Abstraction: Capacity planning determinations affect a signi?cant part of future gross. In equipment intensive industries. these determinations normally need to be made in the presence of both extremely volatile demand and long capacity installing lead times. For a multiple merchandise instance. we present a continuous-time capacity planning theoretical account that addresses jobs of realistic size and complexness found in current pattern. Each merchandise requires speci?c operations that can be performed by one or more tool groups. We consider a figure of capacity allotment policies. We allow tool retirements in add-on to purchases because the stochastic demand prognosis for each merchandise can be diminishing. We present a cluster-based heuristic algorithm that can integrate both discrepancy decrease techniques from the simulation literature and the rules of a generalised upper limit ?ow algorithm from the web optimisation literature. © 2005 Wiley Periodicals. Inc. Naval Research Logistics 53: 137–150. 2006 Keywords: capacity planning ; stochastic demand ; simulation ; submodularity ; semiconducting material industry

Introduction

Because extremely volatile demands and short merchandise life rhythms are platitude in today’s concern environment. capacity investings are of import strategic determinations for makers. In the semiconducting material industry. where the pro?t borders of merchandises are steadily diminishing. makers may pass up to 3. 5 billion dollars for a state-of-the-art works [ 3. 23 ] . The capacity determinations are complicated by volatile demands. lifting costs. and germinating engineerings. every bit good as long capacity procurance lead times. In this paper. we study the buying and retirement determinations of machines ( or interchangeably. “tools” ) .

The early purchase of tools frequently consequences in unneeded capital disbursement. whereas belated purchases lead to lost gross. particularly in the early phases of the merchandise life rhythm when pro?t borders are highest. The procedure of finding the sequence and timing of tool purchases and perchance retirements is referred to as strategic capacity planning. Our strategic capacity be aftering theoretical account allows for multiple merchandises under demand uncertainness. Demand evolves over clip and is modeled by a set of scenarios with associated Correspondence to: W. T. Huh ( [ email protected ]) © 2005 Wiley Periodicals. Inc. chances. We allow for the possibility of diminishing demand. Our theoretical account of capacity ingestion is based on three beds: tools ( i. e. . machines ) . operations. and merchandises. Each merchandise requires a ?xed. product-speci?c set of operations. Each operation can be performed on any tool. The clip required depends on both the operation and the tool.

In our theoretical account clip is a uninterrupted variable. as opposed to the more traditional attack of utilizing distinct clip pails. Our primary determination variables. one for each possible tool purchase or retirement. bespeak the timing of the corresponding actions. In contrast. determination variables in typical discrete-time theoretical accounts are either binary or whole number and are indexed by both tool groups and clip periods. Our aim is to minimise the amount of the lost gross revenues cost and the capital cost. each a map of tool purchase times and retirement times. Our continuous-time theoretical account has the advantage of holding a smaller figure of variables. although it may be dif?cult to ?nd planetary optimum solutions for the ensuing uninterrupted optimisation job. Many makers. chiefly those in hi-tech industries. prefer to keep a negligible sum of ?nished good stock list because engineering merchandises. particularly extremely pro?table 1s. face quickly worsening monetary values and a high hazard of obsolescence. In peculiar. constructing up stock lists in front of demand may non be economically sound for applicationspeci?c integrated circuits.

Because hi-tech merchandises are in a sense “perishable. ” we assume no ?nished goods stock list. In add-on. we assume that no back-ordering is permitted for the undermentioned grounds. First. unsatis?ed demand often consequences in the loss of gross revenues to a rival. Second. delayed order ful?llment frequently consequences in either the lessening or the delay of future demand. The terminal consequence approximates a lost sale. We remark that these premises of no-?nishedgoods and no back-ordering are besides applicable to certain service industries and public-service corporation industries. in which systems do non hold any buffer and necessitate the co-presence of capacity and demand. These premises simplify the calculation of instantaneous production and lost gross revenues since they depend merely on the current demand and capacity at a given minute of clip.

In the instance of multiple merchandises. the aggregative capacity is divided among these merchandises harmonizing to a peculiar policy. This tool-groups-to-products allotment is referred to as tactical production planning. While purchase and retirement determinations are made at the beginning of the planning skyline prior to the realisation of stochastic demand. allotment determinations are recourse determinations made after demand uncertainness has been resolved. When demand exceeds supply. there are two plausible allotment policies for delegating the capacity to merchandises: ( I ) the Lost Gross saless Cost Minimization policy minimising instantaneous lost gross revenues cost and ( two ) the Uniform Fill-Rate Production policy equalising the ?ll-rates of all merchandises. Our theoretical account chiefly uses the former. but can easy be extended to utilize the latter. Our theoretical account is straight related to two togss of strategic capacity planning theoretical accounts. both of which reference jobs of realistic size and complexness arising in the semiconducting material industry.

The ?rst yarn is noted for the three-layer tool-operation-product theoretical account of capacity that we use. arising from IBM’s discrete-time preparations. Bermon and Hood [ 6 ] assume deterministic demand. which is subsequently extended by Barahona et Al. [ 4 ] to pattern scenario-based demand uncertainness. Barahona et Al. [ 4 ] have a big figure of index variables for distinct enlargement determinations. which consequences in a big assorted whole number scheduling ( MIP ) preparation. Standard MIP computational methods such as branch-and-bound are used to work out this challenging job.

Our theoretical account differs from this work in the undermentioned ways: ( I ) utilizing uninterrupted variables. we use a descent-based heuristic algorithm as an option to the standard MIP techniques. ( two ) we model tool retirement in add-on to acquisition. and ( three ) we consider the capital cost in the nonsubjective map alternatively of utilizing the budget restraint. Other noteworthy illustrations of scenario-based theoretical accounts with binary determinations variables include Escudero et Al. [ 15 ] . Chen. Li. and Tirupati [ 11 ] . Swaminathan [ 27 ] . and Ahmed and Sahinidis [ 1 ] ; nevertheless. they do non pattern the operations layer explicitly.

The 2nd yarn of the relevant literature features continuous-time theoretical accounts. Cakanyildirim and Roundy [ 8 ] and Cakanyildirim. Roundy. and Wood [ 9 ] both survey capacity planning for several tool groups for the stochastic demand of a individual merchandise. The former establishes the optimality of a constriction policy where tools from the constriction tool group are installed during enlargements and retired during contractions in the contrary order. The latter uses this policy to jointly optimise tool enlargements along with nested ?oor and infinite enlargements. Huh and Roundu [ 18 ] extend these thoughts to a multi-product instance under the Uniform Fill-Rate Production policy and place a set of suf?cient conditions for the capacity planning job to be reduced to a nonlinear bulging minimisation plan. This paper extends their theoretical account by presenting the bed of operations. the Lost Gross saless Cost Minimization allotment policy and tool retirement.

This consequences in the non-convexity of the ensuing preparation. Therefore. our theoretical account marries the continuous-time paradigm with the complexness of real-world capacity planning. We list a choice of recent documents on capacity planning. Davis et Al. [ 12 ] and Anderson [ 2 ] take an optimum control theory attack. where the control determinations are enlargement rate and work force capacity. severally. Ryan [ 24 ] incorporates autocorrelated merchandise demands with impetus into capacity enlargement. Ryan [ 25 ] minimizes capacity enlargement costs utilizing option pricing expressions to gauge deficits. Besides. Birge [ 7 ] utilizations option theory to analyze capacity deficits and hazard. An extended study of capacity planning theoretical accounts is found in the article by Van Mieghem [ 28 ] . Our computational consequences suggest that the descent algorithm. with a proper low-level formatting method. delivers good solutions and sensible calculation times.

Furthermore. preliminary computational consequences indicate that capacity programs are non really sensitive to the pick of allotment policy. and both policies perform comparably. With the Uniform FillRate Production policy. an instantaneous gross computation that is used repeatedly by the subprograms of the heuristic algorithm can be formulated as a generalised upper limit ?ow job ; the solution of this job can be obtained by a combinative polynomial-time estimate strategy that consequences in a potentially dramatic addition in the velocity of our algorithm.

We assume that the stochastic demand is given as a ?nite set of scenarios. This demand theoretical account is consistent with current pattern in the semiconducting material industry. We besides explore. in Section 5. the possibility that demand is alternatively given as a uninterrupted distribution. e. g. . the Semiconductor Demand Forecast Accuracy Model [ 10 ] . Borrowing consequences from the literature on Monte Carlo estimates of stochastic plans. we point out the being of an built-in prejudice in the optimum cost of the estimate when the scenario sample size is little. We besides describe applicable discrepancy decrease techniques when samples are drawn on an ad hoc footing.

This paper is organized as follows. Section 2 lays out our strategic capacity preparation under two capacity allotment policies. Section 3 describes our heuristic algorithm. and its computational consequences are reported in Section 4. Section 5 presents how our package can be ef?ciently used when the demand is a set of uninterrupted distributions that evolve over clip. We brie?y conclude with Section 6. 2. 2. 1. MODEL Formulation

Ds. P ( T ) Instantaneous demand of merchandise P in scenario s at clip t. ?s Probability of scenario s. We eliminate inferiors to build vectors or matrices by naming the statement with different merchandises p. operations w. and/or tool indices m. For illustration. Bacillus: = ( biological warfare. P ) is the production-to-operation matrix and H: = ( hectometer. tungsten ) is the machine-hours-per-operation matrix. Note that we concatenate merely p. w. or m indices. Therefore. Ds ( T ) = ( Ds. P ( T ) ) for demand in scenario s. and degree Celsius ( T ) = ( cp ( T ) ) for per-unit lost gross revenues cost vectors at clip T.

We assume the continuity of cp P R and Ds. P and the uninterrupted differentiability of Pm and Pm. Primary Variables ?m. n The clip of the n-th tool purchase within group m. ?m. n The clip of the n-th tool retirement within group m. Auxiliary Variables Xs. w. m ( T ) Number of merchandises that pass through operation tungsten on tool group m in scenario s at clip t. Capacity of tool group m at clip t. Unmet demand of merchandise P in scenario s at clip t. Satis?ed demand of merchandise P in scenario s at clip t. Thus. V s. T ( T ) = Ds. P ( T ) ? Vs. P ( T ) .

Let the uninterrupted variable T stand for a clip between 0 and T. the terminal of the planning skyline. We use p. w. and m to index merchandise households in P. operations in W. and tool groups in M. severally. All tools in a tool group are indistinguishable ; this is how tool groups are really de?ned. We denote by M ( tungsten ) the set of tools that can execute operation tungstens and by W ( m ) the set of operations that tool group m can execute. DurP R ing the planning skyline. 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 merchandise P is given by Dp ( T ) = Ds. P ( T ) . where s indexes a ?nite figure of scenarios S. Our preparation uses input informations and variables presented below.

We reserve the use of the word clip for the calendar clip t. as opposed to the treating continuance of operations or productive tool capacities available. To avoid confusion. we refer to the continuance of operations or tool capacities available at a given minute of clip utilizing the phrase machine-hours. Input Data biological warfare. p Number of operations of type tungsten required to bring forth a unit of merchandise P ( typically integer. but fractional values are allowed ) . Sum of machine-hours required by a tool in group m to execute operation w. Entire capacity ( productive machine-hours per month ) of tool group m at the beginning of the clip skyline. Capacity of each tool in group m ( productive machine-hours per month ) . Purchase monetary value of a tool in group m at clip T ( a map of the uninterrupted scalar T ) . Sale monetary value for retiring a tool in group m at clip t. May be positive or negative. Per-unit lost gross revenues cost for merchandise P at clip T.