Capacity Expansion Model Research Articles

A Multi-Stage Stochastic Integer Programming Approach for Capacity Expansion under Uncertainty

This paper addresses a multi-period investment model for capacity expansion in an uncertain environment. Using a scenario tree approach to model the evolution of uncertain demand and cost parameters, and fixed-charge cost functions to model the economies of scale in expansion costs, we develop a multi-stage stochastic integer programming formulation for the problem. A reformulation of the problem is proposed using variable disaggregation to exploit the lot-sizing substructure of the problem. The reformulation significantly reduces the LP relaxation gap of this large scale integer program. A heuristic scheme is presented to perturb the LP relaxation solutions to produce good quality integer solutions. Finally, we outline a branch and bound algorithm that makes use of the reformulation strategy as a lower bounding scheme, and the heuristic as an upper bounding scheme, to solve the problem to global optimality. Our preliminary computational results indicate that the proposed strategy has significant advantages over straightforward use of commercial solvers.

On the Poisson Equation for Piecewise-Deterministic Markov Processes

In this paper we study the problem of the existence of a solution for the Poisson equation (PE) associated to a piecewise-deterministic Markov process (PDP). It is well known that the long run average cost of a stochastic process can be obtained through a solution of the PE associated with the process. Our first result will show that the existence of a solution for the PE of a PDP is equivalent to the existence of a solution for the PE of an embedded discrete- time Markov chain associated with the PDP. It is important to point out that, due to the possibility of jumps from the boundary, the differential formula for the PDPs has a special form, so that general results for the PE of continuous- time stochastic processes cannot be directly applied. Usually discrete-time conditions for the existence of a solution of a PE of a Markov chain are easier to apply than the continuous-time counterpart. We follow this approach to derive our second result, which establishes sufficient conditions for the existence of a PE to the embedded Markov chain, and consequently for the PE of the PDP. The condition is illustrated with an application to the capacity expansion model.

A bicriteria stochastic programming model for capacity expansion in telecommunications

We consider capacity expansion of a telecommunications network in the face of uncertain future demand and potential future failures of network components. The problem is formulated as a bicriteria stochastic program with recourse in which the total cost of the capacity expansion and the probability of future capacity requirements to be violated are simultaneously minimized. Assuming the existence of a finite number of possible future states of the world, an algorithm for the problem is elaborated. The algorithm determines all non-dominated solutions to the problem by a reduced feasible region method, solving a sequence of restricted subproblems by a cutting plane procedure. Computational results are reported for three different problem instances, one of which is a real-life problem faced by SONOFON, a Danish communications network operator.

Singularly Perturbed Markov Chains with Two Small Parameters: A Matched Asymptotic Expansion

This work is concerned with asymptotic properties of solutions to forward equations for singularly perturbed Markov chains with two small parameters. It is motivated by the model of a cost-minimizing firm involving production planning and capacity expansion and a two-level hierarchical decomposition. Our effort focuses on obtaining asymptotic expansions of the solutions to the forward equation. Different from previous work on singularly perturbed Markov chains, the inner expansion terms are constructed by solving certain partial differential equations. The methods of undetermined coefficients are used. The error bound is obtained.

Capacity expansion problem of multisite batch plants with production and distribution

In this study we develop a capacity expansion model of an integrated production and distribution system comprised of multisite batch plants and several warehouses. Such an integration of the production and distribution system can generate significant savings by trading off the costs associated with the whole, rather than minimizing production and distribution costs separately. The study presented here to be considered as two or more multiproduct batch plants producing several products to supply several warehouses. An objective function is defined as a net profit in a chemical company — total expected selling price minus the cost of new units and the expected shipping cost. We determine a revised plant configuration, sizes and operating modes of newly illustrated with two case studies. The case studies are solved with GAMS/DICOPT + +.

Response surface analysis of two-stage stochastic linear programming with recourse

We apply the techniques of response surface methodology (RSM) to approximate the objective function of a two-stage stochastic linear program with recourse. In particular, the objective function is estimated, in the region of optimality, by a quadratic function of the first-stage decision variables. The resulting response surface can provide valuable modeling insight, such as directions of minimum and maximum sensitivity to changes in the first-stage variables. Latin hypercube (LH) sampling is applied to reduce the variance of the recourse function point estimates that are used to construct the response surface. Empirical results show the value of the LH method by comparing it with strategies based on independent random numbers, common random numbers, and the Schruben-Margolin assignment rule. In addition, variance reduction with LH sampling can be guaranteed for an important class of two-stage problems which includes the classical capacity expansion model. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 753–776, 1999

FLEXIBILITY, ADAPTABILITY, AND EFFICIENCY IN MANUFACTURING SYSTEMS

There are many different approaches and conceptualizations of flexibility in the present literature. This lends itself to a lack of clarity and can result in confusion on the central concept. Here we identify shortcomings in the existing literature. To clarify some of the important concepts, we distinguish between flexibility (ability to change states), adaptability (ability to change within a state), and efficiency (resources utilized). To illustrate such distinctions, we present a capacity expansion model that takes flexibility and adaptability into consideration. Numerical examples show how the concepts of flexibility and adaptability can be used in manufacturing.

A continuous-time integrated model for discrete control of production flows in a multi-level bills of material environment

An integrated dynamic model for discrete production scheduling and continuous capacity expansion is presented in this paper. The modeled manufacturing system, based on multi-level bills of materials, is characterized by flexible machines with negligible setups and production rates fixed for the current capacity. Make-to-stock formulation of the problem is studied and optimal behavior of the system is determined with the help of the maximum principle. A fast time-decomposition algorithm is suggested to locate the optimal solution.

A continuous‐time integrated model for discrete control of production flows in a multi‐level bills of material environment

AbstractAn integrated dynamic model for discrete production scheduling and continuous capacity expansion is presented in this paper. The modeled manufacturing system, based on multi‐level bills of materials, is characterized by flexible machines with negligible setups and production rates fixed for the current capacity. Make‐to‐stock formulation of the problem is studied and optimal behavior of the system is determined with the help of the maximum principle. A fast time‐decomposition algorithm is suggested to locate the optimal solution.

Stability of Piecewise-Deterministic Markov Processes

In this paper, we study a form of stability for a general family of nondiffusion Markov processes known in the literature as piecewise-deterministic Markov process (PDMP). By stability here we mean the existence of an invariant probability measure for the PDMP. It is shown that the existence of such an invariant probability measure is equivalent to the existence of a $\sigma$-finite invariant measure for a Markov kernel G linked to the resolvent operator U of the PDMP, satisfying a boundedness condition or, equivalently, a Radon--Nikodým derivative. Here we generalize existing results of the literature [O. Costa, J. Appl. Prob., 27, (1990), pp. 60--73; M. Davis, Markov Models and Optimization, Chapman and Hall, 1993] since we do not require any additional assumptions to establish this equivalence. Moreover, we give sufficient conditions to ensure the existence of such a $\sigma$-finite measure satisfying the boundedness condition. They are mainly based on a modified Foster--Lyapunov criterion for the case in which the Markov chain generated by G is either recurrent or weak Feller. To emphasize the relevance of our results, we study three examples and in particular, we are able to generalize the results obtained by Costa and Davis on the capacity expansion model.

A model of clinker capacity expansion

This paper describes a model which has been applied in practice to determine an optimal plan for clinker capacity expansion. The problem was formulated as an integer linear program aiming to determine the optimal number, size and location of kilns to be introduced each year during a given planning horizon. The optimal solution is defined as the one which maximises the net present value (NPV) of the cash flow generated by the introduction and operation of new kilns subject to capacity and demand constraints. Various demand scenarios and capital expenditure structures were tested by means of the model and a number of possible locations for new kilns were evaluated. Finally, the benefits derived by the company where the application took place are listed.

Robust investment model for long-range capacity expansion of chemical processing networks under uncertain demand forecast scenarios

The problem of long-range capacity expansion planning for chemical processing networks under uncertain demand forecast scenarios is addressed. This optimization problem involves capacity expansion timing and sizing of each chemical processing unit to maximize the expected net present value while considering the deviation of net present values and the excess capacity over a given time horizon. A multiperiod mixed integer nonlinear programming optimization model that is both solution and model robust for any realization of demand scenarios is developed using the two-stage stochastic programming modeling framework. Two example problems are considered to illustrate the effectiveness of the model. Especially, the use of the model is illustrated on a real problem arising from investment planning in Korean petrochemical industry.

Global warming: least-cost electricity planning to meet CO2 emissions limits

This paper explores least-cost strategies for reducing CO2 emissions from the electric power industry. It uses an economic-engineering model for electric generation capacity expansion to investigate expansion plans which meet alternative CO2 emissions constraints in the lowest cost. The model selects the mix of various energy technologies, which are either presently in use or will possibly be in use in the future, to meet a specific carbon emissions limit and, furthermore, estimates the optimal tax required to achieve the least-cost strategy for reducing emissions to the desired level. Using Greece as a case study, the study suggests that to stabilize CO2 emissions at their 1990 level by the year 2005 and thereafter, the industry should move away from lignite generation to hydro and renewables and to coal or lignite technologies with CO2 removal capabilities. An optimal tax of $105 per ton of carbon is required to achieve this target in the lowest cost. © 1998 John Wiley & Sons, Ltd.

Electric capacity expansion under uncertain demand: decomposition approaches

The authors present a stochastic model of electric capacity expansion planning under uncertainty in demand. The goal of this problem is to determine the most interesting investments (plants and capacity levels) over the considered planning time (up to several years). Periods are divided into smaller subperiods (e.g. weekly or monthly) for which demand is assumed uncertain and modeled as a continuous probability distribution function. This leads to consider the risk associated to each decision for the capacity to be used (electricity generation). A first approach as a nonlinear continuous model is presented. Benders decomposition and Lagrangean relaxation-decomposition are proposed as solution methods, where the structures of the related sub-problems are exploited to speed up the convergence. The authors provide a large computational experience and comparisons within these methods and other general purpose optimization packages, and focus the report on the advantages of each.

Investigating strategies to reduce CO 2 emissions from the electricity sector: the case of Greece

This paper investigates strategies to reduce CO 2 emissions from the electricity sector. In particular, it explores possibilities on the supply side and the significance of demand-side changes. It uses an economic engineering model for electricity generation capacity expansion which combines stochastic simulation for production-cost calculations and dynamic programming for selecting the optimum expansion plan. The model is used to investigate optimal strategies for stabilizing CO 2 emissions from the electricity sector in Greece; these accounted for 50% of CO 2 emissions produced in the country in 1990. It is also used to estimate the optimal tax required to achieve the optimal strategy for controlling CO 2 emissions to the desired level.

Capacity Acquisition and Disposal with Discrete Facility Sizes

We consider some key features of capacity acquisition, disposal, and replacement decisions in this paper that are seldom captured in capacity expansion models in the literature. First, capacity is often purchased in the form of equipment which comes only in a few discrete sizes. Second, some or all of the capacity may be replaced periodically due to the availability of better and cheaper equipment, or due to deterioration and increased operating costs of older equipment. Finally, some capacity may be disposed due to declining demand. We present integer programming formulations that model all of these aspects. We identify special structures in these formulations that are then exploited to develop efficient procedures for solving the linear relaxation optimally. The solution to the linear relaxation together with a heuristic interchange procedure are used in a branch-and-bound procedure to obtain optimal solutions. Computational results are presented that establish the effectiveness of the procedures in solving realistic problems to optimality.

A stochastic optimization model for planning capacity expansion in a service industry under uncertain demand

A stochastic optimization model for capacity expansion for a service industry that incorporates uncertainty in future demand is developed. Based on a weighted set of possible demand scenarios, the model generates a recommended schedule of capacity expressions, and calculates the resulting sales under each scenario. The capacity schedule specifies the size, location, and timing of these expansions that will maximize the company's expected profit. The model includes a budget constraint on available resources. By using Lagrangian relaxation and exploiting the special nested knapsack structure in the sub-problems, an algorithm was developed for its solution. Based on the initial computational results, this algorithm appears to be more efficient than linear programming for this special problem. © 1994 John Wiley & Sons, Inc.

Capacity Expansion and Contraction Of A Facility With Economies Of Scale

We present a model of capacity expansion in which capacity can be either expanded or contracted over time. Our model allows for economies of scale often associated with capacity expansion. Furtherm...

Flexible versus dedicated technology: A capacity expansion model

Over the past two decades, flexible manufacturing systems have been adopted in a variety of industries. Shorter product lives have necessitated acquiring flexible facilities. Further, the economies of scope that can be derived from producing products with different seasonality have made flexible facilities attractive. At the same time, dedicated facilities (such as transfer lines) have not disappeared because they offer economies of scale beneficial in high volume manufacturing environments. Hence, when producing a wide variety of products with different seasonality and demand growth rates, firms need to consider both economies of scale and economies of scope in deciding whether to acquire capacity of flexible or dedicated technology or both. In this paper, we consider the problem of making capacity acquisition decisions, when there is a choice of flexible and dedicated technologies available in a deterministic multi-product manufacturing environment with demand assumed to be nondeclining over time. We present a novel formulation of the problem and show the equivalence of the formulation to the uncapacitated plant location formulation. This enables us to use a very efficient solution procedure (Erlenkotter 1978) to solve even large problems optimally. We then present the results of an extensive computational study performed to evaluate the impact of key problem parameters on the proportion of flexible and dedicated capacity acquired. An interesting result of the study is that the proportion of flexible capacity acquired decreases with increasing scale economies. We also find that investment in flexible technology is significant even when it is relatively more expensive than dedicated technology.

A network reformulation of an electric utility expansion planning model

We reformulate a multiperiod capacity expansion model of electric utilities as a network model. We show how to reconstruct the dual solution of the original mathematical program from the network model solution. To formulate the network model, we use information about the properties of the optimal solution of the mathematical program to reduce the number of constraints. The remaining constraints are then readily converted into network constraints. © 1993 John Wiley & Sons. Inc.