Lateral capacity exchange and its impact on capacity investment decisions

We study the problem of capacity exchange between two firms in anticipation of the mismatch between demand and capacity, and its impact on firm's capacity investment decisions. For given capacity investment levels of the two firms, we demonstrate how capacity price may be determined and how much capacity should be exchanged when either manufacturer acts as a Stackelberg leader in the capacity exchange game. By benchmarking against the centralized system, we show that a side payment may be used to coordinate the capacity exchange decisions. We then study the firms' capacity investment decisions using a biform game framework in which capacity investment decisions are made individually and exchange decisions are made as in a centralized system. We demonstrate the existence and uniqueness of the Nash equilibrium capacity investment levels and study the impact of firms' share of the capacity exchange surplus on their capacity investment levels.© 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Capacity expansion and cost efficiency improvement in the warehouse problem

The warehouse problem with deterministic production cost, selling prices, and demand was introduced in the 1950s and there is a renewed interest recently due to its applications in energy storage and arbitrage. In this paper, we consider two extensions of the warehouse problem and develop efficient computational algorithms for finding their optimal solutions. First, we consider a model where the firm can invest in capacity expansion projects for the warehouse while simultaneously making production and sales decisions in each period. We show that this problem can be solved with a computational complexity that is linear in the product of the length of the planning horizon and the number of capacity expansion projects. We then consider a problem in which the firm can invest to improve production cost efficiency while simultaneously making production and sales decisions in each period. The resulting optimization problem is non‐convex with integer decision variables. We show that, under some mild conditions on the cost data, the problem can be solved in linear computational time. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 367–373, 2016     

Collaboration in contingent capacities with information asymmetry

We study the optimal contracting problem between two firms collaborating on capacity investment with information asymmetry. Without a contract, system efficiency is lost due to the profit‐margin differentials among the firms, demand uncertainty, and information asymmetry. With information asymmetry, we demonstrate that the optimal capacity level is characterized by a newsvendor formula with an upward‐adjusted capacity investment cost, and no first‐best solution can be achieved. Our analysis shows that system efficiency can always be improved by the optimal contract and the improvement in system efficience is due to two factors. While the optimal contract may bring the system's capacity level closer to the first‐best capacity level, it prevents the higher‐margin firm from overinvesting and aligns the capacity‐investment decisions of the two firms. Our analysis of a special case demonstrates that, under some circumstances, both firms can benefit from the principal having better information about the agent's costs. © 2007 Wiley Periodicals, Inc. Naval Research Logistics 54:, 2007     

Optimal cutoff strategies in capacity expansion problems

Capacity expansion refers to the process of adding facilities or manpower to meet increasing demand. Typical capacity expansion decisions are characterized by uncertain demand forecasts and uncertainty in the eventual cost of expansion projects. This article models capacity expansion within the framework of piecewise deterministic Markov processes and investigates the problem of controlling investment in a succession of same type projects in order to meet increasing demand with minimum cost. In particular, we investigate the optimality of a class of investment strategies called cutoff strategies. These strategies have the property that there exists some undercapacity level M such that the strategy invests at the maximum available rate at all levels above M and does not invest at any level below M. Cutoff strategies are appealing because they are straightforward to implement. We determine conditions on the undercapacity penalty function that ensure the existence of optimal cutoff strategies when the cost of completing a project is exponentially distributed. A by-product of the proof is an algorithm for determining the optimal strategy and its cost. © 1995 John Wiley & Sons, Inc.     

Capacity investment decisions under risk aversion

This article studies the optimal capacity investment problem for a risk‐averse decision maker. The capacity can be either purchased or salvaged, whereas both involve a fixed cost and a proportional cost/revenue. We incorporate risk preference and use a consumption model to capture the decision maker's risk sensitivity in a multiperiod capacity investment model. We show that, in each period, capacity and consumption decisions can be separately determined. In addition, we characterize the structure of the optimal capacity strategy. When the parameters are stationary, we present certain conditions under which the optimal capacity strategy could be easily characterized by a static two‐sided (s, S) policy, whereby, the capacity is determined only at the beginning of period one, and held constant during the entire planning horizon. It is purchased up to B when the initial capacity is below b, salvaged down to Σ when it is above σ, and remains constant otherwise. Numerical tests are presented to investigate the impact of demand volatility on the optimal capacity strategy. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 218–235, 2016     

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.     

AN ALGORITHM FOR THE SEGREGATED STORAGE PROBLEM

The segregated storage problem involves the optimal distribution of products among compartments with the restriction that only one product may be stored in each compartment. The storage capacity of each compartment, the storage demand for each product, and the linear cost of storing one unit of a product in a given compartment are specified. The problem is reformulated as a large set-packing problem, and a column generation scheme is devised to solve the associated linear programming problem. In case of fractional solutions, a branch and bound procedure is utilized. Computational results are presented.     

A problem in network interdiction

Suppose we are given a network G=(V,E) with arc distances and a linear cost function for lengthening arcs. In this note, we consider a network-interdiction problem in which the shortest path from source node s to sink node t is to be increased to at least τ units via a least-cost investment strategy. This problem is shown to reduce to a simple minimum-cost-flow problem. Applications and generalizations are discussed, including the multiple-destination case.     

Revenue sharing contracts in a supply chain with uncontractible actions

We consider a supplier–customer relationship where the customer faces a typical Newsvendor problem of determining perishable capacity to meet uncertain demand. The customer outsources a critical, demand‐enhancing service to an outside supplier, who receives a fixed share of the revenue from the customer. Given such a linear sharing contract, the customer chooses capacity and the service supplier chooses service effort level before demand is realized. We consider the two cases when these decisions are made simultaneously (simultaneous game) or sequentially (sequential game). For each game, we analyze how the equilibrium solutions vary with the parameters of the problem. We show that in the equilibrium, it is possible that either the customer's capacity increases or the service supplier's effort level decreases when the supplier receives a larger share of the revenue. We also show that given the same sharing contract, the sequential game always induces a higher capacity and more effort. For the case of additive effort effect and uniform demand distribution, we consider the customer's problem of designing the optimal contract with or without a fixed payment in the contract, and obtain sensitivity results on how the optimal contract depends on the problem parameters. For the case of fixed payment, it is optimal to allocate more revenue to the supplier to induce more service effort when the profit margin is higher, the cost of effort is lower, effort is more effective in stimulating demand, the variability of demand is smaller or the supplier makes the first move in the sequential game. For the case of no fixed payment, however, it is optimal to allocate more revenue to the supplier when the variability of demand is larger or its mean is smaller. Numerical examples are analyzed to validate the sensitivity results for the case of normal demand distribution and to provide more managerial insights. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

A nonlinear continuous time optimal control model of dynamic pricing and inventory control with no backorders

In this paper, we present a continuous time optimal control model for studying a dynamic pricing and inventory control problem for a make‐to‐stock manufacturing system. We consider a multiproduct capacitated, dynamic setting. We introduce a demand‐based model where the demand is a linear function of the price, the inventory cost is linear, the production cost is an increasing strictly convex function of the production rate, and all coefficients are time‐dependent. A key part of the model is that no backorders are allowed. We introduce and study an algorithm that computes the optimal production and pricing policy as a function of the time on a finite time horizon, and discuss some insights. Our results illustrate the role of capacity and the effects of the dynamic nature of demand in the model. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Scheduling a production–distribution system to optimize the tradeoff between delivery tardiness and distribution cost

We consider a make‐to‐order production–distribution system with one supplier and one or more customers. A set of orders with due dates needs to be processed by the supplier and delivered to the customers upon completion. The supplier can process one order at a time without preemption. Each customer is at a distinct location and only orders from the same customer can be batched together for delivery. Each delivery shipment has a capacity limit and incurs a distribution cost. The problem is to find a joint schedule of order processing at the supplier and order delivery from the supplier to the customers that optimizes an objective function involving the maximum delivery tardiness and the total distribution cost. We first study the solvability of various cases of the problem by either providing an efficient algorithm or proving the intractability of the problem. We then develop a fast heuristic for the general problem. We show that the heuristic is asymptotically optimal as the number of orders goes to infinity. We also evaluate the performance of the heuristic computationally by using lower bounds obtained by a column generation approach. Our results indicate that the heuristic is capable of generating near optimal solutions quickly. Finally, we study the value of production–distribution integration by comparing our integrated approach with two sequential approaches where scheduling decisions for order processing are made first, followed by order delivery decisions, with no or only partial integration of the two decisions. We show that in many cases, the integrated approach performs significantly better than the sequential approaches. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Parallel machine scheduling with batch deliveries to minimize total flow time and delivery cost

Motivated by the flow of products in the iron and steel industry, we study an identical and parallel machine scheduling problem with batch deliveries, where jobs finished on the parallel machines are delivered to customers in batches. Each delivery batch has a capacity and incurs a cost. The objective is to find a coordinated production and delivery schedule that minimizes the total flow time of jobs plus the total delivery cost. This problem is an extension of the problem considered by Hall and Potts, Ann Oper Res 135 (2005) 41–64, who studied a two‐machine problem with an unbounded number of transporters and unbounded delivery capacity. We first provide a dynamic programming algorithm to solve a special case with a given job assignment to the machines. A heuristic algorithm is then presented for the general problem, and its worst‐case performance ratio is analyzed. The computational results show that the heuristic algorithm can generate near‐optimal solutions. Finally, we offer a fully polynomial‐time approximation scheme for a fixed number of machines. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 492–502, 2016     

Dynamic pricing and replenishment: Optimality,bounds, and asymptotics

In many applications, managers face the problem of replenishing and selling products during a finite time horizon. We investigate the problem of making dynamic and joint decisions on product replenishment and selling in order to improve profit. We consider a backlog scenario in which penalty cost (resulting from fulfillment delay) and accommodation cost (resulting from shortage at the end of the selling horizon) are incurred. Based on continuous‐time and discrete‐state dynamic programming, we study the optimal joint decisions and characterize their structural properties. We establish an upper bound for the optimal expected profit and develop a fluid policy by resorting to the deterministic version of the problem (ie, the fluid problem). The fluid policy is shown to be asymptotically optimal for the original stochastic problem when the problem size is sufficiently large. The static nature of the fluid policy and its lack of flexibility in matching supply with demand motivate us to develop a “target‐inventory” heuristic, which is shown, numerically, to be a significant improvement over the fluid policy. Scenarios with discrete feasible sets and lost‐sales are also discussed in this article.     

The dynamic transportation problem: A survey

The dynamic transportation problem is a transportation problem over time. That is, a problem of selecting at each instant of time t, the optimal flow of commodities from various sources to various sinks in a given network so as to minimize the total cost of transportation subject to some supply and demand constraints. While the earliest formulation of the problem dates back to 1958 as a problem of finding the maximal flow through a dynamic network in a given time, the problem has received wider attention only in the last ten years. During these years, the problem has been tackled by network techniques, linear programming, dynamic programming, combinational methods, nonlinear programming and finally, the optimal control theory. This paper is an up-to-date survey of the various analyses of the problem along with a critical discussion, comparison, and extensions of various formulations and techniques used. The survey concludes with a number of important suggestions for future work.     

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.     

A single-machine replacement model with learning

The replacement or upgrade of productive resources over time is an important decision for a manufacturing organization. The type of technology used in the productive resources determines how effectively the manufacturing operations can support the product and marketing strategy of the organization. Increasing operating costs (cost of maintenance, labor, and depreciation) over time force manufacturing organizations to periodically consider replacement or upgrade of their existing productive resources. We assume that there is a setup cost associated with the replacement of a machine, and that the setup cost is a nonincreasing function of the number of replacements made so far due to learning in setups. The operating cost of a newer machine is assumed to be lower than the operating cost of an older machine in any given period, except perhaps in the first period of operation of the new machine when the cost could be unusually high due to higher initial depreciation. A forward dynamic programming algorithm is developed which can be used to solve finite-horizon problems. We develop procedures to find decision and forecast horizons such that choices made during the decision horizon based only on information over the forecast horizon are also optimal for any longer horizon problem. Thus, we are able to obtain optimal results for what is effectively an infinite-horizon problem while only requiring data over a finite period of time. We present a numerical example to illustrate the decision/forecast horizon procedure, as well as a study of the effects of considering learning in making a series of machine replacement decisions. © 1993 John Wiley & Sons. Inc.     

OPTIMAL FACILITY LOCATION UNDER RANDOM DEMAND WITH GENERAL COST STRUCTURE

This paper investigates the problem of determining the optimal location of plants, and their respective production and distribution levels, in order to meet demand at a finite number of centers. The possible locations of plants are restricted to a finite set of sites, and the demands are allowed to be random. The cost structure of operating a plant is dependent on its location and is assumed to be a piecewise linear function of the production level, though not necessarily concave or convex. The paper is organized in three parts. In the first part, a branch and bound procedure for the general piecewise linear cost problem is presented, assuming that the demand is known. In the second part, a solution procedure is presented for the case when the demand is random, assuming a linear cost of production. Finally, in the third part, a solution procedure is presented for the general problem utilizing the results of the earlier parts. Certain extensions, such as capacity expansion or reduction at existing plants, and geopolitical configuration constraints can be easily incorporated within this framework.     

The gradual covering problem

In this paper we investigate the gradual covering problem. Within a certain distance from the facility the demand point is fully covered, and beyond another specified distance the demand point is not covered. Between these two given distances the coverage is linear in the distance from the facility. This formulation can be converted to the Weber problem by imposing a special structure on its cost function. The cost is zero (negligible) up to a certain minimum distance, and it is a constant beyond a certain maximum distance. Between these two extreme distances the cost is linear in the distance. The problem is analyzed and a branch and bound procedure is proposed for its solution. Computational results are presented. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

The building evacuation problem with shared information

In this article, the Building Evacuation Problem with Shared Information (BEPSI) is formulated as a mixed integer linear program, where the objective is to determine the set of routes along which to send evacuees (supply) from multiple locations throughout a building (sources) to the exits (sinks) such that the total time until all evacuees reach the exits is minimized. The formulation explicitly incorporates the constraints of shared information in providing online instructions to evacuees, ensuring that evacuees departing from an intermediate or source location at a mutual point in time receive common instructions. Arc travel time and capacity, as well as supply at the nodes, are permitted to vary with time and capacity is assumed to be recaptured over time. The BEPSI is shown to be NP‐hard. An exact technique based on Benders decomposition is proposed for its solution. Computational results from numerical experiments on a real‐world network representing a four‐story building are given. Results of experiments employing Benders cuts generated in solving a given problem instance as initial cuts in addressing an updated problem instance are also provided. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Quadratic bottleneck problems

We study the quadratic bottleneck problem (QBP) which generalizes several well‐studied optimization problems. A weak duality theorem is introduced along with a general purpose algorithm to solve QBP. An example is given which illustrates duality gap in the weak duality theorem. It is shown that the special case of QBP where feasible solutions are subsets of a finite set having the same cardinality is NP‐hard. Likewise the quadratic bottleneck spanning tree problem (QBST) is shown to be NP‐hard on a bipartite graph even if the cost function takes 0–1 values only. Two lower bounds for QBST are derived and compared. Efficient heuristic algorithms are presented for QBST along with computational results. When the cost function is decomposable, we show that QBP is solvable in polynomial time whenever an associated linear bottleneck problem can be solved in polynomial time. As a consequence, QBP with feasible solutions form spanning trees, s‐t paths, matchings, etc., of a graph are solvable in polynomial time with a decomposable cost function. We also show that QBP can be formulated as a quadratic minsum problem and establish some asymptotic results. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011