This study proposes a supply chain system model containing Nash game companies that compete in a common market. Each company adopts a vertical integration strategy and runs offline and online selling channels. To maximise profits, the companies determine optimal production quantities and prices. Using theory of finite-dimensional variational inequality, we prove the existence and uniqueness of the equilibrium pattern and develop a converged algorithm. Numerically, we compute the equilibrium production quantities and prices, and subsequently generate three findings through sensitivity analyses. By changing the transportation cost, we demonstrate that the offline price is conditionally higher than the online price with different transportation and processing costs. All equilibrium prices and profits decrease when substitution intensity is increased, and thus companies should differentiate their products. Altering consumer preference for online purchasing shows that consumer preference significantly influences the production quantities and profits. Online selling is not positively affected by consumer preference for online purchasing but is affected by the complex relationship between consumer preference and its variable cost.

This paper studies a real-life multi-objective two-dimensional single-bin-size bin-packing problem arising in industry. A packing pattern is defined by one bin, a set of items packed into the bin and the packing positions of these items. A number of bins can be placed with the same packing pattern. The objective is not only to minimise the number of bins used, as in traditional bin-packing problems, but also to minimise the number of packing patterns. Based on our previous study of a heuristic stemming from dynamic programming by aggregating states to avoid the exponential increase in the number of states, we further develop this heuristic by decomposing a pattern with a number of bins at each step. Computational results show that this heuristic provides satisfactory results with a gap generally less than 20% with respect to the optimum.

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.

Compact, multi-deep three-dimensional (3D), Automated Storage and Retrieval Systems (AS/RS) are becoming more common, due to new technologies, lower investment costs, time efficiency and compact size. Decision-making research on these systems is still in its infancy. This paper studies a particular compact system with rotating conveyors for the depth movement and a Storage/Retrieval (S/R) machine for the horizontal and vertical movement of unit loads. The optimal storage zone boundaries are determined for this system with two product classes: high- and low-turnover, by minimizing the expected S/R machine travel time. We formulate a mixed-integer non-linear programming model to determine the zone boundaries. A decomposition algorithm and a one-dimensional search scheme are developed to solve the model. The algorithm is complex, but the results are appealing since most of them are in closed-form and easy to apply to optimally layout the 3D AS/RS rack. The results show that the S/R machine travel time is significantly influenced by the zone dimensions, zone sizes and ABC curve skewness (presenting turnover patterns of different products). The presented results are compared with those under random storage and it is shown that significant reductions of the machine travel time are obtainable by using class-based storage.

We sequence storage and retrieval jobs to minimize total travel time of a storage/retrieval (S/R) machine in a two-depot automated storage/retrieval system. These systems include storage systems with aisle-captive S/R machines and storage blocks with bridge cranes. The S/R machine must move retrieval unit loads from their current locations in the system to one of the two depots. In addition, it must move storage unit loads from given depots to given locations in the system. We model the problem as an asymmetric traveling salesman problem, which is in general NP-hard. We develop an algorithm to solve the problem in polynomial time, using the property that the system has two depots and the S/R machine always returns to one of the depots to pick up or deliver a load. Furthermore, we develop additional polynomial time algorithms for the following four special cases: (1) retrieval loads have to be delivered to given depots; (2) the system has one input depot and one output depot; (3) the system has a single depot; and (4) there are arbitrary S/R machine starting and ending locations. The computational results show the effectiveness of the proposed algorithms. Compared to first-come-first-served and nearest neighbor algorithms, commonly used in practice, the total travel time reduces on average by more than 30% and 15%, respectively.

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. (C) 2007 Elsevier B.V. All rights reserved.

This paper addresses single-machine scheduling problem with resource allocation and learning effect in the background of past-sequence-dependent (p-s-d) setup times. In the proposed model of this paper, the actual job processing times are dependent on learning effect and the amount of resource allocated, and the setup times are proportional to the length of the already processed jobs. The resource function used here is a general convex one. The optimal job sequence and the optimal amount of resource allocated to each job are determined jointly for the objective function yielded by a combination of the total completion time, total absolute differences in completion times, and the total resource consumption. Besides, we also discuss some extension and special cases of this problem. It is shown that all the problems under study are polynomially solvable while the complexity results are different.