Zuo-Jun Max Shen

A Joint Location-Inventory Model

We consider a joint location-inventory problem involving a single supplier and multiple retailers. Associated with each retailer is some variable demand. Due to this variability, some amount of safety stock must be maintained to achieve suitable service levels. However, risk-pooling benefits may be achieved by allowing some retailers to serve as distribution centers (and therefore inventory storage locations) for other retailers. The problem is to determine which retailers should serve as distribution centers and how to allocate the other retailers to the distribution centers. We formulate this problem as a nonlinear integer-programming model. We then restructure this model into a set-covering integer-programming model. The pricing problem that must be solved as part of the column generation algorithm for the set-covering model involves a nonlinear term in the retailerdistribution-center allocation terms. We show that this pricing problem can (theoretically) be solved efficiently, in general, and we show how to solve it practically in two important cases. We present computational results on several instances of sizes ranging from 33 to 150 retailers. In all cases, the lower bound from the linear-programming relaxation to the set-covering model gives the optimal solution.

AN INVENTORY-LOCATION MODEL: FORMULATION, SOLUTION ALGORITHM AND COMPUTATIONAL RESULTS

We introduce a distribution center (DC) location model that incorporates working inventory and safety stock inventory costs at the distribution centers. In addition, the model incorporates transport costs from the suppliers to the DCs that explicitly reflect economies of scale through the use of a fixed cost term. The model is formulated as a non-linear integer-programming problem. Model properties are outlined. A Lagrangian relaxation solution algorithm is proposed. By exploiting the structure of the problem we can find a low-order polynomial algorithm for the non-linear integer programming problem that must be solved in solving the Lagrangian relaxation subproblems. A number of heuristics are outlined for finding good feasible solutions. In addition, we describe two variable forcing rules that prove to be very effective at forcing candidate sites into and out of the solution. The algorithms are tested on problems with 88 and 150 retailers. Computation times are consistently below one minute and compare favorably with those of an earlier proposed set partitioning approach for this model (Shen, 2000; Shen, Coullard and Daskin, 2000). Finally, we discuss the sensitivity of the results to changes in key parameters including the fixed cost of placing orders. Significant reductions in these costs might be expected from e-commerce technologies. The model suggests that as these costs decrease it is optimal to locate additional facilities.

Channel Performance Under Consignment Contract with Revenue Sharing

Under a consignment contract with revenue sharing, a supplier decides on the retail price and delivery quantity for his product, and retains ownership of the goods; for each item sold, the retailer deducts a percentage from the selling price and remits the balance to the supplier. In this paper we show that, under such a contract, both the overall channel performance and the performance of individual firms depend critically on demand price elasticity and on the retailers share of channel cost. In particular, the (expected) channel profit loss, compared with that of a centralized system, increases with demand price elasticity and decreases with retailers cost share, while the profit share extracted by the retailer decreases with price elasticity and increases with retailers cost share. With an iso-price-elastic demand model, we show that the channel profit loss cannot exceed 26.4%, and that the retailers profit share cannot be below 50%. When price elasticity is low, or when the retailers cost share approaches 100%, or both, the retailer can extract nearly all the channel profit that is almost equal to the centralized channel profit.

Reliable Facility Location Design Under the Risk of Disruptions

Reliable facility location models consider unexpected failures with site-dependent probabilities, as well as possible customer reassignment. This paper proposes a compact mixed integer program (MIP) formulation and a continuum approximation (CA) model to study the reliable uncapacitated fixed charge location problem (RUFL) which seeks to minimize initial setup costs and expected transportation costs in normal and failure scenarios. The MIP determines the optimal facility locations as well as the optimal customer assignments, and the MIP is solved using a custom-designed Lagrangian Relaxation (LR) algorithm. The CA model predicts the total system cost without details about facility locations and customer assignments, and it provides a fast heuristic to find near-optimum solutions. Our computational results show that the LR algorithm is efficient for mid-sized RUFL problems and that the CA solutions are close to optimal in most of the test instances. For large-scale problems, the CA method is a good alternative to the LR algorithm that avoids prohibitively long running times.

Infrastructure Planning for Electric Vehicles with Battery Swapping

The transportation sector is a major source of greenhouse gas (GHG) emissions. As a step toward a greener environment, solutions involving electric vehicles (EVs) have been proposed and discussed. When powered by electricity from efficient and environmentally-friendly generators, EVs have significantly lower per-mile running costs compared to gasoline cars, while generating lower emissions. Unfortunately, due to the limited capacity of batteries, typical EVs can only travel for about 100 miles on a single charge. Because recharging takes several hours, it is impossible to recharge an EV in the middle of a long (round) trip exceeding 100 miles. Better Place (BP), a start-up based in Palo Alto, CA, proposed a novel strategy that potentially overcomes the recharging problem. In the plan, in addition to charging adaptors at homes, work places and shopping malls, swapping stations"", at which depleted batteries can be exchanged for recharged ones in the middle of long trips, will be located at strategic locations along freeways. With its battery swapping equipment, BP has demonstrated how to effectively refuel an EV in less than two minutes. The possible success of EV solutions based on the idea of battery swapping hinges on the ability of the charging service provider (BP or other similar firms) to deploy a cost-effective infrastructure network with comprehensive coverage. Unfortunately, since the adoption rate of electric vehicles, and thus demand for swapping service, is still highly uncertain, the service provider must make deployment plans with incomplete information on hand. In this paper, we develop models that aid the planning process for deploying battery swapping infrastructure, based on a robust optimization framework. We further show that our models can be tightly approximated by mixed-integer second-order cone programs (MISOCPs), which are readily solvable by commercial solvers. Using these models, we demonstrate the potential impacts of battery standardization and various technology advancements on the optimal infrastructure deployment strategy.

Effective Zero-Inventory-Ordering Policies for the Single-Warehouse Multiretailer Problem with Piecewise Linear Cost Structures

We analyze the problem faced by companies that rely on TL (Truckload) and LTL (Less than Truckload) carriers for the distribution of products across their supply chain. Our goal is to design simple inventory policies and transportation strategies to satisfy time varying demands over a finite horizon, while minimizing system wide cost by taking advantage of quantity discounts in the transportation cost structures. For this purpose, we study the cost effectiveness of restricting the inventory policies to the class of zero-inventory-ordering (ZIO) policies in a single-warehouse multiretailer scenario in which the warehouse serves as a cross-dock facility. In particular, we demonstrate that there exists a ZIO inventory policy whose total inventory and transportation cost is no more than 4/3 (5.6/4.6 if transportation costs are stationary) times the optimal cost. However, finding the best ZIO policy is an NP hard problem as well. Thus, we propose two algorithms to find an effective ZIO policy: An exact algorithm whose running time is polynomial for any fixed number of retailers, and a linear-programming-based heuristic whose effectiveness is demonstrated in a series of computational experiments. Finally, we extend the worst-case results developed in this paper to systems in which the warehouse does hold inventory.

Dynamic Assortment Optimization with a Multinomial Logit Choice Model and Capacity Constraint

We consider an assortment optimization problem where a retailer chooses an assortment of products that maximizes the profit subject to a capacity constraint. The demand is represented by a multinomial logit choice model. We consider both the static and dynamic optimization problems. In the static problem, we assume that the parameters of the logit model are known in advance; we then develop a simple algorithm for computing a profit-maximizing assortment based on the geometry of lines in the plane and derive structural properties of the optimal assortment. For the dynamic problem, the parameters of the logit model are unknown and must be estimated from data. By exploiting the structural properties found for the static problem, we develop an adaptive policy that learns the unknown parameters from past data and at the same time optimizes the profit. Numerical experiments based on sales data from an online retailer indicate that our policy performs well.

Trade-offs Between Customer Service and Cost in Integrated Supply Chain Design

When designing supply chains, firms are often faced with the competing demands of improved customer service and reduced cost. We extend a cost-based location-inventory model (Shen et al. 2003) to include a customer service element and develop practical methods for quick and meaningful evaluation of cost/service trade-offs. Service is measured by the fraction of all demands that are located within an exogenously specified distance of the assigned distribution center. The nonlinear model simultaneously determines distribution center locations and the assignment of demand nodes to distribution centers to optimize the cost and service objectives. We use a weighting method to find all supported points on the trade-off curve. We also propose a heuristic solution approach based on genetic algorithms that can generate optimal or close-to-optimal solutions in a much shorter time compared to the weighting method. Our results suggest that significant service improvements can be achieved relative to the minimum cost solution at a relatively small incremental cost.

A multi-commodity supply chain design problem

We consider a multi-commodity supply chain design problem in which we need to determine where to locate facilities and how to allocate customers to facilities so as to minimize total costs. The cost associated with each facility exhibits economies of scale. We show that this problem can be formulated as a nonlinear integer program and propose a Lagrangian-relaxation solution algorithm. By exploiting the structure of the problem, we find a low-order polynomial algorithm for the nonlinear integer program that must be solved in solving the Lagrangian relaxation subproblems. We also compare our approach with an existing algorithm. Contributed by the Location and Transportation Modeling Department

The Effect of Supply Disruptions on Supply Chain Design Decisions

We study an integrated supply chain design problem that determines the locations of retailers and the assignments of customers to retailers to minimize the expected costs of location, transportation, and inventory. The system is subject to random supply disruptions that may occur at either the supplier or the retailers. Analytical and numerical studies reveal the effects of these disruptions on retailer locations and customer allocations. In addition, we demonstrate numerically that the cost savings from considering supply disruptions at the supply chain design phase (rather than at the tactical or operational phase) are usually significant.