Dolores Romero Morales Saïd Business School, University of Oxford, Park End Street, Oxford OX1 1HP, UK,

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1 Online Supplement to A Heuristic Approach to the Multi-Period Single-Sourcing Problem with Production and Inventory Capacities and Perishability Constraints Ravindra K. Ahuja Department of Industrial and Systems Engineering, University of Florida, P.O. Box , Gainesville, Florida , USA, ahuja@ufl.edu Wei Huang Innovative Scheduling, Gainesville Technology Enterprise Center (GTEC), 2153 Hawthorne Road Suite 128, Gainesville, Florida 32641, USA, huang@innovativescheduling.com H. Edwin Romeijn Department of Industrial and Systems Engineering, University of Florida, P.O. Box , Gainesville, Florida , USA, romeijn@ise.ufl.edu Dolores Romero Morales Saïd Business School, University of Oxford, Park End Street, Oxford OX1 1HP, UK, Dolores.Romero-Morales@sbs.ox.ac.uk INFORMS Journal on Computing A. Stochastic Model We generalize the stochastic model proposed by Romeijn and Romero Morales (2001, 2003, 2004) for other variants of the MPSSP to the problem studied in this paper. In the remainder of this appendix, random variables will be denoted by capital letters, and their realizations by the corresponding lowercase letters. In addition, the symbol E will be used to denote expectation. For each j = 1,..., n, let (D j, C j ) be i.i.d. random vectors in [D, D] T [C, C] m (with D > 0), where D j = (D jt ),...,T and C j = (C ij ),...,m. We assume that the vectors (D j, C j ) (j = 1,..., n) are i.i.d. according to an absolutely continuous probability distribution for each j = 1,..., n. Note that the demands and costs for a particular retailer are allowed to be dependent. If the demands of all retailers share some common seasonality pattern we have D jt = σ t D j for a fixed vector of seasonality factors σ as described in Section of the paper. The production and inventory holding costs p it and h it are assumed to be fixed nonnegative constants. For convenience, let p = min i,t p it, p = max i,t p it, h = min i,t h it, and h = max i,t h it. To allow for sufficient capacity as the number of retailers grows, we let all capacities 1

2 b it, r it, and I it depend linearly on n: b it = β it n r it = ρ it n I it = η it n where β it, ρ it, and η it are positive constants. This way of modeling the capacities is customary in probabilistic models for assignment problems; see Dyer and Frieze (1992), Trick (1992), and Romeijn and Piersma (2000). Observe that m and T are fixed, so the size of the MPSSP depends only on the number of retailers n. In the presence of production capacities only, Romeijn and Romero Morales (2003) show that instances of the MPSSP generated using this model are feasible with probability one if E(D 1t ) < and infeasible with probability one if this inequality is reversed. Note that these conditions are closely related to the inequalities characterizing the domain of H i. In the presence of throughput, physical inventory, and perishability constraints, we can extend these results to the case where the retailer demands share a common seasonality pattern by recalling that the feasible region of (A) is the feasible region of an SSP (see (19) in the paper). In this case, we have B i = B i n where B i = { min min,...,t ( ρit σ t ), β it, min τ = 1,..., T r = k,..., T 2 ( τ+r t=τ β ) i[t] τ+r t=τ σ [t] β it, min τ = 1,..., T r = 0,..., T 2 ( τ+r t=τ β )} i[t] + η i[τ 1] τ+r t=τ σ. [t] The feasibility of the SSP was studied by Romeijn and Piersma (2000). They show that the following assumption ensures that problem instances generated according to the probabilistic model given above are asymptotically (as n ) feasible with probability one. Assumption A.1 The retailer demands share a common seasonality pattern, i.e., D jt = σ t D j for some vector of seasonality factors σ (σ t 0 for all t = 1,..., T and T σ t = 1). In addition, the normalized aggregate capacity exceeds the expected demand per retailer, i.e., E(D 1 ) < B i. (1) 2

3 In addition, Romeijn and Piersma (2000) show that asymptotic infeasibility is guaranteed with probability one if the inequality in the assumption is reversed. B. Asymptotic Analysis of the Greedy Heuristic In Section 3 of the paper we defined the pseudo-cost function ( k f(i, j) = c ij + λ it + ξit where λ, ξ, and α l=1 α i[t l] are, respectively, the vectors of optimal dual multipliers for the inventory-balance constraints, the throughput-capacity constraints, and the perishability constraints in (LP). In this appendix, we will show that, under a suitably defined stochastic model of the problem data, the greedy heuristic with this pseudo-cost function is asymptotically feasible and optimal with probability one as the number of retailers increases. The following lemma shows that the number of fractional assignments in a basic optimal solution to (LP) is relatively small, which suggests that the optimal LP solution and its dual could provide valuable information on the solution to (P) itself. If (x LP, y LP, I LP ) denotes a basic optimal solution for (LP), then the set of retailers that are split between facilities in the optimal solution to (LP) can be defined as ) d jt B = {j = 1,..., n : i such that 0 < x LP ij < 1}. Lemma B.1 For any basic optimal solution for (LP) we have that B 3mT. Proof: Rewrite (LP) with equality constraints and nonnegative variables only, by introducing slack variables, and by eliminating the variables I i0 (i = 1,..., m). The resulting problem contains, in addition to the assignment constraints (3), exactly 5mT equality constraints. Now note that in each of the constraints (1), either y it or the slack variable will be positive. Similarly, in each of the constraints (7), either I it or the slack variable will be positive. In addition, there will be at least one positive assignment variable for each retailer. Therefore, there can be no more than 3mT assignments that are split. The next proposition characterizes the set B consisting of retailers that are split between facilities in the optimal solution to (LP). 3

4 Proposition B.2 Suppose that (LP) is feasible and non-degenerate. Let (x LP, y LP, I LP ) be a basic optimal solution for (LP) and let λ, ξ, and α be the vectors of optimal dual multipliers associated with the inventory balance constraints, throughput capacity constraints, and perishability constraints in (LP) (rewritten as constraints). 1. For each j B, x LP ij and = 1 if and only if f(i, j) = f(i, j) < min f(s, j) s=1,...,m min f(s, j) s=1,...,m; s i 2. For each j B, there exists an index i {1,..., m} such that f(i, j) = min f(s, j). s=1,...,m; s i where ( f(i, j) = c ij + λ it + ξit k l=1 α i[t l] Proof: Similar to the proof of Proposition 2.2 in Romeijn and Romero Morales (2000). ) d jt Proposition B.2 provides the formal justification for using f(i, j) as the pseudo-cost function in our greedy heuristic. In particular, result 1 in the proposition states that the pseudocost function f indeed provides a measure of the desirability of assigning a given retailer j to any of the facilities. In the remainder of this appendix we will analyze the average case behavior of the greedy heuristic when employing this pseudo-cost function under the stochastic model studied in Appendix A. In the remainder of this appendix we prove asymptotic feasibility and optimality of the greedy heuristic with the pseudo-cost function proposed in Section 3 of the paper when the retailer demands exhibit a common seasonality pattern. We denote the objective value of the optimal LP solution by optimal assignments in (LP) in the paper by x LP (as before), and its objective value by v LP. Furthermore, let x G denote the (partial) solution to (A) in the paper given by the greedy heuristic, and v G be its objective value. Note that the integral assignments in x LP as well as x G are (partial) solutions to (A). Let N n be the set of assignments for which x G and x LP do not coincide. The following result shows that the number of differences for which x LP and x G do not coincide can be bounded from above by a constant independent of n. 4

5 Lemma B.3 There exists some constant R, independent of n, such that N n R for all instances of (LP) that are feasible and non-degenerate. Proof: It is obvious that it would be possible to fix all feasible assignments from x LP without violating any capacity constraint. Proposition B.2 ensures that in the first step of the greedy heuristic the most desirable facility for each retailer that is feasibly assigned in x LP is equal to the facility to which it is assigned in x LP. Moreover, the same proposition shows that the initial desirabilities are such that the greedy heuristic starts by assigning retailers that are feasibly assigned in x LP. Now suppose that the greedy heuristic would reproduce all the assignments that are feasible in x LP. Then, because the remaining assignments in x LP are infeasible with respect to the integrality constraints, x G and x LP would differ only in the last ones. By Lemma B.1 we know that then N n 3mT, and the result follows. So in the remainder of the proof we will assume that x G and x LP differ in at least one assignment that is feasible in the latter. While the greedy heuristic is assigning retailers that are feasibly assigned in x LP it may at some point start updating the desirabilities of the assignments still to be made due to the decreasing remaining capacities. This may cause the greedy heuristic to deviate from one of the feasible assignments in x LP. Such an assignment could use some capacity that x LP uses for other (feasible) assignments. In particular, this assignment uses at most D units of capacity. Since the facility that is involved in this assignment may now not be able to accommodate all retailers that were feasibly assigned to it in x LP, other deviations from the feasible assignments in x LP may occur. Since any other assignment requires at least D units, the number of additional deviations is at most equal to D/D. The remainder of the proof, which is based on bounding the number of times that the desirabilities ρ must be recalculated, and then bounding the number of deviations from x LP between these recalculations, is analogous to the proof of Theorem 4.1 in Romeijn and Romero Morales (2003). The following result ensures that under the stochastic model proposed in Appendix A, (LP) is non-degenerate with probability one. Lemma B.4 (LP) is non-degenerate with probability one, under the stochastic model proposed. 5

6 Proof: This follows directly from the fact that the demand parameters are absolutely continuous random variables. We will use the following lemma to prove the asymptotic feasibility result in Theorem B.6. Lemma B.5 Under Assumption A.1, B i 1 n j=1 D j X LP ij > 0 with probability one when n goes to infinity. Proof: Note that B i 1 n j=1 D j X LP ij = = > 0 B i 1 n B i 1 n j=1 j=1 B i E(D 1 ) D j D j m X LP ij with probability one as n by the strong law of large numbers and Assumption A.1. We are now ready to prove that the greedy heuristic yields a solution that is asymptotically feasible and optimal with probability one when the retailer demands follow a common seasonality pattern. Theorem B.6 Under Assumption A.1, the greedy heuristic is asymptotically feasible and optimal with probability one when the retailer demands follow a common seasonality pattern. Proof: (LP) is non-degenerate with probability one (see Lemma B.4) and feasible with probability one when n by using Assumption A.1. From Lemma B.3, we then know that the number of assignments that differ between the optimal solution of the relaxation of (P) and the solution given by the greedy heuristic is bounded from above by a constant independent of n. Moreover, Lemma B.5 ensures us that the remaining capacity in the optimal solution for the relaxation of (A) grows linearly with n. Thus, when n grows to infinity, the solution found by the greedy heuristic is a feasible solution to (A). 6

7 It remains to be shown that the greedy solution is asymptotically optimal. It suffices to show that 1 n V LP n 1 n V G n 0 with probability one as n. Since the bounds on the cost parameters imply that the cost associated with any assignment will always lie in the interval [ C + pd, C + ( p + T h ) D ] it follows that 1 n V n LP 1 n V n G ( C C + ( p p + T h ) D ) N n n. The desired result then follows directly from Lemma B.3. The generalization of Theorem B.6 to the case of general demands is an open issue. Although Lemma B.3 applies to the general case, Assumption A.1 is crucial in our proof of Lemma B.5. The main obstacle to proving asymptotic feasibility (and optimality) for the general case is the fact that the capacity constraints can, unlike in the seasonal demand case, not be summarized in a single constraint for each facility. See also Romeijn and Romero Morales (2004). C. Acyclic Models In the paper, we have focused on tactical-level problems of evaluating the performance of a logistics network. This was represented by not fixing the start and ending period and the corresponding inventory levels, but instead considering a sequence of periods whose demand pattern can be expected to repeat over time. This was modeled by equating, but not fixing, initial and ending inventory levels at all facilities using constraints (4). In this appendix, we will outline the modifications to our model and algorithms that are required to capture problems in which a fixed planning horizon is considered, together with (possibly nonzero) initial inventory levels I i0 0 (i = 1,..., m) and target ending inventory levels I it 0 (i = 1,..., m). This would allow the model to be used in a rolling horizon context (see, e.g., Rohde and Wagner 2002, Fleischmann and Meyr 2003). With respect to the model, this means that constraints (4) are replaced by I i0 I i0 i = 1,..., m (4 ) I it I it i = 1,..., m. (4 ) Note that we impose the initial inventory levels as upper bounds rather than equality constraints: due to the presence of perishability constraints, requiring all initial inventory to 7

8 be used make the problem instance infeasible or could result in the assignment of retailers to a facility when this is not cost-effective. Therefore, we will allow discarding initial inventory. Note that, due to the nonnegativity of all costs, we may then assume without loss of optimality that the ending inventories are no larger than necessary, i.e., I it = I it for i = 1,..., m. Taking into account the finite planning horizon and target ending inventory level, the perishability constraints need to be modified as follows: I it I it t+k τ=t+1 j=1 τ=t+1 j=1 d jτ x ij i = 1,..., m; t = 0,..., T k (8 ) d jτ x ij + I it i = 1,..., m; t = T k + 1,..., T (8 ) where we assume that k T (otherwise these constraints are redundant). Note that feasibility of the problem instance also requires that the target inventory level is small enough to be able to produce that quantity without the goods perishing, which is simply a condition on the input data: I it t=t k+1 b it i = 1,..., m. In an analogous way to the proof of Theorem 2.1, we can show that the domain of H i for this model consists of all z R n + satisfying t=τ τ+r t=τ j=1 τ+r t=τ+k j=1 d jt z j d jt z j d jt z j + I it j=1 t=τ+k j=1 d jt z j + I it τ+r b it + I i,τ 1 τ = 1,..., T 1; r = 0,..., T τ 1 t=τ τ+r b it τ = 1,..., T k 1; r = k,..., T τ 1 t=τ b it + I i,τ 1 t=τ τ = 1,..., T b it τ = 1,..., T k. t=τ In the case of seasonal demands, the feasible region of the assignment formulation of the MPSSP is again that of a GAP with agent-independent requirements, with suitably redefined values of B i obtained using the above characterization of the domain of H i. Since in this case we know the ending inventory level, evaluating the objective function H i ( ) can now be done using a single backwards recursion in O(T ) time. 8

9 As for the greedy heuristic, the pseudo-cost function now becomes min(k,t 1) f(i, j) = c ij + λ it + ξit d jt l=1 α i,t l and it is not hard to extend the asymptotic optimality result of this heuristic to this variant of the MPSSP given our earlier results in this paper. Finally, the extension of the VLSN search improvement heuristic to this variant of the MPSSP is straightforward as well. D. Additional Results Table 6: Basic Case, General Demands, m = 5 greedy heuristic including VLSN search n feas. time error feas. time error iter. (sec.) (%) (sec.) (%) (4) Table 7: Physical Inventory Capacities, General Demands, m = 5, δ = 1.1, δ = 1.5 greedy heuristic including VLSN search n feas. time error feas. time error iter. (sec.) (%) (sec.) (%) (3)

10 Table 8: Basic Case, General Demands, m = 10 greedy heuristic including VLSN search n feas. time error feas. time error iter. (sec.) (%) (sec.) (%) (2) References Dyer, M., A. Frieze Probabilistic analysis of the generalised assignment problem. Mathematical Programming Fleischmann, B., H. Meyr Planning hierarchy, modeling and advanced planning systems. A.G. de Kok, S.C. Graves, eds., Supply Chain Management: Design, Coordination, Operation. Elsevier Science Publishers B.V., Amsterdam, The Netherlands Rohde, J., M. Wagner Master planning. H. Stadtler, C. Kilger, eds., Supply Chain Management and Advanced Planning. Springer, Berlin, Germany Romeijn, H.E., N. Piersma A probabilistic feasibility and value analysis of the generalized assignment problem. Journal of Combinatorial Optimization Romeijn, H.E., D. Romero Morales A class of greedy algorithms for the generalized assignment problem. Discrete Applied Mathematics Romeijn, H.E., D. Romero Morales A probabilistic analysis of the multi-period singlesourcing problem. Discrete Applied Mathematics Romeijn, H.E., D. Romero Morales An asymptotically optimal greedy heuristic for the multi-period single-sourcing problem: the cyclic case. Naval Research Logistics Romeijn, H.E., D. Romero Morales Asymptotic analysis of a greedy heuristic for the multi-period single-sourcing problem: the acyclic case. Journal of Heuristics

11 Trick, M.A A linear relaxation heuristic for the generalized assignment problem. Naval Research Logistics