Multiperiod capacity expansion and shipment planning with linear costs

In this paper we consider a multiperiod deterministic capacity expansion and shipment planning problem for a single product. The product can be manufactured in several producing regions and is required in a number of markets. The demands for each of the markets are non-decreasing over time and must be met exactly during each time period (i.e., no backlogging or inventorying for future periods is permitted). Each region is assumed to have an initial production capacity, which may be increased at a given cost in any period. The demand in a market can be satisfied by production and shipment from any of the regions. The problem is to find a schedule of capacity expansions for the regions and a schedule of shipments from the regions to the markets so as to minimize the discounted capacity expansion and shipment costs. The problem is formulated as a linear programming model, and solved by an efficient algorithm using the operator theory of parametric programming for the transporation problem. Extensions to the infinite horizon case are also provided.     

A heuristic method for facility planning in telecommunications networks with multiple alternate routes

Given point-to-point demand forecasts of transmission facilities for services such as voice or data transmission in each period of a finite planning horizon, a decision has to be made as to which types of transmission facilities—together with the amounts of transmission circuits—are to be installed, if any, on each link of the telecommunications network, in each period of the planning horizon. The availability of alternative transmission systems with significantly different costs and circuit capacities necessitates the determination of a minimum (discounted) cost facility installation scheme. This combinatoric choice problem is complicated by the availability of switching equipments enabling the transmission of some of the traffic through intermediary points. This possibility of alternately routing the traffic or the facility requirements of certain point pairs further complicates the problem while creating the opportunity to benefit from economies of scale. We present here a heuristic method for finding a good solution for the general problem; namely, we consider multiple transmission systems and multiple alternate routes. Numerical examples are given and computational experience is reported.     

Joint pricing and ordering policy for exponentially decaying inventory with known demand

This paper is concerned with the problem of simultaneously setting price and production levels for an exponentially decaying product. Such products suffer a loss in utility which is proportional to the total quantity of stock on hand. A continuous review, deterministic demand model is considered. The optimal ordering decision quantity is derived and its sensitivity to changes in perishability and product price is considered. The joint ordering pricing decision is also computed and consideration of parametric changes of these decisions indicates a non-monotonic response for optimal price to changes in product decay. Issues of market entry and extensions to a model with shortages are also analyzed.     

Coping with shortages caused by disruptive events in automobile supply chains

Unpredictable disruptive events significantly increase the difficulty of the management of automobile supply chains. In this paper, we propose an automobile production planning problem with component chips substitution in a finite planning horizon. The shortage of one chip can be compensated by another chip of the same type with a higher-end feature at an additional cost. Therefore, the automobile manufacturer can divert the on-hand inventory of chips to product lines that are more profitable in the event of shortages caused by supply chain disruptions. To cope with this, we propose a max-min robust optimization model that captures the uncertain supplies of chips. We show that the robust model has a mixed-integer programming equivalence that can be solved by a commercial IP solver directly. We compare the max-min robust model with the corresponding deterministic and two-stage stochastic models for the same problem through extensive numerical experiments. The computational results show that the max-min robust model outperforms the other two models in terms of the average and worst-case profits.     

Optimal design of a multi-item,multi-location,multi-repair type repair and supply system

The design of a system with many locations, each with many items which may fail while in use, is considered. When items fail, they require repair; the particular type of repair being governed by a probability distribution. As repairs may be lengthy, spares are kept on hand to replace failed items. System ineffectiveness is measured by expected weighted shortages over all items and locations, in steady state. This can be reduced by either having more spares or shorter expected repair times. Design consists of a provisioning of the number of spares for each item, by location; and specifying the expected repair times for each type of repair, by item and location. The optimal design minimizes expected shortages within a budget constraint, which covers both (i) procurement of spares and (ii) procurement of equipment and manning levels for the repair facilities. All costs are assumed to be separable so that a Lagrangian approach is fruitful, yielding an implementable algorithm with outputs useful for sensitivity analysis. A numerical example is presented.     

Multiperiod resource allocation with a smoothing objective

We consider a multiperiod resource allocation problem, where a single resource is allocated over a finite planning horizon of T periods. Resource allocated to one period can be used to satisfy demand of that period or of future periods, but backordering of demand is not allowed. The objective is to allocate the resource as smoothly as possible throughout the planning horizon. We present two models: the first assumes that the allocation decision variables are continuous, whereas the second considers only integer allocations. Applications for such models are found, for example, in subassembly production planning for complex products in a multistage production environment. Efficient algorithms are presented to find optimal allocations for these models at an effort of O(T²). Among all optimal policies for each model, these algorithms find the one that carries the least excess resources throughout the planning horizon. © 1995 John Wiley & Sons, Inc.     

Determining optimal growth paths in logistics operations

This paper considers a logistics system modelled as a transportation problem with a linear cost structure and lower bounds on supply from each origin and to each destination. We provide an algorithm for obtaining the growth path of such a system, i. e., determining the optimum shipment patterns and supply levels from origins and to destinations, when the total volume handled in the system is increased. Extensions of the procedure for the case when the costs of supplying are convex and piecewise linear and for solving transportation problems that are not in “standard form” are discussed. A procedure is provided for determining optimal plant capacities when the market requirements have prespecified growth rates. A goal programming growth model where the minimum requirements are treated as goals rather than as absolute requirements is also formulated.     

Nash bargaining over allocations in inventory pooling contracts

When facing uncertain demand, several firms may consider pooling their inventories leading to the emergence of two key contractual issues. How much should each produce or purchase for inventory purposes? How should inventory be allocated when shortages occur to some of the firms? Previously, if the allocations issue was considered, it was undertaken through evaluation of the consequences of an arbitrary priority scheme. We consider both these issues within a Nash bargaining solution (NBS) cooperative framework. The firms may not be risk neutral, hence a nontransferable utility bargaining game is defined. Thus the physical pooling mechanism itself must benefit the firms, even without any monetary transfers. The firms may be asymmetric in the sense of having different unit production costs and unit revenues. Our assumption with respect to shortage allocation is that a firm not suffering from a shortfall, will not be affected by any of the other firms' shortages. For two risk neutral firms, the NBS is shown to award priority on all inventory produced to the firm with higher ratio of unit revenue to unit production cost. Nevertheless, the arrangement is also beneficial for the other firm contributing to the total production. We provide examples of Uniform and Bernoulli demand distributions, for which the problem can be solved analytically. For firms with constant absolute risk aversion, the agreement may not award priority to any firm. Analytically solvable examples allow additional insights, e.g. that higher risk aversion can, for some problem parameters, cause an increase in the sum of quantities produced, which is not the case in a single newsvendor setting. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Integrated capacity,demand, and production planning with subcontracting and overtime options

Models for integrated production and demand planning decisions can serve to improve a producer's ability to effectively match demand requirements with production capabilities. In contexts with price‐sensitive demands, economies of scale in production, and multiple capacity options, such integrated planning problems can quickly become complex. To address these complexities, this paper provides profit‐maximizing production planning models for determining optimal demand and internal production capacity levels under price‐sensitive deterministic demands, with subcontracting and overtime options. The models determine a producer's optimal price, production, inventory, subcontracting, overtime, and internal capacity levels, while accounting for production economies of scale and capacity costs through concave cost functions. We use polyhedral properties and dynamic programming techniques to provide polynomial‐time solution approaches for obtaining an optimal solution for this class of problems when the internal capacity level is time‐invariant. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Optimal policy for a dynamic multi-echelon inventory model

A general multiperiod multi-echelon supply system consisting of n facilities each stocking a single product is studied. At the beginning of a period each facility may order stock from an exogenous source with no delivery lag and proportional ordering costs. During the period the (random) demands at the facilities are satisfied according to a given supply policy that determines to what extent stock may be redistributed from facilities with excess stock to those experiencing shortages. There are storage, shortage, and transportation costs. An ordering policy that minimizes expected costs is sought. If the initial stock is sufficiently small and certain other conditions are fulfilled, it is optimal to order up to a certain base stock level at each facility. The special supply policy in which each facility except facility 1 passes its shortages on to a given lower numbered facility called its direct supplier is examined in some detail. Bounds on the base stock levels are obtained. It is also shown that if the demand distribution at facility j is stochastically smaller (“spread” less) than that at another facility k having the same direct supplier and if certain other conditions are fulfilled, then the optimal base stock level (“virtual” stock out probability) at j is less than (greater than) or equal to that at facility k.     

On the equivalence of jit and mrp as technologies for reducing wastes in manufacturing

JIT (just-in-time) is widely regarded as an excellent tool for reducing costs and cycle times, and for improving quality in manufacturing operations. JIT follows a multistep procedure. First, it identifies and prioritizes wastes or non-value-adding activities. Second, it forces these wastes to be removed. MRP (materials requirements planning) can identify the same wastes and prioritize them in the same way that JIT does, by using data from the MRP database and master production schedule, and a waste identification model. In this article, a model is developed which describes the process by which the classic JIT system identifies and prioritizes waste. An equivalent MRP waste identification model is then developed for the production environment of the classic JIT system. (The classic JIT system was developed to produce many products having low to medium volumes.) The results developed here can be extended to other production environments where adaptations of the classic JIT system are used. An example, taken from an actual application, is presented to illustrate the models and the equivalence of JIT and MRP as systems for identifying and prioritizing wastes in manufacturing. © 1993 John Wiley & Sons, Inc.     

The transshipment fund mechanism: Coordinating the decentralized multilocation transshipment problem

The multilocation replenishment and transshipment problem is concerned with several retailers facing random demand for the same item at distinct markets, that may use transshipments to eliminate excess inventory/shortages after demand realization. When the system is decentralized so that each retailer operates to maximize their own profit, there are incentive problems that prevent coordination. These problems arise even with two retailers who may pay each other for transshipped units. We propose a new mechanism based on a transshipment fund, which is the first to coordinate the system, in a fully noncooperative setting, for all instances of two retailers as well as all instances of any number of retailers. Moreover, our mechanism strongly coordinates the system, i.e., achieves coordination as the unique equilibrium. The computation and information requirements of this mechanism are realistic and relatively modest. We also present necessary and sufficient conditions for coordination and prove they are always satisfied with our mechanism. Numerical examples illustrate some of the properties underlying this mechanism for two retailers. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Cost allocation of capacity investment games

Consider a manufacturer serving a set of retail stores each of which faces deterministic demands in a finite planning horizon. At the beginning of the planning horizon, the production capacity of the manufacturer is built, followed by production, outsourcing to third party manufacturers if necessary and distribution to the retail stores. Because the retail stores are usually managed by different managers who act as independent profit centers, it is desirable that the total cost is divided among the retail stores so that their incentives can be appropriately captured and thus efficient operations can be achieved. Under various conditions, we prove that there is a fair allocation of costs among the retail stores in the sense that no subset of retail stores subsidizes others, or equivalently, the resulting capacity investment game has a nonempty core, that is, the capacity investment game is a balanced game. In addition, our proof provides a mechanism to compute a fair cost allocation. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 512–523, 2013     

Coordinated overhaul scheduling of production units

This article presents a new approach to solve the problem of coordinating the overhaul scheduling of several nonidentical production units. For each production unit, we assume that the operating cost is an n-order polynomial function of the time elapsed since its previous overhaul. We develop an efficient iterative algorithm that generates a near-optimal cyclic overhaul schedule. We also construct a simple algorithm for the case where the overhaul interval for each production unit and the cycle time are restricted to be power-of-two multiples of some base planning period. Finally, we provide a worst-case performance bound for the solution to the problem under the power-of-two restriction. © 1994 John Wiley & Sons, Inc.     

The continuous‐time single‐sourcing problem with capacity expansion opportunities

This paper develops a new model for allocating demand from retailers (or customers) to a set of production/storage facilities. A producer manufactures a product in multiple production facilities, and faces demand from a set of retailers. The objective is to decide which of the production facilities should satisfy each retailer's demand, in order minimize total production, inventory holding, and assignment costs (where the latter may include, for instance, variable production costs and transportation costs). Demand occurs continuously in time at a deterministic rate at each retailer, while each production facility faces fixed‐charge production costs and linear holding costs. We first consider an uncapacitated model, which we generalize to allow for production or storage capacities. We then explore situations with capacity expansion opportunities. Our solution approach employs a column generation procedure, as well as greedy and local improvement heuristic approaches. A broad class of randomly generated test problems demonstrates that these heuristics find high quality solutions for this large‐scale cross‐facility planning problem using a modest amount of computation time. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

Disruption management in production planning

We study the problem of recovering a production plan after a disruption, where the disruption may be caused by incidents such as power failure, market change, machine breakdown, supply shortage, worker no‐show, and others. The new recovery plan we seek after has to not only suit the changed environment brought about by the disruption, but also be close to the initial plan so as not to cause too much customer unsatisfaction or inconvenience for current‐stage and downstream operations. For the general‐cost case, we propose a dynamic programming method for the problem. For the convex‐cost case, a general problem which involves both cost and demand disruptions can be solved by considering the cost disruption first and then the demand disruption. We find that a pure demand disruption is easy to handle; and for a pure cost disruption, we propose a greedy method which is provably efficient. Our computational studies also reveal insights that will be helpful to managing disruptions in production planning. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

Optimal investment,pricing and replacement of computer resources

The problem of multiple-resource capacity planning under an infinite time horizon is analyzed using a nonlinear programming model. The analysis generalizes to the long term the short-run pricing model for computer networks developed in Kriebel and Mikhail [5]. The environment assumes heterogeneous resource capacities by age (vingate), which service a heterogeneous and relatively captive market of users with known demand functions in each time period. Total variable operating costs are given by a continuous psuedoconcave function of system load, capacity, and resource age. Optimal investment, pricing, and replacement decision rules are derived in the presence of economies of scale and exogenous technological progress. Myopic properties of the decision rules which define natural (finite) planning subhorizons are discussed.     

Dynamic inventory management with cash flow constraints

In this article, we consider a classic dynamic inventory control problem of a self‐financing retailer who periodically replenishes its stock from a supplier and sells it to the market. The replenishment decisions of the retailer are constrained by cash flow, which is updated periodically following purchasing and sales in each period. Excess demand in each period is lost when insufficient inventory is in stock. The retailer's objective is to maximize its expected terminal wealth at the end of the planning horizon. We characterize the optimal inventory control policy and present a simple algorithm for computing the optimal policies for each period. Conditions are identified under which the optimal control policies are identical across periods. We also present comparative statics results on the optimal control policy. © 2008 Wiley Periodicals, Inc. Naval Research Logistics 2008     

Computational study of the multiechelon production planning problem

In this article we try to identify appropriate solution procedures for different types of multiechelon production planning problems. We conduct an extensive computational study on uncapacitated multiechelon production planning problems with serial and assembly types of bill-of-material structures. Problems are formulated as both single-source fixed charge network problems and as multicommodity flow problems with fixed charges. Solution procedures considered are branch and cut, Lagrangean relaxation (for the network formulation), and branch and bound (for the multicommodity formulation). Three hundred problems with various problem structures are tested. Our conclusions suggest the best approach for each type of problem structure. © 1997 John Wiley & Sons, Inc.     

Multiperiod allocation of limited resources: A minimax approach

We consider a multiperiod model in which limited resources are allocated among competing activities in each period. The objective is to minimize the maximum weighted deviation of the cumulative activity levels from the cumulative demands among all activities at all periods. All resources are assumed to be storable; that is, surpluses at one period can be used later on. This model is useful, for example, in multiperiod production planning for high-technology industries that assemble a large variety of circuit boards using numerous electronic components. The model is formulated with a minimax objective. We develop an efficient algorithm that can solve large-scale problems very quickly. At each iteration, the algorithm makes use of the solution to a relaxed problem to identify activities that should be permanently set to zero, as well as groups of activities that should have the same value.