A Game-Theoretic Approach to Quantity Discount Problems*

The traditional quantity discount problem is analyzed from the perspective of game theory, including both noncooperative and cooperative models. For the noncooperative case, the Stackelberg equilibrium is derived. For the cooperative case, the Pareto Optimality criteria are used to find a group of optimal strategies. Both scenarios are illustrated through an example which quantifies the benefits resulting from cooperation between the buyer and the seller for game-theoretic solutions using geometric programming.     

A Comparison of Three Joint Ordering Inventory Policies*

This paper evaluates the performance of a joint ordering inventory policy which was first suggested and characterized by Renberg and Planche [14]. This paper shows that the policy is easily characterized for Poisson demands. This policy is then compared with two other joint ordering policies—the well-known (S, c, s) or can-order policy of Balintfy [3] and the recent periodic policies suggested by Atkins and lyogun [2]. For a continuous review operating environment, the Renberg and Planche policy utilizes a group reorder point and a combined order quantity (Q), with each item maintaining an order-up-to level (S). For the can-order policy, each item in the product group has a must-order point (s), a can-order point (c) and an order-up-to level (S). The periodic policies require that item orders be grouped at some fixed scheduled intervals. Using long-run total average costs as the basis, it is shown that no one policy is superior to the others in all the examples tested. In some cases, the Renberg and Planche policy performs surprisingly well.     

Standard Container Size Discounts and the Single-Period Inventory Problem*

This paper analyzes the problem of choosing the optimal order quantity and its associated number of standard containers making up the order for single-period inventory models under standard container size discounts. A range is determined that contains the optimal order quantity. Two algorithms are presented. The first algorithm solves the general case in which there is no restriction on the types of containers included in an order. The second algorithm solves a more restricted policy that requires the buyer to accept the order with successively smaller container sizes.     

A Model for Resource Constrained Production and Inventory Management

We present a general model for multi-item production and inventory management problems that include a resource restriction. The decision variables in the model can take on a variety of interpretations, but will typically represent cycle times, production batch sizes, number of production runs, or order quantities for each item. We consider environments where item demand rates are approximately constant and performing an activity such as producing a batch of a product or placing an order results in the consumption of a scarceresource that is shared among the items. Some examples of shared resources include limited machine capacity, a restriction on the amount of money that can be tied up in stock, orlimited storage capacity. We focus on the case where the decision variables must be integer valued or selected from a discrete set of choices, such as when an integer number of production runs is desired for each item, or in order quantity problems where the items come in pack sizes containing more than one unit and, therefore, the order quantities must be an integer multiple of the pack sizes. We develop a heuristic and a branch and bound algorithm for solving the problem. The branch and bound algorithm includes reoptimization procedures and the heuristic to improve its performance. Computational testing indicates that the algorithms are effective for solving the general model.     

Vendor Selection with Bundling*

Product bundling has become increasingly prevalent not only in consumer goods but also in the industrial sector. We study a purchasing problem in which a buyer must obtain necessary numbers of various stock items from a variety of vendors who charge different prices, have limited capacities and different levels of quality, and offer bundled products at discounted prices. We examine relationships among different bundling scenarios and show that the most general scenario is one in which free items are given to the buyer when sufficient quantities are purchased. We develop a mixed integer linear program that finds the purchasing strategy for the buyer that minimizes the total purchase cost. We present computational results which indicate that the problem is very tractable to solve optimally on a personal computer with standard optimization software. Finally, three extensions of the model are discussed.     

Determining Order Quantity and Selling Price by Geometric Programming: Optimal Solution,Bounds, and Sensitivity*

This paper presents a geometric programming (GP) approach to finding a profit-maximizing selling price and order quantity for a retailer. Demand is treated as a nonlinear function of price with a constant elasticity. The proposed GP approach finds optimal solutions for both no-quantity discounts and continuous quantity discounts cases. This approach is superior to the traditional approaches of solving a system of nonlinear equations. Since the profit function is not concave, the traditional approaches may require an exhaustive search, especially for the continuous discounts schedule case. By applying readily available theories in GP, we easily can find global optimal solutions for both cases. More importantly, the GP approach provides lower and upper bounds on the optimal profit level and sensitivity results which are unavailable from the traditional approaches. These bounding and sensitivity results are further utilized to provide additional important managerial implications on pricing and lot-sizing policies.     

Labor Utilization Effects of Labor Scheduling Flexibility Alternatives in a Tour Scheduling Environment

This study used a factorial experimental design and a new modeling methodology to investigate the impact of a number of labor scheduling flexibility alternatives and labor requirements characteristics on labor utilization within a tour scheduling environment. Break-placement flexibility and shift-length flexibility were found to be extremely effective in improving labor utilization for all labor requirement distributions used. Flexibility with respect to the number of days included in a tour schedule resulted in substantial improvement in labor utilization for all labor requirements distributions exhibiting daily and/or weekly variation. Surprisingly, virtually no improvement in labor utilization was achieved for any labor requirement distribution by the removal of requirements for consecutive days off. In addition, almost no improvement was found by allowing the shift start time to vary across the working days included in tours. High labor requirement amplitude was found to have a strong adverse effect on labor utilization while longer operational days were associated with improved labor utilization for all labor requirement distributions. We discuss the implications of these results for service operations management and provide suggestions for future research.     

A Heuristic Algorithm for the Capacitated Multiple Supplier Inventory Grouping Problem

This paper presents and solves a model for the multiple supplier inventory grouping problem, which involves the minimization of logistics costs for a firm that has multiple suppliers with capacity limitations. The costs included in the model are purchasing, transportation, ordering, and inventory holding, while the firm's objective is to determine the optimal flows and groups of commodities from each supplier. We present an algorithm, which combines subgradient optimization and a primal heuristic, to quickly solve the multiple supplier inventory grouping problem. Our algorithm is tested extensively on problems of various sizes and structures, and its performance is compared to that of OSL, a state-of-the-art integer programming code. The computational results indicate that our approach is extremely efficient for solving the multiple supplier inventory grouping problem.     

PROCEDURES FOR DETERMINING RELATIVE FREQUENCIES OF PRODUCTION/ORDER IN MULTISTAGE ASSEMBLY SYSTEMS*

We show that the problems of determination lot sizes in a multistage assembly system for the case of instantaneous production and constant demand for the end item can be reduced to the problems of determining relative frequenceis of production/order for items at each production stage. We further show that such frequencies are independent of the demand levels. Optimal and near-optimal solution procedures for this reduced problem are provided. The near-optimal procedure successively treats each stage of production as a final production stage while simulatenously incorporating decisions made at lower stages into decisions made at higher stages. Experimental results show that the near-optimal procedure results in optimal solutions 75 percent of the time and performs considerably better than representative heuristics available in the literature. Further, its performance is relatively less susceptible to product/structural characteristics of the system.     

Optimal and Heuristic Decision Strategies for Integrated Production and Marketing Planning

In this study we examine the effects of integrating production and marketing decisions for a short- to medium-range planning horizon in a profit maximizing firm. We formulate two models for determining price, marketing expenditure, demand or production volume, and lot size for a single product with stable demand when economies of scale are present. The full integration (FI) model simultaneously determines all the decisions involved, while the partial integration (PI) model separates the lot sizing decision from the others, as happens frequently in practice. Geometric programming (GP) techniques and marginal analysis are used to compare FI and PI, and obtain important managerial implications regarding the two models.     

SCHEDULING PARALLEL PROCESSORS WITH SETUP COSTS AND RESOURCE LIMITATIONS

An integer linear programming model is presented for the scheduling of n products on m identical machines. The particular problem studied is one that occurs frequently in the fiberglass and textile industries. The model incorporates setup costs, lost production costs, and overtime costs. Due to the structure of the model, integer solutions can be obtained by explicitly restricting only a small number of the integer variables. This allows those responsible for scheduling to solve realistically sized problems in an efficient manner. Computational results are provided for a set of generated test problems.     

An Industrial Ocean-Cargo Shipping Problem

This paper reports the modeling and solution of an industrial ocean-cargo shipping problem. The problem involves the delivery of bulk products from an overseas port to transshipment ports on the Atlantic Coast, and then over land to customers. The decisions made include the number and the size of ships to charter in each time period during the planning horizon, the number and location of transshipment ports to use, and transportation from ports to customers. The complexity of this problem is compounded by the cost structure, which includes fixed charges in both ship charters and port operations. Such a large scale, dynamic, and stochastic problem is reduced to a solvable stationary, deterministic, and cyclical model. The process of modeling the problem and the solution of the resultant mixed integer program are described in detail. Recommendations from this study have been implemented.     

PRODUCT STRUCTURE COMPLEXITY AND MULTILEVEL LOT SIZING USING ALTERNATIVE COSTING POLICIES*

This research evaluates the effect of product structure complexity on the performance of several lot-sizing procedures in a multilevel manufacturing environment. The experiment compares two different costing policies, full value added (FVA) and marginal value added (MVA), for calculating inventory holding cost. The major finding of the research is that product structure complexity has very little effect on the performance of various lot-sizing procedures. A second finding is that when product structures with varying components per parent and stocking points for a particular end item are present, the MVA costing policy emerges as the policy of choice because it favors slightly the Silver-Meal (SM)/least-total-cost (LTC) procedures over the Wagner-Whitin (WW)/LTC procedures.     

“A JOINT ECONOMIC-LOT-SIZE MODEL FOR PURCHASER AND VENDOR”: A COMMENT*

Banerjee [1] developed a joint economic-lot-size model for the case where a vendor produces to order for a purchaser on a lot-for-lot basis under deterministic conditions. The assumption of lot-for-lot bases is restrictive in nature. In this note, a more general joint economic-lot-size model is suggested and it is shown to provide a lower or equal joint total relevant cost as compared to the model of Banerjee.     

Concepts,Theory, and Techniques THE LOT-SIZE REORDER-POINT MODEL WITH UPSTREAM-DOWNSTREAM UNCERTAINTY*

One assumption in the classical lot-size problem is certainty in the amount requisitioned. In many practical situations, however, the amount received may be a random variable. In this paper, we discuss a lot-size inventory problem in which the quantity received does not necessarily match the quantity requisitioned. We develop explicit and approximate solutions for the back-orders case by assuming that the standard deviation of the amount received is linearly related to the quantity requisitioned.     

The Value of Supplier Information to Improve Management of a Retailer's Inventory

The distribution of lead time demand is essential for determining reorder points in inventory systems. Usually, the distribution of lead time demand is approximated directly. However, in some cases it may be worthwhile to take the demand per unit time and lead time into account, particularly when specific information is available. This paper deals with the situation where a supplier, who produces on order in fixed production cycles, provides information on the status of the coming production run. The retailer can use this information to gain insight into the lead-time process. A fixed order (s_vQ) strategy is presented, with a set of reorder points s_v depending on the time t until the first possible delivery, which is determined by the information of the supplier. A Markov model that analyzes a given (s_vQ) strategy is used to quantify the value of the information provided by the supplier. Some numerical examples show that the approach may lead to considerable cost savings compared to the traditional approach that uses only one single reorder point, based on a two-moments approximation. Using this numerical insight, the pros and cons of a more frequent exchange of information between retailers and suppliers can be balanced.     

THE CLASSIC ECONOMIC PRODUCTION QUANTITY MODEL WITH SETUP COST AS A FUNCTION OF CAPITAL EXPENDITURE*

The formulation of the classic economic production quantity (EPQ) model is extended to include setup cost as a function of capital expense. Additional capital will buy reduced setup cost. Thus, the objective now is to balance holding, setup, and capital expenses. This new formulation is solved under conditions where setup cost varies exponentially and linearly as a function of capital expense. Decision rules are formulated to indicate under what conditions setup cost reduction reduces total cost. For the linear function, it is shown that once the decision to reduce setup cost is justified, the optimal choice is the minimum setup cost that is technologically feasible.     

A Mathematical Programming Approach to Production Decisions in the Emerging Aquaculture Industry

The creation of new food sources is an important advancement in the search for solutions to food shortages in both developing and underdeveloped countries. Aquaculture, the production of aquatic animals under controlled conditions, offers exciting new possibilities for alternative protein sources in a wide variety of locales. This paper presents a mathematical programming model designed to aid the manager of an aquaculture facility in production scheduling and the evaluation of new technologies in this rapidly changing environment.     

Vendor Selection with Bundling: A Comment*

In a recent article by Rosenthal, Zydiak, and Chaudhry (1995), a mixed integer linear programming model was introduced to solve the vendor selection problem for the case in which the vendor can sell items individually or as part of a bundle. Each vendor offered only one type of bundle, and the buyer could purchase at most one bundle per vendor. The model employed n(m+ 1) binary variables, where n is the number of vendors and m is the number of products they sell. The existing model can lead to a purchasing paradox: it may force the buyer to pay more to receive less. We suggest a reformulation of the same problem that (i) eliminates this paradox and reveals a more cost-effective purchasing strategy; (ii) uses only n integer variables and significantly reduces the computational workload; and (iii) permits the buyer to purchase more than one bundle per vendor.