Capacity Planning Model Essay Example for Free

Capacity Planning Model Essay Abstract: Capacity planning decisions affect a signiï ¬ cant portion of future revenue. In equipment intensive industries, these decisions usually need to be made in the presence of both highly volatile demand and long capacity installation lead times. For a multiple product case, we present a continuous-time capacity planning model that addresses problems of realistic size and complexity found in current practice. Each product requires speciï ¬ c operations that can be performed by one or more tool groups. We consider a number of capacity allocation policies. We allow 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 incorporate both variance reduction techniques from the simulation literature and the principles of a generalized maximum ï ¬â€šow algorithm from the network optimization literature.  © 2005 Wiley Periodicals, Inc. Naval Research Logistics 53: 137–150, 2006 Keywords: capacity planning; stochastic demand; simulation; submodularity; semiconductor industry INTRODUCTION Because highly volatile demands and short product life cycles are commonplace in today’s business environment, capacity investments are important strategic decisions for manufacturers. In the semiconductor industry, where the proï ¬ t margins of products are steadily decreasing, manufacturers may 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 costs, and evolving technologies, as well as long capacity procurement lead times. In this paper, we study the purchasing and retirement decisions of machines (or interchangeably, â€Å"tools†). The early purchase of tools often results in unnecessary capital spending, whereas tardy purchases lead to lost revenue, especially in the early stages of the product life cycle when proï ¬ t margins are highest. The process of determining the sequence and timing of tool purchases and possibly retirements is referred to as strategic capacity planning. Our strategic capacity planning model allows for multiple products under demand uncertainty. Demand evolves 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 ï ¬ xed, product-speciï ¬ c 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 groups 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 difï ¬ cult to ï ¬ 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 ï ¬ nished good inventory because technology products, especially highly proï ¬ table ones, face rapidly declining prices and a high risk of obsolescence. In particular, building up inventories ahead of demand may not be economically sound for applicationspeciï ¬ c integrated circuits. Because high-tech products are in a sense â€Å"perishable,† we assume no ï ¬ nished goods inventory. In addition, we assume that no back-ordering is permitted for the following reasons. First, unsatisï ¬ ed demand frequently results in the loss of sales to a competitor. Second, delayed order fulï ¬ llment 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-ï ¬ 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 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 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 resolved. When demand exceeds supply, there 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 Uniform Fill-Rate Production policy equalizing the ï ¬ 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 indu stry. The ï ¬ rst thread is noted for the three-layer tool-operation-product model of capacity that we use, originating from IBM’s discrete-time formulations. 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. Çakanyildirim and Roundy [8] and Çakanyildirim, 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 ï ¬â€šoor 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 sufï ¬ cient 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 initialization 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 ï ¬â€šow 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 ï ¬ 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 organized as follows. Section 2 lays out 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 efï ¬ ciently used when the demand is a set of continuous distributions that evolve over time. We brieï ¬â€šy conclude with Section 6. 2. 2.1. MODEL Formulation Ds,p (t) Instantaneous demand of product p in scenario s at time t. Ï€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. 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 . Primary Variables Ï„m,n The time of the nth tool purchase within group m. ÃŽ ³m,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. Satisï ¬ ed 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 deï ¬ ned. 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 ï ¬ 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 avoid 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). 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.