bounds An upper bound at a given node of the B&B tree is obtained by perturbing the optimal solutions of SP1. Recall that the solutions of SP1 may not be feasible in (P) as the total quantity of transportation resources absorbed in period t (S t = m i= 1 r it R it ) may not fall within the shipment size range specified by the λ-values associated to the B&B tree node. The perturbation procedure involves a backward and a forward pass that either reduces shipment sizes to avoid capacity shortages or increases shipment sizes to avoid excess capacity. At each iteration the surplus or missing capacity is calculated for each period. Based on procurement and inventory cost considerations, the surplus or shortage is eliminated by shifting shipment quantities among periods. The result is a feasible solution that constitutes an upper bound for (P). This feasible solution is further improved by a heuristic procedure that seeks higher gains through additional period shifting adustments. A detailed description of the backward procedure is presented in Figure 3 where for a given node the following notations are used:

13 for t = T downto (τ+2) do begin step 1. gap t = S t U t ; if gap t > 0 then {reduce shipment size} repeat step 2. ϕ = {θ < t S θ < U θ }; if (ϕ φ) then begin step 3. Choose a product i 0 in F and a period t 0 < t in ϕ based on the following cost criterion: t (c i0t0 + 1 t θ = t h 0 i0θ - c i0t ) = min { min ((c α + 1 θ= α h θ - c t ))}; α ϕ F step 4. Compute the shipment quantity of product i 0 to shift from period t to period t 0 as follows: = min (R i0t, gap t /r i0t, (U t0 S t0)/r i0t0 ); step 5. Update R i0t, R i0t0, and gap t : R i0t = R i0t -, R i0t0 = R i0t0 +, gap t = gap t (r i0t * ) endif until (gap t 0 or ϕ = φ) else {S t U t 0} begin step 6. gap t = L t S t ; Rt> 0 if gap t > 0 then repeat step 7. ϕ = {θ < t L θ < S θ, and i such that R iθ > 0, step 8. step {increase shipment size} min I iδ >0} θ δ <t if ϕ φ then begin Choose a product i 0 and a period t 0 in ϕ based on the following t c i0t - c i0t0-1 t θ = t 0 h i0θ = min min ( c it - c iα - 1 θ = α h iθ ); α ρ i I(α) where I(α) = {i R iα > 0, min I iθ >0} α θ <t Compute the shipment quantity of product i 0 to shift from period t 0 to period t as follows: = min(r i0t0, gap t/r i0t, (S t0 L t0)/r i0t0, min I iδ ); t 0 δ <t step 10. Update R i0t, R i0t0, and gap t; R i0t = R i0t +, R i0t0 = R i0t0 -, gap t = gap t - r i0t * endif until (gap t 0 or ϕ = φ) endelse endfor Figure 3: Backward Pass Procedure

14 13 Γ = set of periods for which the λ-values have been fixed according to the branching strategy f(t) max 1 γ γ = { λt = 1} if t Γ if t (H - Γ) At a given node of the B&B tree, a feasible solution to the sub-problem represented by the node must be such that the total quantity of resources absorbed in each period t (S t ) is within the shipment size range [L t, U t ] where L t and U t are given by L t = 0 f(t) = 1 1 b if t Γ if not and f(t) U t = = 1 b The backward pass procedure, starting from period T and ending at period (τ+2), verifies for each period t whether the quantity of transportation resources (S t ) is greater than the maximum capacity allowed (U t ), by computing gap t = S t U t (step 1). An iterative procedure is then used to eliminate the capacity shortage by shifting the reception of a certain quantity of a product i 0 from period t to a period t 0 < t. The product i 0 and the period t 0, if any, are chosen in step 3 so as to have the smallest net increase of the procurement and holding costs (c i0t0 + t 1 θ = t 0 h i0θ - c i0t), and the quantity is determined in step 4 in a way not to generate unfeasibility at period t 0 ( (U t0 S t0)/r i0t0 ) and not to be larger than the needed ( gap t /r i0t ). This process is repeated until the shortage capacity is eliminated (gap t 0) or no product i 0 can be moved from the period t to any period t 0 < t so as to reduce capacity shortage without violating the (P) problem constraints. The second part of the backward pass procedure verifies whether the quantity of transportation resources absorbed (S t ) is less than the minimum capacity required (L t ) by computing gap t = L t S t (step 6). If gap t > 0 (S t < L t ), a step by step procedure is used to

15 14 eliminate the shortage of the quantity transported by shifting the reception of certain quantity of product i 0 from a period t 0 < t to the period t. The product i 0 and the period t 0 < t are chosen according to a cost criterion given in step 8, and the quantity is determined in step 9. This process is repeated until the excess capacity is eliminated (gap t 0) or no product i 0 can be moved from any period t 0 < t to period t without violating the problem constraints. Starting from period τ+1 and ending at period T-1, the forward pass procedure follows the same logic as the backward procedure, but this time, the surplus (missing) quantity in period t is shifted to (from) a period t 0 > t. To improve the quality of the feasible solution found by the perturbation procedure, a backward-forward improvement algorithm is then used. A description of this algorithm is given in Figure 4. Starting from period T and ending at period (τ+2), the backward pass of the improvement algorithm verifies for each period t whether the quantity of transportation resources adsorbed (S t ) is greater than the minimum capacity allowed (L t ) (step 1). If S t > L t, for each period t e and for each product i, the quantity of product i that can be shifted from the period t to the period t e without violating the solution feasibility is calculated in step 2. The net increase in the obective function if the quantity of product i is shifted from period t to period t e is calculated in step 3. If the net increase is below 0, the reception quantities of product i in the period t (R it ) and in the period t e (R ite) are respectively decreased and increased by the quantity. Starting from period τ+1 and ending at period T-1, the forward pass of the improvement algorithm procedure follows the same logic, but this time, the shift, if advantageous, is done from period t to a period t e > t.

16 for t = T downto (τ+2) do begin step 1. step 2. if S t > L t then begin for t e = t-1 downto (τ+1) do begin if U te > S te then begin for i =1 to m do begin compute the shipment quantity of product i to shift from period t to period t e as follows: = min ( R it, (S t - L t ) / r it, (U te S te) / r ite ); if > 0 then begin step 3. compute the net increase on the obective function if the quantity of product i is shifted from the period t to the period t e as follows : ao = CT CT, {where CT: is the obective function value of the current solution. CT : is the obective function value of the solution obtained from the current solution by shifting a quantity of product i from the period t to the period t e } if ao < 0 then begin step 4. R it = R it -, R ite = R ite +, endif endif endfor endif endfor endif endfor Figure 4: Backward Pass of Improvement Procedure 15 Lastly, it is important to note that the sub-problems obtained when all the binary variables are fixed (i.e. for the nodes at the bottom of the B&B tree) are linear programs. We use a linear programming library (CPLEX) to solve these sub-problems. Since the B&B procedure usually finds the optimal solution before these lowest level nodes are explored, a powerful heuristic is

17 16 obtained by stopping the exploration as soon as the difference between the lower and upper bounds does not exceed ε% of the upper bound value. This heuristic procedure thus guarantees an ε%-optimal solution for any ε TESTS RESULTS Testing was done to compare the computational times and the quality of the solutions obtained for different values of ε (0%, 1%, and 3%) as well as to evaluate the efficiency of our perturbation procedure. We programmed the algorithm in Pascal in a Delphi 2.0 environment and implemented it on a Pentium 90 computer. The tests were done on 560 problems of different sizes and with different characteristics. For each ε-value, five problems of each type were solved. The experimental design is based on the following factors. The structure of the cost function: The transportation/purchasing cost function considered are those presented in Figure 5. The number of products: Two groups of problems were studied. The first group has a number of products that is uniformly distributed between 50 and 100 (m U[50, 100]). The number of products in the second group is uniformly distributed between 250 and 300 (m U[250, 300]). The number of periods: Planning horizons of 13 and 26 periods were tested. The variation of demands: Product demands are generated from a Normal distribution (d i N(µ i U[10, 100]), σ i = µ i / CV) where the coefficient of variation CV is equal to either 0.25 or 0.6 in order to evaluate the impact of demand variation amplitudes. The importance of the inventory holding cost relative to the item cost. Test problems with small inventory costs (h i U[0.1% c i, 0.2% c i ]) and large inventory costs (h i U[0.5% c i,

18 0.6% c i ]) relative to a time invariant purchasing cost c i (c i U[10, 40]) are considered in our experiments 17 z(s) z(s) 1) Private fleet 2) Public fleet with 3 intervals z(s) z(s) 3) Public fleet with 2 intervals and a lower bound 4) Public fleet with 1 interval z(s) z(s) 5) Public fleet with 1 interval and an upper bound 6) Public fleet with 2 intervals and lower and upper bounds z(s) 7) Public fleet with 3 intervals and an upper bound Figure 5: Structure of the cost functions used Overall we considered 7 different transportation/procurement cost functions, 2 different ranges for the number of products, 2 different planning horizon lengths, 2 different demand variations, and 2 different inventory costs scenarios. Thus, we have 112 ( ) different problem instances. Since we generated five problems for each instance, 560 problems (5 112) were solved for each ε-value. All together, the problems solved can be classified in the 15 problem types described in Table1.

19 Notation Description Nbr. Pbs P1**** Private fleet 16 P2**** Public fleet with 3 intervals (continuous) 16 P3**** Public fleet with 2 intervals and a lower bound (not continuous) 16 P4**** Public fleet with 1 interval 16 P5**** Public fleet with 1 interval and an upper bound 16 P6**** Public fleet with 2 intervals and a lower bound and an upper bound (non continuous) 16 P7**** Public fleet with 3 intervals and an upper bound (continuous) 16 P*1*** Family of 50 to 100 products 56 P*2*** Family of 250 to 300 products 56 P**1** 13 periods horizon 56 P**2** 26 periods horizon 56 P***1* Demands generated with CV = P***2* Demands generated with CV = P****1 Inventory costs generated within [(0.1%)ci, (0.2%)c i ] 56 P****2 Inventory costs generated within [(0.5%)ci, (0.6%)ci] 56 Table 1: Classification of Problems Solved 18 The computational results for each ε-value are presented in Table 2. The table gives computational times and % deviations from the optimal solution cost, for the various types of problems solved. Times are not given for the perturbation procedure used to obtain the initial solution because they were negligible. The results indicate that our Lagrangean decomposition heuristic yields high quality solutions. When ε is set to 1%, instead of 0%, the computational time reduces by 59% on average, but the deviation from the optimum is only of 0.16%. When ε is set to 3%, a 95% reduction in computation time is observed but the average deviation from the optimum is only of 0.32%. The perturbation procedure used to get a first feasible solution is also a very good heuristic. Even if the CPU time required to compute these solutions is negligible, for the test problems the perturbation procedure gave the optimal solution four times out of six. A more detailed look at the results reveal that some problems are much harder than others. The main discriminating criteria is the presence of a tight upper bound on the cost function. All the problems with an upper bound (cost functions 1, 5, 6 and 7) were solved optimally by the initial perturbation procedure, independently of the shape of the cost function. The time required to solve the problems with unbounded cost functions is much higher and the quality of the

20 19 solution given by the heuristics is not as high. This is due to the fact that since the perturbation algorithm does not perform as well in those cases, the upper bounds obtained during the branch and bound procedure are not as good. The quality of the solution obtained with the heuristics, in terms of deviation from the optimum, does not seem to be affected very much by the other experimental factors considered. Problem type ε = 0 % ε = 1% ε = 3% Initial solution Aver. time (sec) Aver. time (sec) % Devi./opt.sol Aver. time (sec) % Devi./opt.sol % Devi./opt.sol P1**** 2,30 2,30 0,00 2,30 0,00 0,00 P2**** 102,60 36,96 0,16 33,66 0,47 5,98 P3**** 337,49 42,24 0,21 9,99 0,62 0,24 P4**** 824,34 433,90 0,83 2,29 1,25 1,28 P5**** 2,46 2,38 0,00 2,31 0,00 0,00 P6**** 5,34 4,78 0,00 4,23 0,00 0,00 P7**** 4,14 3,75 0,00 3,30 0,00 0,00 P*1*** 74,90 22,35 0,18 4,60 0,32 0,73 P*2*** 290,43 128,02 0,14 12,00 0,31 1,22 P**1** 21,99 10,65 0,15 3,82 0,27 0,96 P**2** 343,34 139,72 0,18 12,78 0,37 1,00 P***1* 188,68 65,46 0,18 8,60 0,30 0,99 P***2* 176,66 84,92 0,15 8,00 0,34 0,97 P****1 152,68 49,17 0,16 8,20 0,23 1,00 P****2 212,66 101,20 0,16 8,40 0,40 0,94 All problems 182,67 75,19 0,17 8,30 0,32 0,98 Table 2 : Computational Results As could be expected, however, the time required to solve the problems depends on the number of products and on the length of the planning horizon. The planning horizon effect is much more pronounced. The computation time is almost directly proportional to the number of products: on average, when the number of products is multiplied by 3.6 the time required to solve the problems to optimality is multiplied by 3.8. On the other end, doubling the number of planning periods multiplies the time required to obtain the optimal solution by 15. Problems with high holding costs are a bit longer to solve, but the impact of this factor is not as marked as the previous ones. Finally, the variability of the demand does not seem to influence computation times.

21 20 5. CONCLUSION We have formulated a mixed-integer linear programming model to plan the procurement of a family of items under piecewise linear economies of scale. The model accommodates switches between transportation modes based on volumes, and various types of discounts that may be offered in practice on transportation and/or purchasing prices. Mathematical properties of a Lagrangean relaxation of the model were developed and a branch-and-bound procedure that exploits these properties was presented. The latter includes a powerful heuristic for generating upper bounds. The B&B procedure appears to be efficient for solving moderate size problems, and the heuristic method can be used to obtain near-optimal solutions for large real-life supply planning problems. 6. REFERENCES - ATKINS, D. AND P. IYOGUN, "A Heuristic with Lower Bound Performance Guarantee for the Multi-Product Dynamic Lot-Size Problem", IIE Trans., 20 (1988), BARANY, I., T. Van ROY AND L. WOLSEY, "Strong Formulations for Multi-Item Capacitated Lot Sizing", Man. Sci., 30 (1984), BITRAN, G. AND H. YANASSE, "Computational Complexity of the Capacitated Lot Size Problem", Man. Sci., 28 (1982), BUFFA, F. AND J. MUNN, "A Recursive Algorithm for Order Cycle-time that Minimizes Logistics Cost", J. Opl Res. Soc., 40 (1989), CATTRYSSE, D., J.MAES AND L.N. VANWASSENHOVE, "Set Partitioning and Column Generation Heuristics for Capacitated Dynamic Lot Sizing", E.J.O.R., 46 (1990), CHAKRAVARTY, A., "Joint Inventory Replenishments with Group Discounts Based on Invoice Value", Man. Sci., 30 (1984),

22 21 - DIABY, M., H. BAHL, M. KARWAN AND S. ZIONTS, "Capacitated Lot-Sizing and Scheduling by Lagrangean Relaxation", E.J.O.R., 59-3 (1992), DIABY, M., AND A. MARTEL, "Dynamic Lot-Sizing in Multi-Echelon Distribution Systems with Purchasing and Transportation Price Discounts", Ops. Res., vol 41-1 (1993), GELDERS, L., J. MAES AND L.N. VANWASSENHOVE, "A Branch and Bound Algorithm for the Multi-Item Single Level Capacitated Dynamic Lotsizing Problem", in Axsater, Schneeweiss and Silver (eds), Multi-Stage Production Planning and Inventory Control, Springer-Verlag, 1986, GUIGNARD, M. AND S. KIM, "Lagrangean Decomposition: A Model Yielding Stronger Lagrangean Bounds", Math. Prog., 39 (1987), HASEBORG, F., "On the Optimality of Joint Ordering Policies in a Multi-Product Dynamic Lot Size Model with Individual and Joint Set-up Costs", EJOR, 9 (1982), KAO, E., "A Multi-Product Dynamic Lot-Size Model with Individual and Join Set-up Costs", Ops. Res., 27 (1979), MAES, J. AND L. VAN WASSENHOVE, "Multi Item Single Level Capacitated Dynamic Lotsizing Heuristics: A Computational Comparison (Part I and II)", IIE Trans., 18 (1988), MARTEL, A., M. DIABY AND F. BOCTOR, Multiple Items Lot-sizing under Stochastic Nonstationary Demands, E.J.O.R., 87 (1995), SILVER, E., "Inventory Control Under a Probabilistic Time-Varying Demand Pattern", AIIE Trans., 10 (1978), SILVER, E., "Coordinated Replenishments of Items Under Time-varying Demand: Dynamic Programming Formulation", NRLQ, 26 (1979),

23 22 - SILVER, E. AND R. PETERSON, Decision Systems for Inventory Management and Production Planning, 2 nd ed., Wiley, THIZY, J. AND L. VAN WASSENHOVE, "Lagrangean Relaxation for the Multi-item Capacitated Lot-Sizing Problem: A Heuristic Implementation", IIE Trans., 17 (1985), TRIGEIRO, W., J. THOMAS AND J. MCCLAIN, "Capacitated Lot Sizing with Setup Times", Man. Sci., 35 (1989), VEINOTT, A. F., "Minimum Concave-Cost Solution of Leontief Substitution Models of Multifacility Inventory Systems", Ops. Res., 17 (1969), WAGNER, H. AND T.M. WHITIN, "Dynamic Version of the Economic Lot Size Model", Man. Sci., 5 (1958), ZANGWILL, W.I., "A Deterministic Multiproduct, Multifacility Production and Inventory Model", Ops. Res., 14 (1966), ZANGWILL, W.I., "A Backlogging Model and a Multi-Echelon Model of a Dynamic Economic Lot Size Production System", Man. Sci., 15 (1969), ZOLLER, K. AND A. ROBRADE, "Efficient Heuristics for Dynamic Lot Sizing", Int. J. Prod. Res., 26 (1988),