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Capacity Planning Model EssayAbstract Capacity planning stopping points affect a signi raiset portion of future revenue. In equipment intense industries, these decisions usually need to be made in the presence of both highly volatile demand and long depicted object installation lead condemnations. For a multiple produce case, we present a continuous- term energy planning lay that addresses problems of realistic size and complexity found in on-line(prenominal) practice. from each one product requires specic military actions that can be consummateed by one or more tool assemblages. We consider a number of capacity allocation policies. We waive tool retirements in addition to purchases because the stochastic demand forecast for each product can be decreasing. We present a cluster-based heuristic algorithm that can incorpo order both variance reduction techniques from the simulation literature and the principles of a generalized maximum ow algorithm from the network o ptimization literature. 2005 Wiley Periodicals, Inc. Naval Research Logistics 53 137150, 2006 Keywords capacity planning stochastic demand simulation submodularity semiconductor industryINTRODUCTIONBecause highly volatile demands and short product life cycles are everyday in todays business environment, capacity investments are important strategic decisions for manufacturers. In the semiconductor industry, where the prot margins of products are steadily decreasing, manufacturers whitethorn spend up to 3.5 billion dollars for a state-of-the-art plant 3, 23. The capacity decisions are complicated by volatile demands, rising comprises, and evolving technologies, as well as long capacity procurement lead times. In this composing, we plain the purchasing and retirement decisions of machines (or interchangeably, tools).The early purchase of tools often results in unnecessary great(p) spending, whereas drowsy purchases lead to confounded revenue, especially in the early stages of t he product life cycle when prot margins are highest. The process of determining the sequence and clock of tool purchases and possibly retirements is referred to as strategic capacity planning. Our strategic capacity planning model allows for multiple products under demand un authenticty. requirement germinates over time and is modeled by a set of scenarios with associated Correspondence to W.T. Huh (emailprotected) 2005 Wiley Periodicals, Inc. probabilities. We allow for the possibility of decreasing demand. Our model of capacity consumption is based on three layers tools (i.e., machines), operations, and products. Each product requires a xed, product-specic set of operations. Each operation can be performed on any tool. The time required depends on both the operation and the tool.In our model time is a continuous variable, as opposed to the more traditional approach of using discrete time buckets. Our primary decision variables, one for each potential tool purchase or retirement, indicate the timing of the corresponding actions. In contrast, decision variables in typical discrete-time models are either binary or integer and are indexed by both tool root words and time periods. Our objective is to minimize the sum of the lost sales cost and the capital cost, each a function of tool purchase times and retirement times. Our continuous-time model has the advantage of having a smaller number of variables, although it may be dif cultus to nd global best solutions for the resulting continuous optimization problem. Many manufacturers, primarily those in high-tech industries, prefer to maintain a negligible amount of nished intimately inventory because technology products, especially highly protable ones, face rapidly declining prices and a high risk of obsolescence. In particular, building up inventories onwards of demand may not be economically sound for applicationspecic integrated circuits.Because high-tech products are in a sense perishable, we assume no n ished goods inventory. In addition, we assume that no back-ordering is permitted for the following reasons. First, unsatised demand frequently results in the loss of sales to a competitor. Second, delayed order fulllment often results in either the decrease or the postponement of future demand. The end result approximates a lost sale. We remark that these assumptions of no-nishedgoods and no back-ordering are also applicable to certain service industries and utility industries, in which systems do not have any buffer and require the co-presence of capacity and demand. These assumptions simplify the computation of instantaneous production and lost sales since they depend only on the current demand and capacity at a given moment of time.In the case of multiple products, the compound capacity is divided among these products according to a particular policy. This tool-radicals-to-products allocation is referred to as tactical production planning. While purchase and retirement decisions are made at the beginning of the planning horizon prior to the realization of stochastic demand, allocation decisions are recourse decisions made after demand uncertainty has been re kneadd. When demand exceeds supply, on that point are two plausible allocation policies for assigning the capacity to products (i) the Lost Sales Cost Minimization policy minimizing instantaneous lost sales cost and (ii) the undifferentiated Fill-Rate Production policy equalizing the ll-rates of all products. Our model primarily uses the former, but can easily be extended to use the latter. Our model is directly related to to two threads of strategic capacity planning models, both of which address problems of realistic size and complexity arising in the semiconductor industry.The rst thread is noted for the three-layer tool-operation-product model of capacity that we use, originating from IBMs discrete-time grammatical constructions. Bermon and Hood 6 assume deterministic demand, which is later ext ended by Barahona et al. 4 to model scenario-based demand uncertainty. Barahona et al. 4 have a large number of index number variables for discrete expansion decisions, which results in a large mixed integer programming (MIP) formulation. Standard MIP computational methods such as branch-and-bound are used to solve this challenging problem.Our model differs from this work in the following ways (i) using continuous variables, we use a descent-based heuristic algorithm as an alternative to the ensample MIP techniques, (ii) we model tool retirement in addition to acquisition, and (iii) we consider the capital cost in the objective function instead of using the budget constraint. separate notable examples of scenario-based models with binary decisions variables include Escudero et al. 15, Chen, Li, and Tirupati 11, Swaminathan 27, and Ahmed and Sahinidis 1 however, they do not model the operations layer explicitly.The second thread of the relevant literature features continuous-time models. akanyildirim and Roundy 8 and akanyildirim, Roundy, and Wood 9 both study capacity planning for several tool groups for the stochastic demand of a single product. The former establishes the optimality of a bottleneck policy where tools from the bottleneck tool group are installed during expansions and retired during contractions in the reverse order. The latter uses this policy to jointly optimize tool expansions along with nested oor and space expansions. Huh and Roundu 18 extend these ideas to a multi-product case under the Uniform Fill-Rate Production policy and identify a set of sufcient conditions for the capacity planning problem to be reduced to a nonlinear convex minimization program. This paper extends their model by introducing the layer of operations, the Lost Sales Cost Minimization allocation policy and tool retirement.This results in the non-convexity of the resulting formulation. Thus, our model marries the continuous-time paradigm with the complexity of real- world capacity planning. We list a selection of recent papers on capacity planning. Davis et al. 12 and Anderson 2 take an optimal control theory approach, where the control decisions are expansion rate and workforce capacity, respectively. Ryan 24 incorporates autocorrelated product demands with drift into capacity expansion. Ryan 25 minimizes capacity expansion costs using option pricing formulas to estimate shortages. Also, Birge 7 uses option theory to study capacity shortages and risk. An extensive survey of capacity planning models is found in the article by Van Mieghem 28. Our computational results suggest that the descent algorithm, with a proper initialisation method, delivers good solutions and reasonable computation times.Furthermore, preliminary computational results indicate that capacity plans are not very sensitive to the choice of allocation policy, and both policies perform comparably. With the Uniform FillRate Production policy, an instantaneous revenue calculation that is used repeatedly by the subroutines of the heuristic algorithm can be formulated as a generalized maximum ow problem the solution of this problem can be obtained by a combinatorial polynomial-time approximation scheme that results in a potentially dramatic increase in the speed of our algorithm.We assume that the stochastic demand is given as a nite set of scenarios. This demand model is consistent with current practice in the semiconductor industry. We also explore, in Section 5, the possibility that demand is instead given as a continuous distribution, e.g., the Semiconductor Demand Forecast Accuracy Model 10. Borrowing results from the literature on Monte Carlo approximations of stochastic programs, we point out the existence of an inherent bias in the optimal cost of the approximation when the scenario sample size is small. We also describe applicable variance reduction techniques when samples are drawn on an ad hoc basis.This paper is form as follows. Section 2 lays ou t our strategic capacity formulation under two capacity allocation policies. Section 3 describes our heuristic algorithm, and its computational results are reported in Section 4. Section 5 presents how our software can be efciently used when the demand is a set of continuous distributions that evolve over time. We briey conclude with Section 6. 2. 2.1. MODEL FormulationDs,p (t) Instantaneous demand of product p in scenario s at time t. s Probability of scenario s. We eliminate subscripts to construct vectors or matrices by listing the argument with different products p, operations w, and/or tool indices m. For example, B = (bw,p ) is the production-to-operation matrix and H = (hm,w ) is the machine-hours-per-operation matrix. scar that we concatenate only p, w, or m indices. Thus, Ds (t) = (Ds,p (t)) for demand in scenario s, and c(t) = (cp (t)) for per-unit lost sales cost vectors at time t.We assume the continuity of cp P R and Ds,p and the continuous differentiability of Pm and Pm . Primary Variables m,n The time of the nth tool purchase within group m. m,n The time of the nth tool retirement within group m. Auxiliary Variables Xs,w,m (t) Number of products that pass through operation w on tool group m in scenario s at time t. Capacity of tool group m at time t. Unmet demand of product p in scenario s at time t. Satised demand of product p in scenario s at time t. Thus, V s,t (t) = Ds,p (t) Vs,p (t).Let the continuous variable t represent a time between 0 and T , the end of the planning horizon. We use p, w, and m to index product families in P, operations in W, and tool groups in M, respectively. all told tools in a tool group are identical this is how tool groups are actually dened. We denote by M(w) the set of tools that can perform operation w and by W (m) the set of operations that tool group m can perform. DurP R ing the planning horizon, we purchase Nm (retire Nm ) tools 1 belonging to tool group m. Purchases or retirements of tools P R in a tool group are indexed by n, 1 n Nm , or 1 n Nm . Random demand for product p is given by Dp (t) = Ds,p (t), where s indexes a nite number of scenarios S. Our formulation uses input data and variables presented below.We reserve the usage of the word time for the calendar time t, as opposed to the processing duration of operations or deep tool capacities available. To avoid confusion, we refer to the duration of operations or tool capacities available at a given moment of time using the style machine-hours. Input Data bw,p Number of operations of type w required to produce a unit of product p (typically integer, but fractional determine are allowed). Amount of machine-hours required by a tool in group m to perform operation w. Total capacity (productive machine-hours per month) of tool group m at the beginning of the time horizon. Capacity of each tool in group m (productive machine-hours per month). Purchase price of a tool in group m at time t (a function of the continuous scalar t). Sale price for retiring a tool in group m at time t. May be positive or negative. Per-unit lost sales cost for product p at time t.