Capacity Planning Model Essay Example for Free

expertness Planning Model EssayAbstract capacitor supplying decisions affect a signi placet per centum of future revenue. In equipment intensive industries, these decisions usually need to be made in the presence of both highly volatile contend and long condenser installation lead quantifys. For a multiple merchandise case, we present a invariable- conviction content readiness simulation that solicites problems of realistic size and complexity found in current practice. Each overlap requires specic functions that can be performed by one or more joyride groups. We consider a number of capacity allocation policies. We allow animate being hideaways in addition to leveragings beca hold the stochastic film forecast for each point of intersection can be decreasing. We present a cluster-establish heuristic algorithm that can incorporate both variance reduction techniques from the simulation literary works and the principles of a generalized utmost ow algorithm f rom the network optimization literature. 2005 Wiley Periodicals, Inc. Naval Research Logistics 53 137150, 2006 Keywords capacity plan stochastic demand simulation submodularity semiconductor industryINTRODUCTIONBecause highly volatile demands and short product life cycles be commonplace in todays business environment, capacity investments are crucial 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, hike costs, and evolving technologies, as well as long capacity procurement lead times. In this paper, we study the acquire and retirement decisions of machines (or interchangeably, tools).The early purchase of tools often results in unnecessary detonator spending, whereas tardy purchases lead to deep in thought(p) revenue, especially in the early sta ges of the product life cycle when prot margins are highest. The swear out of determining the sequence and timing of tool purchases and possibly retirements is referred to as strategic capacity planning. Our strategic capacity planning assume allows for multiple products under demand uncertainty. Demand evolves over time and is pretenseed 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 whatsoever tool. The time involve regards on both the operation and the tool.In our model time is a continuous variable, as opposed to the more traditional approach of using clear-cut time buckets. Our primary decision variables, one for each potential tool purcha se or retirement, argue the timing of the corresponding actions. In contrast, decision variables in typical discrete-time models are either binary or whole number and are indexed by both tool groups and time periods. Our objective is to minimize the sum of the muzzy 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 difcult to nd global optimal solutions for the resulting continuous optimization problem. Many manufacturers, primarily those in high-tech industries, prefer to maintain a negligible amount of nished good inventory because technology products, especially highly protable ones, prospect rapidly declining prices and a high risk of obsolescence. In particular, building up inventories ahead of demand may not be economically sound for applicationspecic integrated circuits.Because high-tech products are in a sense perishable, we assume no nished goods inventory. In addition, we assume that no back-ordering is permitted for the following reasons. First, unsatised demand much results in the loss of sales to a competitor. Second, delayed order fulllment often results in either the abate 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 jiffy of time.In the case of multiple products, the aggregate capacity is divided among these products according to a particular policy. This tool-groups-to-products allocation is referred to as tactical production planning. While purchase and retirement decisi ons 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 resolved. When demand exceeds supply, there are two plausible allocation policies for assigning the capacity to products (i) the befogged Sales Cost Minimization policy minimizing instantaneous lost sales cost and (ii) the Uniform Fill-Rate employment 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 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 looks. Bermon and Hood 6 assume deterministic demand, which is later extended by Barahona et al. 4 to model scenario-based demand uncertainty. Barahona et al. 4 have a large number of indicator 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 standard 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. Other 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 akanyi ldirim, 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 preference 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 initialization method, delivers good solutions and bonny computation times.Furthermore, preliminary computational results indicate that capacity plans are not very sensitive to the plectrum of allocation policy, and both policies perform comparably. With the Uniform FillRate Production policy, an instantaneous revenue calculation that is used repeatedly by the subrout ines 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 appraisal turning away 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 three-card 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 organized as follows. Section 2 lays out our strategic cap acity formulation under two capacity allocation policies. Section 3 describes our heuristic algorithm, and its computational results are account 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) instant(prenominal) demand of product p in scenario s at time t. s Probability of scenario s. We buy the farm 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. Note 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 . Prim ary 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 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 index ed 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 productive tool capacities available. To void confusion, we refer to the duration of operations or tool capacities available at a given moment of time using the phrase machine-hours. Input Data bw,p Number of operations of type w required to produce a unit of product p (typically integer, but fractional values are allowed). come of machine-hours required by a tool in group m to perform operation w. append 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 pri ce 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.