Optimal resource allocation across related channels

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Abstract

In this paper, we consider resource allocation strategies of a limited resource across two related channels in a multi-period setting. We study a stochastic control problem where the objective is to determine the optimal limited resource allocation policy across two related channels and optimal transshipment policy between these two channels. We characterize some structural results of the optimal resource allocation policy and show that it is determined by three monotone curves.

Introduction

There are many scenarios where a rational agent (e.g. a firm, a public organization) allocates available resources to his different branches in order to complete the task or obtain social rewards. For example, in manufacturer industry, a centralized firm has to find an effective way to allocate the limited capacity to different regions every day to minimize the total cost where each region faces stochastic demand and can make some transshipment if necessary after demand realization. In retailing industry, as digital online channel is playing more and more important role in firms’ sales, many firms begin to consider so called “omni-channel” strategy, thus an omni-channel retailer has to figure out a way to configure appropriate allocation and transshipment strategies in the offline and online channels. In health care community, the hospital needs to allocate some common limited resources, such as equipments, doctors and nurses, to different departments every day, which also have demand uncertainty due to random patients’ arrivals. In this paper, we consider a resource allocation problem across two related channels in a multi-period setting, where demand from each channel during each period is uncertain and there exists a fixed available resource constraint during each period due to the possible capacity limit, transportation factor or budget constraint. The centralized rational agent has to make the decision of allocating the right quantity for both channels and possible transshipment after demand realization during each period.

One stream of literature related to ours is the resource allocation. One class is termed Network Revenue Management Problems (Topaloglu [13]) and dynamic fleet management (Godfrey and Powell [[5], [6]], Topaloglu and Powell [14]), where the resources correspond to transport vehicles and demands come from shipping request. However, the literatures above assume the resource is nonrenewable, which means it gradually decreases over the consumption in each period, and the key tradeoff in the decision process is to balance the usage between the current period and the future. In our model, we make the limited resource renewable at the beginning of each period, and two channels are related since they can transfer the available resources to each other to get more revenue if necessary. There is also a growing body of literature on capacity allocation in the field of operations management (e.g. [[8], [11], [15]]). In their typical setting, a monopolistic firm allocates available capacity to multiple downstream channels under certain mechanisms. However, they generally use a game theory model to study the impact of the competition among the downstream channels on the allocation decision during a single period, which is different from our multi-period formulation.

Another stream closely related to ours is the literature on capacitated multi-period inventory problem and lateral transshipment. In this context, the allocation decision to multi-channel corresponds to order decisions from the upstream supplier with a limited capacity.  Federgruen and Zipkin [[3], [4]] are the first to study an inventory system with capacity constraints and show that order-up-to level policies are optimal for the infinite horizon case. A multi-product version is analyzed by DeCroix and Arreola-Risa [2], who also demonstrate that a modified base-stock policy is optimal for infinite-horizon case and present some heuristics, however, they do not consider the second transshipment stage between two channels. The papers listed above study single product or multi-product inventory system but just one location case. In multi-location stochastic inventory systems, ordering decisions and post-transshipment are considered (e.g. Axsäter [1], Paterson et al. [9], Ramakrishna et al. [10]). Hu et al. [7] characterize some structural results of the optimal replenishment policy and lateral transshipment policies under random supply capacity. Actually, the transshipment stage after demand realization in our model is exactly the same as the paper of Hu et al. [7], but the first resource allocation stage of our model is totally different from theirs. In their setting, each location has an independent supplier with uncertain capacity, however, our paper assumes one common limited resource for two channels. In this paper, we focus on the resource allocation stage and show that some properties of the value-to-go function and its preservation can still hold in our setting. The most closely related to our paper is Shaoxiang [12], who develops a notion called μ-difference monotone and proves that the hedging point policy is optimal. The structural results given in this paper can also be described as a simple priority rule (hedging point policy) given by Shaoxiang [12], but the proof technique is totally different. We provide a more concise and natural way to prove the preservation of function properties.

The purpose of this paper is to apply some existing results in Hu et al. [7] to tackle the multi-period resource allocation problem with lateral transshipment described above, where the limited resource is renewable and it can cover many different applications such as manufacturer, retailing and health care industry in different context. Our model is set in a two-stage dynamic programming framework and despite its technical difficulty, we can still use a concise way and some existing results to characterize some structural results of the optimal resource allocation policy.

Throughout this paper, we use decreasing, increasing, and monotonicity in a weak sense. The boldface small letter is denoted as the vector unless indicated.

Section snippets

Model

We consider an allocation decision of limited resources to two channels over a finite planning horizon of length T. Let i=1,2 denote each channel, respectively. Each channel faces uncertain demand D and the stochastic demand distributions are independent in each period and channel with expectation d0(E[D]=d0). At the beginning of each period t, each channel has some inventory of resources x1,x2 on hand (here we omit the time index t in our model formulation for ease of exposition), the decision

Structure of the optimal resource allocation policy

In this section, we characterize the structure of the optimal resource allocation policy for the model given in (1)–(3). As the initial step of induction, we first assume Gt+1(x1,x2) has some properties as follows

Assumption 1

Gt+1(x1,x2) is jointly concave in (x1,x2) and submodular, and (Gt+1)ii(x1,x2)(Gt+1)12(x1,x2)=(Gt+1)21(x1,x2)0 for i,j=1,2.

Here we show that the preservation of the properties can still hold when one common limited resource is

Future research

There are several directions for future research on multi-period renewable resource allocation problem. For example, we do not consider the possible uncertainty of the limited resource level, which is common in practice subject to transportation risk, limited budget risk, deterioration risk, etc. Another important topic is to consider the price decision for both channels, whereas here we assume the prices of both channels are exogenous. Finally, we consider a centralized decision maker that

Acknowledgments

The authors sincerely thank the associate editor and anonymous referees for their helpful comments and suggestions that helped to significantly improve the paper. This work was supported in part by National Natural Science Foundation of China (Nos. 71632008, 71371123, 71702106). This support is gratefully acknowledged.

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