Electronic Companion to Initial Shipment Decisions for New Products at Zara

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1 e-companion to Gallien et al: Initial Shipment Decisions for New Products at Zara ec1 Electronic Companion to Initial Shipment Decisions for New Products at Zara This document includes appendices to Initial Shipment Decisions for New Products at Zara References herein point to that document, and we use notation specified there EC1 Modeling Dependence Among Stores An assumption underlying our dynamic optimization formulation in 33 is that demands and forecast errors are statistically independent across stores Here we extend the formulation to allow for positive correlation of an article s demand across stores We first extend our model of demand and learning by introducing a clustering G of articles, where each element g G specifies a probability mass function Pr{D OB = d g} over observation period sales at each store As an example, we might let G = { hot seller, cold seller, mixed seller }, where the hot seller cluster indicates optimistically biased pmfs for D OB across stores, the cold seller indicates pessimistically biased pmfs, etc When an article is introduced, we assume that nature draws a g from G, and demands at the stores are assumed to be drawn independently of each other, conditional on g The identity of g is assumed to be revealed to the decision maker at the end of the observation period, before second shipment decisions are made Given this setup, the second period forecasts Pr{D Pr(k g) = LT +2 k } are constructed the same as before, but the probabilities d supp D OB of the partitions k become conditional on g Pr{k D OB = d} Pr{D OB = d g} We have not tried to estimate this model from Zara data, but a basic estimation scheme follows A basic item clustering could separate historical articles into three clusters given suitable a and b: hot sellers : articles for which at least a% of stores exceeded demand forecasts, cold sellers : articles for which at least b% of stores fell below demand forecasts, mixed sellers : items that fall in neither of the above two categories

2 ec2 e-companion to Gallien et al: Initial Shipment Decisions for New Products at Zara Our model requires estimation of the following, which could potentially be obtained by examination of the historical data in each cluster: For each g G, we require the probability Pr(g) of a new article falling into cluster g For each g G, we require the conditional pmf of observation period demand (ie, Pr{D OB Given the A/F ratio approach we use to generate the distributional forecasts of observation period demand, it is natural to define these conditional pmfs at the A/F ratio level Our dynamic optimization formulation in 33 relies on store independence mainly through its use of Lagrangian relaxation When demands are independent across stores, stocking and replenishment at different stores are linked only through the constraint that we cannot ship more units than we have available When correlations are introduced, however, stores become linked both through the constraint and through statistical dependence We can still make use of Lagrangian relaxation if we model correlation among stores using the clustering G When evaluating costs in the first period it is sufficient to consider expected costs, thus it is unnecessary to condition on g in the first period We do, however, need to condition on g when computing the cost-to-go: g}) V (w 2, v 2, k) = g G Pr(g)V g (w 2, v 2, k), where V g (w 2, v 2, k) = min q 2 : E second period inventory cost g q2 w2 There is now one constraint q2 w 2 for each cluster g We can apply a Lagrangian relaxation to each such constraint, with λ g the Lagrange multiplier corresponding to cluster g The optimization formulation (8) then requires the following modifications Whereas the previous formulation had only a variable V representing the expected secondperiod cost for store, now we require a variable V g, for each store and cluster The expectations in constraints 8 must be conditioned on g, and the variables b 2,d, y,d and z,d must be indexed by g: b 2 g,,d, y g,,d, z g,,d In the obective function, the term λw 2 cost-to-go is Pr(g)V g G g, must be replaced by g Pr(g)λ gw 2 The expected

3 e-companion to Gallien et al: Initial Shipment Decisions for New Products at Zara ec3 These modifications have a moderate impact on the difficulty of the mixed integer program They add a number of additional variables and constraints numbering linearly in G Most notable is an increase in the number of binary variables b 2 g,,d However, the solution heuristic of 34 could potentially be applied to this formulation with only minor modifications to the heuristic Where the modification does bring additional computational complexity is in the criteria for choosing the multipliers λ g A subgradient minimization approach could be employed for choosing suitable λ g s, though this would likely add additional iterations to the solution procedure While we believe the modeling of dependence among stores may merit future consideration, Zara elected not to implement it as part of the current study because of the additional estimation and computational complexity it brings EC2 Technical Development of Solution Heuristic In this appendix we provide some technical background and analysis of the greedy solution heuristic described in 34 EC21 Procedure for Computing P λ (q 1 ) We first describe an efficient procedure, referenced in 34, for computing P λ (q 1 ) By fixing q 1, several of the decision variables in problem (12) are determined Specifically, u,d = max{0, d q 1 } for all d supp D 1, v +,d = max{0, q1 d} for all d supp D OB, b 1 = 1l{q 1 > 0} It remains to choose optimal values for y,d, z,d, and b 2,d for each J and d supp D OB Recalling formulation P λ and considering v +,d and b1 as fixed, these variables must solve for each J and d supp D OB, min (h 2 λ)e b 2,d (c 2 + λ)y,d + (c 2 + h 2 )z,d st v +,d y,d b 2,d(M e ) z,d α k,l y,d + β k,l, l L k (d), b 2,d b 1

4 ec4 e-companion to Gallien et al: Initial Shipment Decisions for New Products at Zara We solve this math program by branching on b 2,d Setting b 2,d = 0 (which is only feasible if b 1 = 0 and v +,d = 0) implies y,d = v +,d = 0 and z,d = max l Lk (d), β k (d),l Setting b 2,d = 1, the decision variables y,d and z,d solve the linear program: min (c 2 + λ)y,d + (c 2 + h 2 )z,d st v +,d y,d M e z,d α k,l y,d + β k,l, l L k (d), An optimal solution (y,d, z,d ) must occur at an extreme point of this linear program, either at y,d = v +,d, at y,d = M e, or at one of the kinks of max l Lk (d), αk (d),ly + β k (d),l Let Y,d indicate the values of y at these kinks, and let Y,d = Y,d {v +,d, M e } We can thus search for the optimal choice of y,d by a finite enumeration of L k (d), + 2 values Formally, the procedure for setting b 2,d, y,d, and z,d is as follows Define z d (y) = max αk (d),ly + β k (d),l l L k (d), 1 if b 1 b 2 = 1 and/or y > 0 (y) = 0 otherwise For all d supp D OB, we set y,d = arg min y Y,d (h 2 λ)e b 2 (y) (c 2 + λ)y + (c 2 + h 2 )z d (y), z,d = z d (y,d ), and b 2,d = b 2 (y,d ) EC22 Characterizing the Heuristic s Traectory Here we characterize the traectory followed by the heuristic by proving several lemmas, the last of ( ) which will be useful in our asymptotic analysis Define Γ n Γ (n), ˆq n, (n) ˆqn+1 (n) as the minimizing ratio in step n of the heuristic Let N indicate the terminating step in the heuristic; it follows that the ratio Γ(, ˆq N, q ) is nonnegative for all feasible (, q ) The first lemma shows that {Γ n } is a non-decreasing sequence in n Lemma EC1 For 2 n N 1, Γ n Γ n 1

5 e-companion to Gallien et al: Initial Shipment Decisions for New Products at Zara ec5 Proof Fix n At step n 1, the heuristic only modifies the store (n 1) that minimizes Γ(, ˆq n 1, q ) In step n, the set of feasible steps at stores (n 1) will be the same or reduced Therefore the lemma holds when (n) (n 1) Now consider the case (n) = (n 1) = and suppose Γ n < Γ n 1 By the definitions of Γ n and Γ n 1, P λ (ˆq n ) P λ (ˆq n 1 ) = Γ n 1 X (ˆq n ) X (ˆq n 1 ) (EC1) P λ (ˆq n+1 ) P λ (ˆq n ) = Γ n X (ˆq n+1 ) X (ˆq n ) (EC2) We also have Γ n X (ˆq n+1 ) X (ˆq n ) < Γ n 1 X (ˆq n+1 ) X (ˆq n ), (EC3) because X (ˆq n+1 ) X (ˆq n ) is positive and Γ n < Γ n 1 by assumption Adding (EC1), (EC2), and (EC3) yields P λ (ˆq n+1 ) P λ (ˆq n 1 ) < Γ n 1 X (ˆq n+1 ) X (ˆq n 1 ), which is equivalent to P λ (ˆq n+1 ) P λ (ˆq n 1 ) X (q n+1 ) X (q n 1 ) < Γn 1 (EC4) Inequality (EC4) is a contradiction because it implies that the heuristic should have moved from ˆq n 1 to ˆq n+1 Γ n Γ n 1 in step n 1, thereby skipping ˆq n Therefore if (n) = (n 1) then we must have A second lemma bounds the Γ-ratio between any step of the heuristic and any possible smaller shipment quantity to the same store Lemma EC2 P λ (ˆqn ) P λ (q ) X (ˆq n ) X (q ) Γn 1 for all J, n N, q {q : q Q, q < ˆq n } Proof Fix a q {q : q Q, q < ˆq n } and consider two cases: (1) in which q is on the heuristic s path, and (2) in which q is not on the heuristic s path First, suppose that q is on the heuristic s path so that q = ˆq n for some n < n Because ˆq n and ˆq n are on the heuristic s path, the heuristic moved from ˆq n to ˆq n in a sequence of steps in which

6 ec6 e-companion to Gallien et al: Initial Shipment Decisions for New Products at Zara either there was no incremental commitment to store or else an incremental commitment with Γ-ratio less than or equal to Γ n 1, by Lemma EC1 This implies P λ (ˆq n +1 ) P λ (ˆq n ) Γ n 1 X (ˆq n +1 ) X (ˆq n ) P λ (ˆq n +2 ) P λ (ˆq n +1 ) Γ n 1 X (ˆq n +2 ) X (ˆq n +1 ) P λ (ˆq n ) P λ (ˆq n 1 ) Γ n 1 X (ˆq n ) X (ˆq n 1 ) Adding these inequalities yields P λ (ˆq n ) P λ (ˆq n ) Γ n 1 X (ˆq n ) X (ˆq n ), which is equivalent to the desired result Second, suppose q is not on the heuristic s path We prove the desired result by contradiction Suppose P λ (ˆq n ) P λ (q ) X (ˆq n ) X (q ) > Γn 1 (EC5) and let ˆq m 1 and ˆq m be the points on the heuristic s path immediately below and above q so that ˆq m 1 < q < ˆq m ˆq n Define P P λ (ˆq m ) P λ (ˆq m ) P λ (ˆq m 1 X (ˆq m ) X (ˆq m 1 ) ) X (ˆq m ) X (q ), then P λ (ˆq m ) P X (ˆq m ) X (q ) = P λ (ˆq m ) P λ (ˆq m 1 ) X (ˆq m ) X (ˆq m 1 ) Γn 1, where the inequality follows from Lemma EC1 This implies P λ (ˆq m ) P Γ n 1 X (ˆq m ) X (q ) (EC6) The first case considered in this proof implies P λ (ˆq n ) P λ (ˆq m ) Γ n 1 X (ˆq n ) X (ˆq m ) (EC7)

7 e-companion to Gallien et al: Initial Shipment Decisions for New Products at Zara ec7 Adding (EC6) and (EC7), we have P λ (ˆq n ) P Γ n 1 X (ˆq n ) X (q ) Combining this with our original assumption (EC5) yields P λ (ˆq n ) P < P λ (ˆq n ) P λ (q ) P λ (q ) < P P λ (q ) P λ (ˆq m 1 ) X (q ) X (ˆq m 1 ) < P P λ (ˆq m 1 ) X (q ) X (ˆq m 1 ) = P λ (ˆq m ) P λ (ˆq m 1 ) X (ˆq m ) X (ˆq m 1 ), where the final equality follows from the definition of P This implies that the heuristic would have stepped from ˆq m 1 to q rather than to ˆq m, which is a contradiction Define N as the index of the last shipment commitments before the heuristic is impacted by the constraint X (q ) W 1 l X l(ˆq n l ) That is, N is the largest n for which min,q Q Γ(, ˆq n 1, q ) = min,q Q Γ(, ˆq n 1, q ) st X (q ) W 1 X l l(ˆq n 1 l ) st q > q n 1 q > q n 1 Also, for convenience let ˆP λ = P λ λ(w 1 + W 2 ) c 2 Pr(k k K )E ˆDLT +1 k (That is, problem ˆP λ is problem P λ without the constant leading term) A third lemma proves that {ˆq N } solves a particular Lagrangian relaxation of problem ˆP λ : P λ,µ : min P λ (q ) + µ W 1 q:q Q = µw 1 + min P λ (q 1 ) µx (q 1 ) q Q Lemma EC3 The vector {q N } solves P λ,µ for µ = Γ N 1 X (q ) Proof Consider the Lagrangian subproblem min P λ (q ) Γ N 1 X (q ) q Q

8 ec8 e-companion to Gallien et al: Initial Shipment Decisions for New Products at Zara and observe that moving from q to any feasible q + for any q + q yields an obective reduction if and only if P λ (q + ) Γ N 1 X (q + ) P λ (q ) Γ N 1 X (q ), or equivalently, P λ (q + ) P λ (q ) X (q + ) X (q ) 1 ΓN Therefore optimality of {ˆq N } in P λ,γn 1 holding: (A) P λ (ˆqN ) P λ (q ) ) X (q ) ΓN 1 for all J, q {q : q Q, q < ˆq N X (ˆq N (B) P λ (q ) P λ (ˆqN ) X (q ) X (ˆq N ) ΓN 1 for all J, q {q : q Q, q > ˆq N is equivalent to both of the following two conditions Condition (A) holds because of Lemma EC2 Condition (B) is true because, if not, the heuristic would have attempted to step to such a q in step N 1, thus contradicting the definition of {q N } as the last set of shipment commitments not impacted by the warehouse inventory constraint The following lemma shows that the heuristic is optimal when the constraint (10) does not impact the heuristic solution Let ˆP λ X (ˆq N ) denote problem ˆP λ with the value W 1 replaced by X (ˆq N ) Lemma EC4 } } N {ˆq } is an optimal solution to problem ˆP λ X (ˆq N ) Proof We restate the lemma as ˆP λ X (ˆq N ) P λ (ˆq N and so P λ (ˆq N ) ˆP ( λ X (ˆq N ) ) = ˆP λ X (ˆq N ) ˆq N ) is feasible in To prove the converse, consider the Lagrangian problem P λ,µ with W 1 replaced by X (ˆq N ) Observe that {q N } remains the last shipment commitments unimpeded by the constraint for this modified problem Lemma EC3 implies that {ˆq N } is optimal for this problem with µ = Γ N 1 Therefore we have ˆP λ ( X (ˆq N ) ) P λ (ˆq N ) + Γ N 1 X (ˆq N ) X (ˆq N ) = P λ (ˆq N ), where the first inequality follows because the optimal obective of the Lagrangian relaxation lower bounds ˆP λ X (ˆq N )

9 e-companion to Gallien et al: Initial Shipment Decisions for New Products at Zara ec9 EC23 Proof of Asymptotic Optimality In this section we prove that the heuristic solution is asymptotically optimal in the problem P λ as the number of stores J is scaled up For the purpose of this section, we assume that the Lagrange multiplers λ are chosen to maximize the Lagrangian lower bound We also define a constant P that upper bounds the difference max λ c 2,0,q 1 Q P λ (q 1 ) min λ c 2,0,q 1 Q P λ (q 1 ) It is straightforward to show that such a constant must exist since q 1 is bounded and the decision variables y,d and z,d in P λ (q 1 ) can be bounded using solutions to well-defined newsvendor problems with positive overage cost The following lemma says that P upper bounds the heuristic s optimality gap Let ˆP λ (W 1 ) indicate the optimal cost of problem ˆP λ and let ˆQ λ indicate the cost of the heuristic solution in problem ˆP λ Lemma EC5 For fixed λ c 2, 0, ˆQ λ ˆP λ P Proof Consider a modified version of the heuristic which allows one step beyond {q N }, namely ˆq N +1 = arg min J,q Q Γ(, ˆq N, q ) st q q N without requiring feasibility in the constraint X (q ) W 1 X l l(q l ) This solution {q N +1 } may be infeasible in the original problem, but by Lemma EC4, it is optimal in problem ˆP λ X (ˆq N +1 ) We next argue that ˆP λ ( X (ˆq N ) ) ˆQ λ (W 1 ) ˆP λ (W 1 ) ˆP λ ( X (ˆq N +1 ) ) The first inequality follows because (by Lemma EC4) the heuristic is optimal for problem ˆP λ X (ˆq N ) and because expanding the capacity from X (ˆq N ) to W 1 may permit the heuristic to take one or more cost-reducing steps The second inequality follows follows because ˆQ λ (W 1 ) is a heuristic solution to a problem for which ˆP λ (W 1 ) is optimal The third inequality

10 ec10 e-companion to Gallien et al: Initial Shipment Decisions for New Products at Zara { } ( follows because, by Lemma EC4, ˆq N +1 ) is an optimal solution to problem P λ X (ˆq N +1 ), which can be viewed as a relaxation of problem P λ (W 1 ) ( λ ) The difference between ˆP X (ˆq N ) = P λ (ˆq N ) and ˆP λ X (ˆq N +1 ) P λ (ˆq N +1 ) is at most one obective-reducing shipment to a single store, which is bounded by the potential cost at a store Therefore P λ (ˆq N ) P λ (ˆq N +1 ) P, which implies the lemma Because ˆP λ and P λ differ only by a constant, Lemma EC5 implies that the optimality gap of the heuristic in the problem P λ is also bounded by P In the remainder of this section we consider a sequence of instances of problem P λ indexed by M Instance M has J = M stores, and we permit initial inventory W 1 = W 1 M to vary with M in an arbitrary way Denote the optimal cost for problem M using multiplier λ = λm c 2, 0 by P λ M and denote the cost corresponding to the final heuristic solution as Q λ M Since the heuristic always generates a feasible solution to P λ, we naturally have Q λ M P λ M for any λ = Lemma EC6 Assuming the condition of Theorem 1, P λm M MV for all M Proof Because we are assuming that λm is chosen as λm = arg max λ P λ M, it follows that P λm M P λ=0 M Therefore, it is sufficient to show that P λ=0 M MV In the obective of P λ=0 M, observe that c 1 Pr(D 1 = d)u,d 0, h 2 d supp D 1 d supp D OB λw 2 = 0, Pr(D OB = d)b 2,de 0, therefore P λ=0 M is lower-bounded by the term V, minimized subect to the other constraints in P λ=0 M Consider a relaxation that includes only the constraints 8 and 11; P λ=0 M is

11 e-companion to Gallien et al: Initial Shipment Decisions for New Products at Zara ec11 lower-bounded by P λ=0 M : min V st 8 V = c 2 k K Pr(k )E + d supp D OB Pr(D OB ˆD LT +2 k = d) c 2 y,d + (c 2 + h 2 )z,d J 11 z,d α k (d),ly,d + β k (d),l J, d supp D OB, l L k (d), V, y 0 J This problem decomposes by store The constraints 8 and 11 implement the 2nd-period storelevel newsvendor problem exactly, so P λ=0 M is equivalent by construction (see equation (5)) to E k min E c 2 LT +2 ( ˆD y ) + + h 2 LT +2 (y ˆD ) + k y 0 By the theorem condition, this is greater than or equal to MV, which completes the proof Theorem 1 of 34 follows from Lemmas EC5 and EC6, since Q λm M P λm M P lim lim M P λm M M MV = 0