Multi-period price promotions in a single-supplier, multi-retailer supply chain under asymmetric demand information

Abstract

This paper considers a two-stage supply chain in which a supplier serves a set of stores in a retail chain. We consider a two-stage Stackelberg game in which the supplier must set price discounts for each period of a finite planning horizon under uncertainty in retail-store demand. As a mechanism to stimulate sales, the supplier offers periodic off-invoice price discounts to the retail chain. Based on the price discounts offered by the supplier, and after store demand uncertainty is resolved, the retail chain determines individual store order quantities in each period. Because the supplier offers store-specific prices, the retailer may ship inventory between stores, a practice known as diverting. We demonstrate that, despite the resulting bullwhip effect and associated costs, a carefully designed price promotion scheme can improve the supplier’s profit when compared to the case of everyday low pricing (EDLP). We model this problem as a stochastic bilevel optimization problem with a bilinear objective at each level and with linear constraints. We provide an exact solution method based on a Reformulation-Linearization Technique (RLT). In addition, we compare our solution approach with a widely used heuristic and another exact solution method developed by Al-Khayyal (Eur. J. Oper. Res. 60(3):306–314, 1992) in order to benchmark its quality.

Appendix

1.1 A.1 Decomposition of uncapacitated minimum cost flow problem

This section provides a proof that an uncapacitated minimum cost flow problem can be decomposed into a set of shortest path problems.

Theorem 1

A single-origin, N-destination uncapacitated minimum cost flow problem can be decomposed into N shortest path problems that do not depend on the demand level.

Proof

Assume without loss generality that we have a directed network G=(V,E), where V={0,1,…,N} is the set of nodes and E is the set of arcs. Each arc (i,j)∈E has an associated cost c _ij that denotes the cost per unit flow on that arc. We also assume that each arc has an infinite capacity, i.e., there is no upper bound on the maximum amount that can flow on any arc. The network has a unique node 0, called the source, with supply $\sum_{i=1}^{N} d_{i}$. For each nonsource node i∈V, we associate with it a demand level d _i (≥0). The single-origin N-destination uncapacitated minimum cost flow problem is to determine a minimum cost flow through the uncapacitated network in order to satisfy demands at the N nonsource nodes from available supplies at the source node 0. The decision variables in this problem are arc flows, and we represent the flow on an arc (i,j)∈E by x _ij . The single-origin N-destination uncapacitated minimum cost flow problem can be formulated as follows:

$$\begin{aligned} &(\mathit{UMCF}) \quad \min\ \sum _{(i,j)\in E} c_{ij}x_{ij} \\ &\hphantom{(\mathit{UMCF}) \quad} \mbox{s.t.}\quad \sum_{j:(0,j)\in E} x_{0j} - \sum_{j:(j,0)\in E} x_{j0} =\sum _{i=1}^N d_i \\ &\hphantom{\hphantom{(\mathit{UMCF}) \quad} \mbox{s.t.}\quad} \sum_{j:(i,j)\in E} x_{ij} - \sum _{j:(j,i)\in E} x_{ji} =-d_i, \quad \forall i \in \{1,\ldots,N\} \\ &\hphantom{\hphantom{(\mathit{UMCF}) \quad} \mbox{s.t.}\quad} x_{ij} \geq 0 \quad \forall (i,j) \in E. \end{aligned}$$

The flow decomposition theorem (Ahuja et al. 1993) implies that the flow x _ij on each arc (i,j)∈E can be decomposed by destination, with $x_{ij}^{l}$ representing of flow on arc (i,j) used to satisfy demand at node l. With this notation, we can reformulate the single-origin N-destination uncapacitated minimum cost flow problem as follows:

$$\begin{aligned} &\bigl(\mathit{UMCF}'\bigr)\quad \min\ \sum _{l=1}^N\sum_{(i,j)\in E} c_{ij}x_{ij}^l \\ &\hphantom{\bigl(\mathit{UMCF}'\bigr)\quad} \mbox{s.t.}\quad \sum_{j:(0,j)\in E} x_{0j}^l - \sum_{j:(j,0)\in E} x_{j0}^l =d_l, \quad \forall l\in \{1,\ldots,N\} \\ &\hphantom{\bigl(\mathit{UMCF}'\bigr)\quad \mbox{s.t.} \quad} \sum_{j:(i,j)\in E} x_{ij}^l - \sum_{j:(j,i)\in E} x_{ji}^l = \begin{cases}0 & \text{if } i \neq l, \\ -d_l & \text{if } i = l, \end{cases} \quad \forall i,l \in \{1,\ldots,N\} \\ &\hphantom{\bigl(\mathit{UMCF}'\bigr)\quad \mbox{s.t.} \quad} x_{ij}^l \geq 0, \quad \forall (i,j) \in E, l \in \{1,\ldots,N\}. \end{aligned}$$

By definition, we have $x_{ij} = \sum_{l=1}^{L} x_{ij}^{l}$. Suppose we define variables $y_{ij}^{l}=\frac{x_{ij}^{l}}{d_{l}}$. Using these, the single-origin N-destination uncapacitated minimum cost flow problem can be expressed as follows:

$$\begin{aligned} &\bigl(\mathit{UMCF}''\bigr) \quad \min\ \sum_{l=1}^N d_l \sum_{(i,j)\in E} c_{ij}y_{ij}^l \\ &\hphantom{\bigl(\mathit{UMCF}''\bigr) \quad} \mbox{s.t.}\quad \sum_{j:(0,j)\in E} y_{0j}^l - \sum_{j:(j,0)\in E} y_{j0}^l =1, \quad \forall l\in \{1,\ldots,N\} \\ &\hphantom{\bigl(\mathit{UMCF}''\bigr) \quad \mbox{s.t.}\quad} \sum_{j:(i,j)\in E} y_{ij}^l - \sum_{j:(j,i)\in E} y_{ji}^l = \begin{cases}0, & \text{if } i \neq l, \\ -1, & \text{if } i = l, \end{cases}\quad \forall i,l \in \{1,\ldots,N\} \\ &\hphantom{\bigl(\mathit{UMCF}''\bigr) \quad \mbox{s.t.}\quad} y_{ij}^l \geq 0,\quad \forall (i,j) \in E, l \in \{1,\ldots,N\}. \end{aligned}$$

In this above formulation, the set of feasible solutions Y can be decomposed into N sets Y ¹,Y ²,…,Y ^N, where

$$Y^l \equiv \left\{y^l: \sum _{j:(i,j)\in E} y_{ij}^l - \sum _{j:(j,i)\in E} y_{ji}^l =\begin{cases}1 & \text{if } i =0,\\ 0 & \text{if } i \neq l, \\ -1 & \text{if } i = l, \end{cases} \quad \forall (i,j)\in E \right\}. $$

By observation, the variables for each subset do not appear in any other set. Consequently, we can decompose UMCP″ into N shortest path problems which do not depend on the demand level in the following formulation:

$$\begin{aligned} &\bigl(\mathit{SP}^l\bigr)\quad \min\ d_l \sum_{(i,j)\in E} c_{ij}y_{ij}^l = d_l \min \sum_{(i,j)\in E} c_{ij}y_{ij}^l \\ &\hphantom{\bigl(\mathit{SP}^l\bigr)\quad} \mbox{s.t.} \quad \sum_{j:(0,j)\in E} y_{0j}^l - \sum_{j:(j,0)\in E} y_{j0}^l =1 \\ &\hphantom{\bigl(\mathit{SP}^l\bigr)\quad \mbox{s.t.} \quad} \sum_{j:(i,j)\in E} y_{ij}^l - \sum_{j:(j,i)\in E} y_{ji}^l = \begin{cases} 0, & \text{if } i, \neq l \\ -1, & \text{if } i = l, \end{cases} \quad \forall i \in \{1,\ldots,N\} \\ &\hphantom{\bigl(\mathit{SP}^l\bigr)\quad \mbox{s.t.} \quad} y_{ij}^l \geq 0 \quad \forall (i,j) \in E. \end{aligned}$$

After solving all of these N shortest path problems, we can obtain the solutions to the original UMCP using the following relationship:

$$x_{ij}=\sum_{l=1}^L x_{ij}^l=\sum_{l=1}^L d_l y_{ij}^l \quad \forall (i,j)\in E. $$

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1.2 A.2 Detailed Al-Khayyal approach

The detailed Al-Khayyal branch-and-bound algorithm is shown in Algorithm 2. After initializing the branch-and-bound tree in line 1, the algorithm repeats the Main Step in the while loop in lines 2–29 until the optimality gap is less than a predetermined value ϵ. At the start of each iteration k of the while loop, a node (u,v) is selected and removed from tree T _k in line 3. In lines 4 and 5, the linear program LP(Ω ^(u,v)) is solved to obtain the solution $\bar{\varLambda}$ and the corresponding optimal value $\varPhi (\bar{\varLambda})$ is assigned to UB _k . In lines 6–16, the current iteration lower bound LB _k is obtained and the partitioning index (p,q) is found by checking the optimality and the feasibility of the solution $\bar{\varLambda}$. After finding (p,q), partition the region Ω ^(u,v) into two mutually exclusive and exhaustive subregions in lines 18–21. At the end of the while loop, the branch-and-bound tree is updated by three types of operations in lines 22–27. Included here are

• 1. update the lower bound of the problem if LB<LB _k as follows:

  $$\mathit{LB} \gets \mathit{LB}_k \text{ and } \bigl(x^*,y^*,z^*,\pi^*\bigr) \gets ( \bar{x},\bar{y},\bar{z},\bar{\pi}); $$

• 2. update the node set at iteration k as

  $$T_k \gets T_k -\bigl\{ (m,n)\in T_k: \mathit{UB}^{(m,n)} \leq \mathit{LB}\bigr\} ; $$

• 3. update the upper bound of the problem as

  $$\mathit{UB} =\max \bigl\{ \mathit{UB}^{(m,n)}, \forall (m,n) \in T_{k+1}\bigr\} . $$

Algorithm 2

Al-Khayyal’s Branch-and-Bound algorithm