gang Yu

Stackelberg game-theoretic model for optimizing advertising, pricing and inventory policies in vendor managed inventory (VMI) production supply chains

This paper discusses how a manufacturer and its retailers interact with each other in order to optimize their individual net profits by adjusting product marketing (advertising and pricing) and inventory policies in an information-asymmetric VMI (vendor managed inventory) supply chain. The manufacturer produces and supplies a single product at the same wholesale price to multiple retailers who then sell the product in dispersed and independent markets at retail prices. The demand rate in each market is an increasing and concave function of the advertising investments of both local retailers and the manufacturer, but a decreasing and convex function of the retail prices. The manufacturer determines its wholesale price, its advertising investment, replenishment cycles for the raw materials and finished product, and backorder quantity to maximize its profit. Retailers in turn consider the replenishment policies and the manufacturers promotion policies and determine the optimal retail prices and advertisement investments to maximize their profits. This problem is modeled as a Stackelberg game where the manufacturer is the leader and retailers are followers. An algorithm has been proposed to search the Stackelberg equilibrium. A numerical study has been conducted to demonstrate how the algorithm works and to understand the influences of decision variables and/or parameters. Several research questions are examined, including under what circumstances the retailers and manufacturer should increase their advertising expenditures and/or reduce the retail prices and what actions should be taken if the prices of raw materials or their holding costs increase.

A new model and hybrid approach for large scale inventory routing problems

This paper studies an inventory routing problem (IRP) with split delivery and vehicle fleet size constraint. Due to the complexity of the IRP, it is very difficult to develop an exact algorithm that can solve large scale problems in a reasonable computation time. As an alternative, an approximate approach that can quickly and near-optimally solve the problem is developed based on an approximate model of the problem and Lagrangian relaxation. In the approach, the model is solved by using a Lagrangian relaxation method in which the relaxed problem is decomposed into an inventory problem and a routing problem that are solved by a linear programming algorithm and a minimum cost flow algorithm, respectively, and the dual problem is solved by using the surrogate subgradient method. The solution of the model obtained by the Lagrangian relaxation method is used to construct a near-optimal solution of the IRP by solving a series of assignment problems. Numerical experiments show that the proposed hybrid approach can find a high quality near-optimal solution for the IRP with up to 200 customers in a reasonable computation time.

Designing an optimal turnover-based storage rack for a 3D compact automated storage and retrieval system

Compact, multi-deep (3D) automated storage and retrieval systems (AS/RS) are becoming increasingly popular for storing products. We study such a system where a storage and retrieval (S/R) machine takes care of movements in the horizontal and vertical directions of the rack, and an orthogonal conveying mechanism takes care of the depth movement. An important question is how to layout such systems under different storage policies to minimize the expected cycle time. We derive the expected single-command cycle time under the full-turnover-based storage policy and propose a model to determine the optimal rack dimensions by minimizing this cycle time. We simplify the model, and analytically determine optimal rack dimensions for any given rack capacity and ABC curve skewness. A significant cycle time reduction can be obtained compared with the random storage policy. We illustrate the findings of the study by applying them in a practical example.

Scheduling Twin Yard Cranes in a Container Block

Annually, millions of containers enter and exit the stacking area of a terminal. If the stacking operations are not efficient, long ship, train, and truck delays will result. To improve the stacking operations, new container terminals, especially in Europe, decouple the landside and seaside by deploying twin automated stacking cranes. The cranes cannot pass each other and must be separated by a safety distance. We study how to schedule twin automated cranes to carry out a set of container storage and retrieval requests in a single block of a yard. Storage containers are initially located at the seaside and landside input/output I/O points of the block. Each must be stacked in a specific location of the block, selected from a set of open locations suitable for stacking the storage container. Retrieval containers are initially located in the block and must be delivered to the I/O points. Based on the importance and acceptable waiting times of different modes of transport, requests have different priorities. The problem is modeled as a multiple asymmetric generalized traveling salesman problem with precedence constraints. The objective is to minimize the makespan. We have developed an adaptive large neighborhood search heuristic to quickly compute near-optimal solutions. We have performed extensive computational experiments to assess the performance of the heuristic including validation at a real terminal. It obtains near-optimal solutions for small instances. For large instances, it is shown to yield better solutions than CPLEX truncated after four hours, and it outperforms other heuristics from practice by more than 24% in terms of makespan. The average gaps between our heuristic and optimal solutions for relaxed problems are less than 3%.

Newsvendor Model for a Dyadic Supply Chain with Nash Bargaining Fairness Concerns

The paper investigates newsvendor problem for a dyadic supply chain in which both the supplier and the retailer have the preference of status-seeking with fairness concerns. Nash bargaining solution is introduced as the fairness reference point and equilibrium results are derived. The effects of fairness-concerned status-seeking behaviors on optimal decisions as well as channel efficiency are further analyzed. It is shown that the channel efficiency will decrease because of such behavioral preference. The retailer’s share will be larger when the supplier concerns fairness less, and the supplier’s sensitivity to fairness plays a relatively more important role for the channel efficiency. Additionally, another interesting managerial insight is concluded that fairness concerns will not change the status of channel coordination in certain conditions. More specifically, those contracts able (unable) to coordinate fairness-neutral supply chain, based on affine transformations with scale factors within certain ranges, still succeed (fail) to coordinate the fairness-concerned. Furthermore, several insights on bargaining powers are given as well.

An exact method for scheduling a yard crane

This paper studies an operational problem arising at a container terminal, consisting of scheduling a yard crane to carry out a set of container storage and retrieval requests in a single container block. The objective is to minimize the total travel time of the crane to carry out all requests. The block has multiple input and output (I/O) points located at both the seaside and the landside. The crane must move retrieval containers from the block to the I/O points, and must move storage containers from the I/O points to the block. The problem is modeled as a continuous time integer programming model and the complexity is proven. We use intrinsic properties of the problem to propose a two-phase solution method to optimally solve the problem. In the first phase, we develop a merging algorithm which tries to patch subtours of an optimal solution of an assignment problem relaxation of the problem and obtain a complete crane tour without adding extra travel time to the optimal objective value of the relaxed problem. The algorithm requires common I/O points to patch subtours. This is efficient and often results in obtaining an optimal solution of the problem. If an optimal solution has not been obtained, the solution of the first phase is embedded in the second phase where a branch-and-bound algorithm is used to find an optimal solution. The numerical results show that the proposed method can quickly obtain an optimal solution of the problem. Compared to the random and Nearest Neighbor heuristics, the total travel time is on average reduced by more than 30% and 14%, respectively. We also validate the solution method at a terminal.

Optimal selection of retailers for a manufacturing vendor in a vendor managed inventory system

A Vendor Managed Inventory (VMI) system consists of a manufacturing vendor and a number of retailers. In such a system, it is essential for the vendor to optimally determine retailer selection and other related decisions, such as the product’s replenishment cycle time and the wholesale price, in order to maximize his profit. Meanwhile, each retailer’s decisions on her willingness to enter the system and retail price are simultaneously considered in the retailer selection process. However, the above interactive decision making is complex and the available studies on interactive retailer selection are scarce. In this study, we formulate the retailer selection problem as a Stackelberg game model to help the manufacturer, as a vendor, optimally select his retailers to form a VMI system. This model is non-linear, mixed-integer, game-theoretic, and analytically intractable. Therefore, we further develop a hybrid algorithm for effectively and efficiently solving the developed model. The hybrid algorithm combines dynamic programming (DP), genetic algorithm (GA) and analytical methods. As demonstrated by our numerical studies, the optimal retailer selection can increase the manufacturer’s profit by up to 90% and the selected retailers’ profits significantly compared to non-selection strategy. The proposed hybrid algorithm can solve the model within a minute for a problem with 100 candidate retailers, whereas a pure GA has to take more than 1h to solve a small sized problem of 20 candidate retailers achieving an objective value no worse than that obtained by the hybrid algorithm.

Sequencing heuristics for storing and retrieving unit loads in 3D compact automated warehousing systems

Sequencing unit-load retrieval requests has been extensively reported on in the literature for conventional single-deep automated warehousing systems. A proper sequence can greatly reduce the makespan when carrying out a group of such requests. Although the sequencing problem is NP-hard, some very good heuristics exist. Surprisingly, the problem has not yet been investigated for compact (multi-deep) storage systems, which have greatly increased in popularity the last decade. This article studies how to sequence a group (or block) of storage and retrieval requests in a multi-deep automated storage system with the objective to minimize the makespan. Currently utilized sequencing heuristics for the multi-deep system are adapted in this article and in addition a new heuristic, Percentage Priority to Retrievals with Shortest Leg (PPR-SL), is proposed and evaluated. It is shown that the PPR-SL heuristic consistently outperforms all of the other heuristics. Generally, it can outperform the benchmark First-Come First-Served (FCFS) heuristic by between 20 and 70%. The nearest neighbor heuristic that performs very well in conventional single-deep storage systems appears to perform poorly in the multi-deep system, even worse than FCFS. In addition, based on FCFS and PPR-SL, robust rack dimensions that yield a short makespan, regardless of the number of storage and retrieval requests, are found.

A decision-tree stacking heuristic minimising the expected number of reshuffles at a container terminal

Reshuffling containers, one of the daily operations at a container terminal, is time consuming and increases a ship’s berthing time. We propose a decision-tree heuristic to minimise the expected number of reshuffles when arriving containers should be stacked in a block of containers with an arbitrary number of piles. The heuristic algorithm uses the optimal solutions of a stochastic dynamic programming model. Since the total number of states of the dynamic programming model increases exponentially, the model can only solve small-scale problems in a reasonable time. To solve large-scale problems, the heuristic uses the results of the exact model for small-scale problems to generate generalised decision trees. These trees can be used to solve problems with a realistic number of piles. The numerical experiments show the effectiveness of the algorithm. For small-scale problems, the trees can quickly make optimal decisions. For large-scale problems, the decision-tree heuristic significantly outperforms stacking policies commonly used in practice. Using the decision trees, we can compare the performance of a shared-stacking policy, which allows containers of multiple ships to be stacked on top of each other, with a dedicated-stacking policy. Shared-stacking appears to outperform dedicated-stacking.

Small is Beautiful: A Framework for Evaluating and Optimizing Live-Cube Compact Storage Systems

Warehouses occupy much space and land, which has become increasingly scarce in many parts of Europe, Asia, and the United States, particularly close to areas where demand is generated, such as large cities. This paper studies live-cube compact storage systems that may solve this space shortage problem as they do not require travel aisles. Each stored unit load is accessible individually and can be moved in x and y directions by a shuttle as long as an empty location is available, comparable to the well-known 15-puzzle in which 15 numbered tiles slide within a 4 × 4 grid. When multiple empty locations are available on a level, the shuttles can cooperate to create a virtual aisle for fast retrieval of a desired unit load. A lift moves the unit loads across different levels in z direction. Such storage systems are increasingly used in different service sectors like car parking, warehousing, and container handling, but so far they have hardly been studied. For live-cube systems, many research questions still have to be answered, including cycle time calculations, cost comparisons, and energy requirements. In this paper, we first derive simple to use closed-form formulas for expected retrieval time of an arbitrary unit load and validate the quality of these formulas by comparing them with a real application. Second, we propose and solve a mixed-integer nonlinear model to optimize system dimensions by minimizing the retrieval time. We obtain closed-form expressions for minimum retrieval time that are simple to apply in practice. Third, we compare the investment, operational costs, and energy consumption of live-cube systems with traditional systems based on a real application.