Stochastic programming based capacity planning for semiconductor wafer fab with uncertain demand and capacity

Capacity planning is a challenging problem in semiconductor manufacturing industry due to high uncertainties both in market and manufacturing systems, short product life cycle, and expensive capital invest. To tackle this problem, this paper proposes a scenario-based stochastic programming model which considers demand and capacity uncertainties via scenarios, where the overall equipment efficiency is employed to describe the uncertain capacity for the first time. Based on the decentralized structure of tool procurement, production, stockout, and inventory decision-making processes, recourse approximation strategies are presented with varying degree of information share. The computational experiments show that the resulting tool set is robust enough to cope with the changes in capacity with the expected profits being maximized for different scenarios, and the scheme can generate pretty good solutions in reasonable computational time.     

Competitive capacity expansion under demand uncertainty

We analyze the capacity expansion behavior of firms in a duopoly faced with an uncertain new market. The market demand may be high or low with a given probability mass function. Firms obtain private information about the market size and build their capacity before the market demand is known. Once the demand is revealed, firms enter a capacity constrained price competition phase which determines their revenues. Two scenarios are considered: first, when firms choose their capacities simultaneously in the investment phase, and second, when they do so sequentially. For each case, we determine the unique symmetric Nash equilibrium. Excess capacity can occur in equilibrium in the industry. It is seen that preempting the competitor in the capacity expansion phase offers first mover benefits. We argue that the sequential moves game is more prone to equilibrium excess capacity compared to the simultaneous case. We show that preemption is a good strategy if the investing environment is either highly optimistic or highly pessimistic. If the industry outlook is only moderately optimistic, a capacity planner is still better off preempting his competitor, however, the industry may encounter overcapacity as a consequence.     

Models for planning capacity expansion in local access telecommunication networks

The rapid progress of communications technology has created new opportunities for modeling and optimizing the design of local telecommunication systems. The complexity, diversity, and continuous evolution of these networks pose several modeling challenges. In this paper, we present an overview of the local telephone network environment, and discuss possible modeling approaches. In particular, we (i) discuss the engineering characteristics of the network, and introduce terminology that is commonly used in the communications industry and literature; (ii) describe a general local access network planning model and framework, and motivate different possible modeling assumptions; (iii) summarize various existing planning models in the context of this framework; and (iv) describe some new modeling approaches. The discussion in this paper is directed both to researchers interested in modeling local telecommunications systems and to planners interested in using such models. Our goal is to present relevant aspects of the engineering environment for local access telecommunication networks, and to discuss the relationship between engineering issues and the formulation of economic decision models. We indicate how changes in the underlying switching and transmission technology affect the modeling of the local telephone network. We also review various planning issues and discuss possible optimization approaches for treating them.This research was initiated through a grant from GTE Laboratories, IncorporatedSupported in part by an AT&T research award.Supported in part by Grant No. ECS-8316224 from the Systems Theory and Operations Research Program of the National Science Foundation.     

Tool capacity planning for semiconductor fabrication facilities under demand uncertainty

This research is motivated by issues faced by a large manufacturer of semiconductor devices. Semiconductor manufacturing companies allocate millions of dollars every year for new types of machine tools for their facilities. Typically these are special purpose machine tools which are made to order. The rate of change in products and technology makes it difficult for manufacturers to have a good estimate of future tool requirements. Further, manufacturers experience a long lead time while procuring these tools. In this paper, we model the tool capacity planning problem under uncertainty in demand. The number of tools required in a facility is sufficiently large (nearly hundred or more tools) to make it nearly impossible to obtain efficient exact algorithms. We provide heuristics to find efficient tool procurement plans and test their quality using lower bounds on the formulation.     

Singular value decomposition expansion for electrical demand analysis

Email: fan.li{at}ngc.co.uk To analyse electricity-demand data, appropriate mathematicalmodels and algorithms have been developed. Some theoreticalproperties of the singular value decomposition (SVD) and SVDexpansion have been found very useful during these developments.These properties, which have generic implications in data miningand numerical analysis, are presented in this paper. Followinga discussion of the theoretical development, this paper reportsapplications of the SVD expansion in demand analysis and forecastingwith three illustrative practical algorithms, that have beendeveloped by the National Grid Company in recent years.     

Modelling and analysis for capacity expansion planning in warehousing

In this paper, a dynamic programming model is developed for the purpose of establishing a warehouse capacity expansion schedule and underlying multi-item inventory policy that are jointly optimal. The optimal warehouse size over any segment of the planning horizon is obtained by solving a nonlinear optimisation problem, this being accomplished efficiently by exploiting the Kuhn–Tucker conditions. Repeating this procedure between each pair of time periods results in a discrete state space, so that the optimal capacity expansion schedule corresponds to a shortest path in a network. Managerial insights are provided through experimentation with the model.     

Optimization of capacity expansion planning for gas transportation networks

This paper presents techniques for solving the problem of minimizing investment costs on an existing gas transportation network. The goal of this program is to find, first, the optimal location of pipeline segments to be reinforced and, second, the optimal sizes (among a discrete commercial list of diameters) under the constraint of satisfaction of demands with high enough pressure for all users.     

The value of information for price dependent demand

Predicting demand and determining optimal pricing are essential components of operations management. It is often useful to think in terms of the price elasticity of demand when reasoning about the demand curve. Firms wishing to invest in demand prediction and information gathering should reason about the relationship between the expected value of perfect information (EVPI) on demand and demand elasticity. Should firms pay more/less for information on demand if elasticity is high/low? Furthermore, when considering different product prices, correlation may exist between demand at different prices. Should firms pay more/less for information if the correlation between demand at different prices is high or low? This paper derives analytic and numeric results to answer these questions. We start with the assumption that demand is uncertain and follows a uniformly distributed band around a deterministic demand curve where the upper and lower bounds of the demand distribution vary with price. This formulation enables a closed form expression for EVPI that provides a useful benchmark. We find nuanced behavior of EVPI that depends on both the elasticity and the initial price preference. The EVPI approaches zero as elasticity increases (decreases) for a firm that initially prefers the low (high) price. Numerical results using the truncated normal and beta distributions relax assumptions about the uniform distribution and show EVPI is similar when the distribution variances are similar. Finally, we relax the assumption of perfect information and show the expected value of imperfect information (EVOI) follows similar patterns as EVPI with respect to demand elasticity.     

Outsourcing and capacity planning in an uncertain global environment

Firms employing offshore outsourcing strategies may face both exchange rate and demand uncertainties. In this paper, we show that the firms may benefit from operational option to switch production by keeping capacities with both domestic and foreign suppliers. The value of the operational option increases as the exchange rate uncertainty or demand uncertainty increases. In addition, when the firms become risk-averse, they may use domestic capacity to hedge against offshore capacity. As a result, the firms may choose to hold local capacity even if it exhibits negative marginal contribution to the profit. Furthermore, risk-averse firms may keep more total capacity than risk-neutral firms.     

Doubly uncertain transportation network: Degradable capacity and stochastic demand

This study developed a methodology to model doubly uncertain transportation network with stochastic link capacity degradation and stochastic demand. We consider that the total travel demand comprises of two parts, infrequent travelers and commuters. The traffic volume of infrequent travelers is stochastic, which adds to the network traffic in a random manner based on fixed route choice proportions. On the other hand, the traffic volume of commuters is stable or deterministic. Commuters acquire the network travel time variability from past experiences, factor them into their route choice considerations, and settle into a long-term habitual route choice equilibrium in which they have no incentive of switching away. To define this equilibrium, we introduce the notion of “travel time budget” to relate commuters’ risk aversion on route choices in the presence of travel time variability. The travel time budget varies among commuters according to their degrees of risk aversion and requirements on punctual arrivals. We then developed a mixed-equilibrium formulation to capture these stochastic considerations and illustrated its properties through some numerical studies.     

A simple recourse model for power dispatch under uncertain demand

Optimal power dispatch under uncertainty of power demand is tackled via a stochastic programming model with simple recourse. The decision variables correspond to generation policies of a system comprising thermal units, pumped storage plants and energy contracts. The paper is a case study to test the kernel estimation method in the context of stochastic programming. Kernel estimates are used to approximate the unknown probability distribution of power demand. General stability results from stochastic programming yield the asymptotic stability of optimal solutions. Kernel estimates lead to favourable numerical properties of the recourse model (no numerical integration, the optimization problem is smooth convex and of moderate dimension). Test runs based on real-life data are reported. We compute the value of the stochastic solution for different problem instances and compare the stochastic programming solution with deterministic solutions involving adjusted demand portions.This research is supported by the Schwerpunktprogramm Anwendungsbezogene Optimierung und Steuerung of the Deutsche Forschungsgemeinschaft.     

A primal-dual subgradient method for time staged capacity expansion planning

We model capacity expansion problems as nondifferentiable convex programs. A dual to this problem is also formulated as a nondifferentiable convex program. The solution methodology we utilize is a primal-dual subgradient approach. Here the primal-dual pair is utilized in step length determination. The special structure of these capacity expansion problems also results in other simplifications. In particular, unlike the application of subgradient optimization for general convex programs, the test for feasibility in certain capacity expansion problems is straightforward. Further, quadratic programs associated with projection operators are also avoided by using the special problem structure. The algorithm is shown to be convergent. In order to illustrate the applicability of our methodology, we discuss its application to a time dynamic power generation capacity planning problem. Computational results with this application is also provided.     

A non-stationary periodic review production–inventory model with uncertain production capacity and uncertain demand

In this paper we consider a nonstationary periodic review dynamic production–inventory model with uncertain production capacity and uncertain demand. The maximum production capacity varies stochastically. It is known that order up-to (or base-stock, critical number) policies are optimal for both finite horizon problems and infinite horizon problems. We obtain upper and lower bounds of the optimal order up-to levels, and show that for an infinite horizon problem the upper and the lower bounds of the optimal order up-to levels for the finite horizon counterparts converge as the planning horizons considered get longer. Furthermore, under mild conditions the differences between the upper and the lower bounds converge exponentially to zero.     

Bounds on the value of information in uncertain decision problems

The cost of obtaining good information regarding the various probability distributions needed for the solution of most stochastic decision problems is considerable. It is important to consider questions such as: (1) what minimal amounts of information are sufficient to determine optimal decision rules; (2) what is the value of obtaining knowledge of the actual realization of the random vectors; and (3) what is the value of obtaining some partial information regarding the actual realization of the random vectors. This paper is primarily concerned with questions two and three when the decision maker has an a priori knowledge of the joint distribution function of the random variables. Some remarks are made regarding results along the lines of question one. Mention is made of assumptions sufficient so that knowledge of means, or of means, variances, co-variances and n-moments are sufficient for the calculation of optimal decision rules. The analysis of the second question leads to the development of bounds on the value of perfect information. For multiperiod problems it is important to consider when the perfect information is available. Jensen's inequality is the key tool of the analysis. The calculation of the bounds requires the solution of nonlinear programs and the numerical evaluation of certain functions. Generally speaking, tighter bounds may be obtained only at the expense of additional information and computational complexity. Hence, one may wish to compute some simple bounds to decide upon the advisability of obtaining more information. For the analysis of the value of partial information it is convenient to introduce the notion of a signal. Each signal represents the receipt of certain information, and these signals are drawn from a given probability distribution. When a signal is received, it alters the decision maker's perception of the probability distributions inherent in his decision problem. The choice between different information structures must then take into account these probability distributions as well as the decision maker's preference function. A hierarchy of bounds may be determined for partial information evaluation utilizing the tools of the multiperiod perfect information case. However, the calculation of these bounds is generally considerably more dicult than the calculation of similar boulids in the perfect information case. Most of the analysis is directed towards problems in which the decision maker has a linear utility function over profits, costs or some other numerical variable. However, some of the bounds generalize to the case when the utility function is strictly increasing and concave.     

Two-stage workforce planning under demand fluctuations and uncertainty

We address a multi-category workforce planning problem for functional areas located at different service centres, each having office-space and recruitment capacity constraints, and facing fluctuating and uncertain workforce demand. A deterministic model is initially developed to deal with workforce fluctuations based on an expected demand profile over the horizon. To hedge against the demand uncertainty, we also propose a two-stage stochastic program, in which the first stage makes personnel recruiting and allocation decisions, while the second stage reassigns workforce demand among all units. A Benders’ decomposition-based algorithm is designed to solve this two-stage stochastic mixed-integer program. Computational results based on some practical numerical experiments are presented to provide insights on applying the deterministic versus the stochastic programming approach, and to demonstrate the efficacy of the proposed algorithm as compared with directly solving the model using its deterministic equivalent.     

On capacity expansion planning under strategic and operational uncertainties based on stochastic dominance risk averse management

A new scheme for dealing with uncertainty in scenario trees is presented for dynamic mixed 0–1 optimization problems with strategic and operational stochastic parameters. Let us generically name this type of problems as capacity expansion planning (CEP) in a given system, e.g., supply chain, production, rapid transit network, energy generation and transmission network, etc. The strategic scenario tree is usually a multistage one, and the replicas of the strategic nodes root structures in the form of either a special scenario graph or a two-stage scenario tree, depending on the type of operational activity in the system. Those operational scenario structures impact in the constraints of the model and, thus, in the decomposition methodology for solving usually large-scale problems. This work presents the modeling framework for some of the risk neutral and risk averse measures to consider for CEP problem solving. Two types of risk averse measures are considered. The first one is a time-inconsistent mixture of the chance-constrained and second-order stochastic dominance (SSD) functionals of the value of a given set of functions up to the strategic nodes in selected stages along the time horizon, The second type is a strategic node-based time-consistent SSD functional for the set of operational scenarios in the strategic nodes at selected stages. A specialization of the nested stochastic decomposition methodology for that problem solving is outlined. Its advantages and drawbacks as well as the framework for some schemes to, at least, partially avoid those drawbacks are also presented.     

Short-term liner ship fleet planning with container transshipment and uncertain container shipment demand

This paper proposes a short-term liner ship fleet planning problem by taking into account container transshipment and uncertain container shipment demand. Given a liner shipping service network comprising a number of ship routes, the problem is to determine the numbers and types of ships required in the fleet and assign each of these ships to a particular ship route to maximize the expected value of the total profit over a short-term planning horizon. These decisions have to be made prior to knowing the exact container shipment demand, which is affected by some unpredictable and uncontrollable factors. This paper thus formulates this realistic short-term planning problem as a two-stage stochastic integer programming model. A solution algorithm, integrating the sample average approximation with a dual decomposition and Lagrangian relaxation approach, is then proposed. Finally, a numerical example is used to evaluate the performance of the proposed model and solution algorithm.     

Stochastic and minimum regret formulations for transmission network expansion planning under uncertainties

After deregulation of the Power sector, uncertainty has increased considerably. Vertically integrated utilities were unbundled into independent generation, transmission and distribution companies. Transmission network expansion planning (TNEP) is now performed independent from generation planning. In this environment TNEP must include uncertainties of the generation sector as well as its own. Uncertainty in generation costs affecting optimal dispatch and uncertainty in demand loads are captured through composite scenarios. Probabilities are assigned to different scenarios. The effects of these uncertainties are transferred to the objective function in terms of total costs, which include: generation (dispatch), transmission expansion and load curtailment costs. Two formulations are presented: stochastic and minimum regret. The stochastic formulation seeks a design with minimum expected cost. The minimum regret formulation seeks a design with robust performance in terms of variance of the operational costs. Results for a test problem and a potential application to a real system are presented.     

A multi-objective facility location model for closed-loop supply chain network under uncertain demand and return

A closed-loop supply chain (CLSC) network consists of both forward and reverse supply chains. In this paper, a CLSC network is investigated which includes multiple plants, collection centres, demand markets, and products. To this aim, a mixed-integer linear programming model is proposed that minimizes the total cost. Besides, two test problems are examined. The model is extended to consider environmental factors by weighed sums and ε-constraint methods. In addition, we investigate the impact of demand and return uncertainties on the network configuration by stochastic programming (scenario-based). Computational results show that the model can handle demand and return uncertainties, simultaneously.     

Optimal control for uncertain discrete-time singular systems under expected value criterion

Fuzzy Optimization and Decision Making - Optimal control problems governed by two different types of uncertain discrete-time singular systems are investigated under expected value criterion. The...