an innovative approach for strategic capacity portfolio planning under uncertainties

European Journal of Operational Research 207 (2010) 1002–1013

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Innovative Applications of O.R.

An innovative approach for strategic capacity portfolio planning under uncertainties

Cheng-Hung Wu *, Ya-Tang ChuangInstitute of Industrial Engineering, National Taiwan University, Taipei, Taiwan

a r t i c l e i n f o a b s t r a c t

Article history:Received 30 December 2008Accepted 12 May 2010Available online 23 May 2010

Keywords:Technology choiceMulti-capacity expansionDynamic programmingDemand uncertainty

0377-2217/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.ejor.2010.05.015

* Corresponding author. Tel.: +886 2 33669505; faxE-mail address: [email protected] (C.-H. Wu).

This research studies multi-generation capacity portfolio planning problems under various uncertaintyfactors. These uncertainty factors include price uncertainties, demand fluctuation and uncertain productlife cycle. The objective of this research is to develop an efficient algorithm that generates capacity port-folio policies robust to aforementioned uncertainties.

We model this capacity portfolio planning problem using Markov decision processes (MDP). In thisMDP model, we consider two generation of manufacturing technology. The new generation capacityserves as a flexible resource that can be used to downward fulfill the deficiency of old generation capac-ity. The objective of this MDP model is to maximize the expected profit under uncertainties. An efficientalgorithm is proposed to solve the problem and provide an optimal capacity expansion policy for bothtypes of capacity. Moreover, we show that the optimal capacity expansion policy can be characterizedby a monotone structure. We verify our results by detail simulation study.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Production technology is evolving rapidly in high-tech industries. The introduction of new production technology is usually accompa-nied with new product lines and improvement of production efficiency. As a result, in high-tech industries, product life cycle is short andnew production technology is introduced frequently. For example, in semiconductor industry, new manufacturing technology is introducedevery 18 months by the well known Moore’s law. Following the Moore’s law, Intel first introduced the 65 nm microprocessors in 2006 andthen the 45 nm microprocessors in 2008.

In many industries, new production technology can also be used to produce existing products. As an example, Samsung gradually shiftstheir production of DRAM from 80 nm technology to 60 nm in 2007 and the new 60 nm processes can be used to produce both new andexisting DRAM product lines.

In high-tech industry, manufactures have to keep investing in new products and technology. The decision of adopting new technologydepends on expected future revenues generated from new product lines, which is difficult to estimate under market uncertainties. More-over, the adoption of new technology usually comes with high costs, which sometimes defer the investment of new production technologyfrom best timing.

This paper studies multiple generation capacity portfolio planning problems in rapid changing high-tech industries. In general, ad-vanced machines are relatively expensive and have the flexibility for producing different products. Existing or old generation machinesare less expensive and can only be used to produce limited product types. Thus, manufactures usually use machines of both generationsand gradually change the capacity portfolio over time. However, it remains unclear how to optimally adjust capacity portfolios over theplanning horizon under uncertainties.

In particular, it is difficult to estimate revenue generated from new products under demand uncertainties. Capacity expansion plansbased on biased estimation of demands which leads to unnecessary capacity shortage and idling costs. In addition to demand uncertainties,prices also change rapidly in high-tech industries. To address the above mentioned issues, this paper considers capacity portfolio planningproblems under demand, price and yield uncertainties.

1.1. Literature review

Strategic capacity planning problem has been studied in many research. While some research considers only deterministic models,uncertainty factors affect the quality of capacity planning. Van Mieghem (2003) provides an excellent review of literature on capacity

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C.-H. Wu, Y.-T. Chuang / European Journal of Operational Research 207 (2010) 1002–1013 1003

investment and management under uncertainties. In literature, the most popular issue is concerned on demand uncertainty. Most researchfocuses on determining the quantities, types and timing of capacity investment under demand uncertainties. However, other uncertaintyfactors, such as yield and price uncertainties, are rarely discussed and also exercise influence on capacity expansion plans.

Capacity planning under uncertainties is complex and depends on many uncertainty factors. Demand scenarios are extensively used instochastic capacity planning problem to represent different demand profiles. Mean and variance of demands under each scenario is esti-mated by marketing department of a company and given to capacity planners. Swaminathan (2000) develop a heuristic algorithm to solvethe stochastic mixed integer programming which considers multiple tools in two stage planning problems. Their objective is to minimizethe gap between capacity and actual demand. By similar algorithm, Barahona et al. (2005) study the multiple product capacity planningproblem and assume budget constraints. Their aim is to find a good compromised capacity expansion plan for all demand scenarios. Fur-thermore, planning horizon also affects computational complexity. Huang and Ahmed (2010) consider an infinite horizon capacity plan-ning problem. They show that optimal policy structures can guide the convergence of an infinite horizon model.

Because of continuous evolving of technology, product life cycle is now shorter. This phenomenon leads to high obsolescence rate. Thus,the characteristic of short product life cycle critically affect the optimal timing of capacity investment. According to Meixell and Wu (2001),demands for technology products are volatile but follow few life cycle patterns. Demand can be considered as an internal factor, and de-mand information is shared between manufacturers and suppliers. Moreover, manufacturer has better information than supplier. Chen(2005) and Cachon and Lariviere (2001) assume the informed demand follows two states (high or low) in each period. Manufacturers adjustcapacity and production planning according to be shared demand information.

Another important issue in capacity planning processes is the obsolescence of equipments. Michael and Sundaresan (2009) mentionthat short product life cycles lead to a rapid equipment obsolescence rate. The obsolescence of equipments and product life cycles are alsoconsidered in this research. In this research, based on different product life cycle patterns, demand and price distributions are generatedfrom business future outlook. This business future outlook is usually generated by the marketing department of a company to capture thedynamic change of business environment.

When new manufacturing technologies are introduced frequently, the selection of production technology is another strategic capacityplanning problem. Li and Tirupati (1995) consider technology choice problem. They depict that many firms classify products and capacityinto two product families. While dedicated equipment is used to produce only one product family, advanced flexible equipment can bemade available for multiple products. Chod and Rudi (2005) studied the use of flexible resource under demand uncertainty and responsivepricing. In their research, the capacity expansion decision is made under demand uncertainty.

Demand and prices relationship also affects capacity planning processes. Bish and Wang (2004) consider capacity investment and allo-cation problems with two price setting firms. Modarres and Sharifyazdi (2009) study capacity allocation problems under price and demanduncertainties. In their research, two demand types are considered while the prices for different demand types are different. Geunes et al.(2009) discuss revenue and operation planning problems with price-sensitive demand functions.

Since the main purpose of capacity investment is to supply outputs for future demands, production yield rate is another issue that con-nects capacity planning with demand fulfillment. Karabuk and Wu (2002) consider a two stage capacity allocation problem under bothdemand and capacity uncertainties. Capacity decision is made according to information shared between marketers and manufacturers.Bernstein and DeCroix (2004) analyze multi-assembly stage capacity problem by game theory. In their study, delay of raw materials, ma-chine failures or quality of manufacturing processes lead to delay or scraps in a production line. Therefore, yield uncertainties affect theoutput quantities of systems.

This paper uses stochastic dynamic programming to model the strategic capacity planning problem. The main purpose is to provide abetter strategic capacity expansion plan. The closest works to our research are Eberly and Van Mieghem (1997), Rajagopalan et al. (1998)and Narongwanich et al. (2002). Eberly and Van Mieghem (1997) consider dynamic programming under production information and priceuncertainties. They show the optimality of ISD (Invest/Stay put/Disinvest) policy. Under ISD policy structure, the state space is partitionedinto several sub-regions. Each state in a region is associated with a specific optimal decision until the boundary of adjacent region has beenreached. Rajagopalan et al. (1998) assume new technology will be introduced with unknown magnitude and timing. They focused on deter-mining the level of capacity expansion. Narongwanich et al. (2002) consider capacity investment and selling problem in a manufacturingsystem with dedicated equipment and reconfigurable equipment. In their research, only one product is produced in each period. When anew product is introduced, old product will obsolete and dedicated machine will be sold.

To the best of our knowledge, none of pervious research considers demand, price and yield uncertainties simultaneously. Moreover, weconsider two different generations of products and equipments. Multiple product generations lead to higher demand fluctuation. We mod-el the multi-generation capacity planning problem by Markov Decision Processes (MDP). Based on the MDP model, an efficient new algo-rithm is developed to solve the problem.

The rest of this paper is organized as follows: Section 2 presents the problem and model. In Section 3, an efficient algorithm is proposed.The optimality of this new algorithm is proved in the same section. In addition, we show the existence of monotone structure that char-acterizes the optimal policy. By conducting simulation study, Section 4 verifies the need and the advantage of considering uncertainty fac-tors explicitly. We conclude this paper in Section 5.

2. Problem description and model formulation

In this research, decision makers can invest in equipments of different technology. Advanced equipments provide flexibility of produc-ing multiple products but higher cost. Existing or current generation equipments cost less but produce only one product. Therefore, in therest of this paper, we refer to the advanced and existing generation equipments as flexible and dedicated equipments, respectively.

Throughput, price, demand and yield uncertainties from different sources are considered in this research, as shown in Fig. 2.1. Amongwhich price and demand are affected by external market environments. In recent research, demand and price uncertainties have becomeimportant issues in supply chain management. Throughput, yield or quality uncertainties are caused by unexpected material delay ormachine failures within the production chain. In our model, we assume the probability distributions of yield, price and demand are givenfrom business future outlook. The objective is to find a capacity expansion policy that maximizes expected total profit in each period.

1004 C.-H. Wu, Y.-T. Chuang / European Journal of Operational Research 207 (2010) 1002–1013

In this manufacture system, we consider two products and assume product one and product two are in different stage of their productlife cycle. For example, product one is a mature product and is going to phase out by the end of planning horizon. Product two is a newproduct and its demand will keep increasing in the near future. Dedicated equipments are used to manufacture product one only, and flex-ible equipments can be used to manufacture both products, as shown in Fig. 2.2.

According to the description above, we structure a multi-generation capacity planning problem by using Markov Decision Processes(MDP). The following notations are used in the model:

Indicest time, t 2 {1,2, . . . ,T}d discount factor, 0 < d 6 1e product families, e 2 {1,2}D dedicated equipmentF flexible equipment

Production system parametersxD capacity of each dedicated equipmentxF capacity of each flexible equipmentUCD the maximum number of dedicated equipmentsUCF the maximum number of flexible equipments

State variablesit the number of dedicated equipment at period tjt the number of flexible equipment at period tkt market state of product 1 in period tlt market state of product 2 in period t

System internal/external uncertaintiesyt yield rate in period t, y 2 (0,1)de,t demand of product e in period tpe,t unit price of product e in period t

Cost related parametersbe shortage cost rate of product ehD idling cost rate of dedicated capacityhF idling cost rate of flexible capacitycD,t cost of dedicated equipment in period tcF,t cost of flexible equipment in period tsD salvage value of dedicated equipment in TsF salvage value of flexible equipment in T

In this MDP model, we consider a finite planning horizon t = {1,2, . . . ,T}, where T is the length of planning horizon. Available action ineach period is to increase the quantity of dedicated or flexible equipments.

Let st be a possible system state at time t in our MDP formulation, the state space S can be defined by:

st ¼ ðit ; jt; kt ; ltÞ 2 S ¼ fS1 � S2 � S3 � S4g:

We consider market states and the numbers of equipments in the MDP model. The number of dedicated and flexible equipments areit 2 S1 = {0,1,2, . . . ,UCD} and jt 2 S2 = {0,1,2, . . . ,UCF}, respectively. In this MDP model, market state variables provide information on currentmarket environment. Market state variables are usually linked with global or national wide macroeconomic status. For example, marketstates for product one and product two are defined by kt 2 S3 = {L,M,H} and lt 2 S4 = {L,M,H}, respectively. In which, {L,M,H} indicates lowerthan forecast average, meet forecast average, and higher than forecast average, respectively. According to the information provided by ourpartners in high-tech industry, a large number of market states is hard to define and may not be practical in general. Therefore, in our mod-el and numerical study, we use three market states for each product. However, our MDP model and theoretical result can be extended toany system with a larger number of market states.

Yield uncertainty Demand uncertainty

MaterialManufacturer

Demand 1

Demand 2

Profit 1

Profit 2

Price uncertainty

Profit uncertainty

Fig. 2.1. Source of uncertainties in a supply chain.

man

ufac

ture

manufacture

Dedicated

Demand 1Product 1

manufacture

Product 2Demand 2

Flexible

Fig. 2.2. Manufacture process.


In our model, market states of different products may or may not be independent from each other. Demand and price of each productwill be affected by the associated market state. Different market states correspond to different demand and price distributions.

In the MDP model, a capacity expansion action is represented by at = (at,bt), where at and bt are the number of dedicated and flexibleequipment purchased. Due to budget and facility size constraints, we set upper bounds UCD and UCF for the number of dedicated and flex-ible equipments. Therefore, at and bt are constrained by at 6 UCD � it and bt 6 UCF � jt. The lead time of machine procurement and instal-lation is assumed to be one time period in our model. A longer lead time is possible with the expansion of state space.

Transitions between two decision epochs are defined by the following condition probability equation P(st+1jst,at). To simplify notation,we use (i, j) to represent (it, jt) and (i0, j0) to represent (it+1, jt+1) in this section. The transition probability is thereforePðstþ1jst ; atÞ ¼ Pðstþ1 ¼ ði0; j0; k0; l0Þjst ¼ ði; j; k; lÞ; atÞÞ. By assuming that the market state transition is independent of the equipment state tran-sition, the one step transition equation can be written by:

Ptði0; j0; k0; l0ji; j; k; l; atÞ ¼ Pði0; j0ji; j; atÞ � Pðk0; l0jk; lÞ;

where P(i0, j0ji, j,at) is solely controlled by initial states (i,j) and action (at,b t) at time t. P(i0, j0ji, j,at) = 1 when i0 = i + at and j0 = j + bt. Otherwise,P(i0, j0ji, j,at) equals to 0. P(k0, l0jk, l) is the market state transition probability, which follows a Markov transition probability matrix.

The one period reward function of this MDP model is Rt(i, j,k, l) � cD,t � cF,t, where Rt(i, j,k, l) is the one period expected profit from prod-uct selling and cD,t + cF,t is the capacity expansion costs. Define rt(p1,t,p2,t,d1,t,d2,t,yjst) be the immediate revenue or profit function fromproduct selling. The revenue function depends on random variables p1,t, p2,t, d1,t, d2,t, y and current system state st. We assume unfilleddemands are lost and shortage cost will be accrued. Based on the current capacity sufficiency status, immediate revenue function canbe categorized into the following five cases:

Case 1: d1,t 6 ytxDi, d2,t 6 ytxFj

rtðp1;t;p2;t ;d1;t; d2;t ; yjsÞ ¼ p1;td1;t þ p2;td2;t � hDðytxDi� d1;tÞ � hFðytxF j� d2;tÞ:

Case 2: d1,t 6 ytxDi, ytxFj < d2,t

rtðp1;t;p2;t ;d1;t; d2;t ; yjsÞ ¼ p1;td1;t þ p2;tðytxF jÞ � hDðytxDi� d1;tÞ � b2ðd2;t � ytxFjÞ:

Case 3: ytxDi < d1,t, d2,t 6 ytxFj, d1,t � ytxDi 6 ytxFj � d2,t

rtðp1;t;p2;t ;d1;t; d2;t ; ytjsÞ ¼ p1;tðd1;tÞ þ p2;tðd2;tÞ � hFðytxDiþ ytxF j� d1;t � d2;tÞ:

Case 4: ytxDi < d1,t, d2,t 6 ytxFj, ytxF j � d2,t < d1,t � ytxDi

rtðp1;t;p2;t ;d1;t; d2;t ; ytjsÞ ¼ p1;tðytxDiþ ytxFj� d2;tÞ þ p2;tðd2;tÞ � b1ðd1;t þ d2;t � ytxDi� ytxFjÞ:

Case 5: ytxDi < d1,t, ytxFj < d2,t

rtðp1;t;p2;t ;d1;t; d2;t ; ytjsÞ ¼ p1;tðytxDiÞ þ p2;tðytxFjÞ � b1ðd1;t � ytxDiÞ � b2ðd2;t � ytxF jÞ:

Based on the revenue function, the expected one step sales revenue can be calculated from:

Rtði; j; k; lÞ ¼ E½rtðp1;t;p2;t; d1;t ;d2;t ; yt jstÞ� ¼X

p1;t ;p2;t ;dk ;dl ;y

rtðp1;t ;p2;t ;d1;t ;d2;t; yt jstÞPðp1;t ;p2;t;d1;t ;d2;t; yt jstÞ:

The expansion cost of each dedicated equipment and flexible equipment are cD,t and cF,t, respectively. We assume all capacities will besold at the end of planning horizon, and the salvage value of each dedicated and flexible equipments are sD and sF, respectively. The capacityexpansion cost or salvage value cD,t, cF,t given action at = (at,bt) are defined as follow:

cD;t ¼cD;t � at if t < T;

�sD � i if t ¼ T;

�cF;t ¼

cF;t � bt if t < T;

�sF � j if t ¼ T:

�


Therefore, the one period reward function can be calculated from Rt(i, j,k, l) � cD,t � cF,t and the discounted dynamic programming opti-mality equation with discount factor d is defined by:

V�t ði; j; k; lÞ ¼ maxa;bfRtði; j; k; lÞ � cD;t � cF;t þ dE½V�tþ1ði

0; j0; k0; l0Þ�g;

where E½V�tþ1ði0; j0; k0; l0Þ� ¼

Pi0 ;j0 ;k0 ;l0Ptði0; j0; k0; l0ji; j; k; l;at; bt:Þ � V�tþ1ði

0; j0; k0; l0Þg.

In the optimality equation, value function V�t ði; j; k; lÞ is the optimal expected total reward from t to T. Based on the optimality equation,we will introduce an efficient induction algorithm in the next section.

3. An efficient algorithm and optimal policy structure

This section first introduces the original value iteration algorithm. Value iteration algorithm solves optimality equations and generatesoptimal policies and value function. The optimality of value iteration algorithm is shown in Puterman (1994). A revised algorithm is thendeveloped based on our observation on value functions. We prove the optimality of the revised algorithm. The revised algorithm is alsoshown to require less computational resources. In addition, we prove the existence of monotone optimal policy structure.

In the original value iteration algorithm, we check all possible actions for all states in each iteration. For a state (i, j), that means all ac-tions at = (at,bt) that satisfies the capacity expansion constraint a 6 UCD � i and b 6 UCF � j. And the optimality equation selects an actionthat generates the highest expected total reward. The implementation of the original value iteration algorithm is shown below:

The original value iteration algorithm:Step 1: Set t = T, calculate the expected instantaneous reward RT(i, j,k, l) and salvage value (sDi + sFj) for all state (i, j,k, l). Let

V�Tði; j; k; lÞ ¼ RTði; j; k; lÞ þ sDiþ sF j.Step 2: Let t = t � 1. For all states s = (i, j,k, l) 2 S, compute the expected total reward Rt(i, j,k, l). The capacity expansion cost is

(cD,t � a,cF,t � b) for action (a 6 UCD � i,b 6 UCF � j). The expected total reward and optimality equation is given as follows.

V�t ði; j; k; lÞ ¼ Rtði; j; k; lÞ þmaxa;b

�cD;t � a� cF;t � bþ dXk0 ;l0

Ptðiþ a; jþ b; k0; l0ji; j; k; lÞ � V�tþ1ðiþ a; jþ b; k0; l0Þ

8<:

9=;:

For state s = (i, j,k, l), the optimal action a�t ðsÞ ¼ ða�t;s;b�t;sÞ is defined by:

a�t ðsÞ ¼ ða�t;s; b�t;sÞ ¼ arg max

a;brtði; j; k; lÞ � cD;t � a� cF;t � bþ d

Xk0 ;l0

Ptðiþ a; jþ b; k0; l0ji; j; k; lÞ � V�tþ1ðiþ a; jþ b; k0; l0Þ

8<:

9=;:

Step 3: If t > 0, repeat Step 2. Otherwise, stop.

In the original algorithm, Step 1 calculates the reward function for the final period T. In step two, the algorithm compares all possibleactions and selects the one that maximizes expected total rewards to go.

In this paper, we develop an innovative algorithm based on the original value iteration algorithm. This revised algorithm revises theaction set and optimality equation for each state. Instead of evaluating all possible actions, we compare only three actions for each statein the revised algorithm. Again, we let V�s ðsÞ and a�t ðsÞ be the optimal value function and optimal action for a state s. This revised algorithmworks as follow:

An innovative algorithm:

Step 1: Set t = T and calculate the expected instantaneous reward RT(i, j,k, l) and salvage value (sDi + sFj) for all state s = (i, j,k, l) 2 S. LetV�Tði; j; k; lÞ ¼ RTði; j; k; lÞ þ sDiþ sF j.

Step 2: Substitute t with t � 1 and let i = UCD, j = UCF and a�s¼ðUCD ;UCF ;k;lÞ ¼ ð0;0Þ for all market state (k, l).Step 2.1: For all market state (k, l), compute the reward Rt(i, j,k, l) and capacity expansion cost (cD,t � a,cF,t � b) under each of the following

three actions. The expected total future reward is given by:
Action 1:a = 0, b = 0
Vtði; j; k; l;0;0Þ ¼ Rtði; j; k; lÞ þ dXk0 ;l0

Ptði; j; k0; l0ji; j; k; lÞ � V�tþ1ði; j; k0; l0Þ:

Action 2: a = 0,b = 1

Vtði; j; k; l;0;1Þ ¼ Rtði; j; k; lÞ � cF;t þ V�t ði; jþ 1; k; lÞ � Rtði; jþ 1; k; lÞ:

Action 3: a = 1,b = 0

Vtði; j; k; l;1;0Þ ¼ Rtði; j; k; lÞ � cD;t þ V�t ðiþ 1; j; k; lÞ � Rtðiþ 1; j; k; lÞ:

Under Action 2 and Action 3, instead of using V�tþ1ði; jþ 1; k; lÞ and V�tþ1ðiþ 1; j; k; lÞ, note that V�t ði; jþ 1; k; lÞ and V�t ðiþ 1; j; k; lÞ areused in the revised algorithm. We then compare the expected reward under these three actions and select the one that maximizethe reward.In conclusion, the revised optimality equation becomes:

V�t ði; j; k; lÞ ¼ maxða;bÞ2 ð0;0Þ;ð0;1Þ;ð1;0Þf g

fVtði; j; k; l;a; bÞg:

Let (a*,b*) maximize the equation above


ða�;b�Þ ¼ arg maxða;bÞ2 ð0;0Þ;ð0;1Þ;ð1;0Þf g

fVtði; j; k; l;a;bÞg:

The optimal action a�t ði; j; k; lÞ ¼ ða�t;s;b�t;sÞ for state s = (i, j,k, l) is defined by:

a�t ði; j; k; lÞ ¼ ða�;b�Þ þ a�t ðiþ a�; jþ b�; k; lÞ:

Step 2.2: Let i = i � 1. If i < 0 go to Step 2.3, otherwise repeat Step 2.1.Step 2.3: Let i = UCD and j = j � 1. If j < 0, go to Step 3. Otherwise, repeat Step 2.1.Step 3: If t > 0 repeat Step 2. Otherwise stop.

In the revised algorithm, Step 1 calculates the boundary condition t = T and is identical to the original algorithm. A significant point inthis algorithm is that we reuse value that has been calculated in the same iteration (time period). Instead of evaluating all possible actions,we significantly decrease the number of action considered.

In Lemma 3.1, we prove that the results from both algorithms are identical. As a result, the revised algorithm gives an optimal solution.Let V�t;1ði; j; k; lÞ be the optimal value function solved by the original value iteration algorithm and V�t;2ði; j; k; lÞ be that by the revisedalgorithm.

Lemma 3.1. For all 15t5T; V�t;1ði; j; k; lÞ ¼ V�t;2ði; j; k; lÞ and the optimal actions are identical in both algorithms.Proof of Lemma 3.1 is conducted by a nested induction. Fig. 3.1 shows the nested structure of induction proof. This nested induction is

over both decision epochs (time) and state space. We first show the desired properties for decision epochs t = T. For all other decisionepochs, based on the induction hypothesis, we first prove that both algorithms are identical at initial state i = UCD, j = UCF. We then showthe properties for all other boundary points i = UCD and j = UCF in Induction 1 and Induction 2. Based on results from Induction 1 and Induction2, we then show the desired properties for all other states by Induction 3.

Proof of Lemma 3.1. Let V�t;1ði; j; k; lÞ be the optimal expected total reward generated from the original value iteration algorithm andV�t;2ði; j; k; lÞ be the optimal reward under the revised algorithm. We prove Lemma 3.1 by nested mathematical induction on both decisionepochs t and state s.

Induction on Decision Epochs:

1. At t = T, Step 1 of both algorithms are identical. It is obvious that V�t;1ði; j; k; lÞ ¼ V�t;2ði; j; k; lÞ when t = T.2. Induction hypothesis: Assume thatV�tþ1;1ði; j; k; lÞ ¼ V�tþ1;2ði; j; k; lÞ holds for some decision epoch 1 6 t + 1 6 T and optimal actions are

identical.We then show V�t;1ði; j; k; lÞ ¼ V�t;2ði; j; k; lÞ holds for decision epoch t by the following inductions over state space.

3. Induction on states: At time t, we first finish the proof for the capacity upper bound state i ¼ UCD; j ¼ UCf . When i ¼ UCD; j ¼ UCf , theexpected total reward of value iteration algorithm and revised algorithm are both

V�t;1ðUCD;UCF ; k; lÞ ¼ RtðUCD;UCF ; k; lÞ þ dE½V�tþ1ðUCD;UCF ; k0; l0Þ�;

V�t;2ðUCD;UCF ; k; lÞ ¼ RtðUCD;UCF ; k; lÞ þ dE½V�tþ1ðUCD;UCF ; k0; l0Þ�:

Fig. 3.1. The nested structure of mathematic induction proof.


Therefore, by the induction hypothesis, V�t;1ði; j; k; lÞ ¼ V�t;2ði; j; k; lÞ is trivial and the optimal action is a = 0 and b = 0 under both algorithms.

Induction 1:We know from the initial condition that V �t;1ði; j; k; lÞ ¼ V�t;2ði; j; k; lÞ for i ¼ UCD; j ¼ UCf . To complete induction 1, we make an inductionhypothesis that the optimal actions are identical and V�t;1ði; j; k; lÞ ¼ V�t;2ði; j; k; lÞ for some 1 6 i = g1 + 1 6 UCD and j = UCF. We then show thesame equality holds for i = g1, j = UCF.For i ¼ g1; j ¼ UCf , the optimal expected total reward of the original value iteration algorithm is:

V�t;1ðg1;UCF ; k; lÞ ¼ Rtðg1;UCF ; k; lÞ þmaxa

�cD;t � aþ dXk0 ;l0

Ptðg1 þ a;UCF ; k0; l0jg1;UCF ; k; lÞ � V�tþ1ðg1 þ a;UCF ; k

0; l0Þ

8<:

9=;

¼ Rtðg1;UCF ; k; lÞ þmaxaf�cD;t � aþ dE½V�tþ1ðg1 þ a;UCF ; k

0; l0Þ�g;

where 0 6 a 6 UCD � g1. And, the value function under the revised algorithm is:

V�t;2ðg1;UCF ; k; lÞ ¼ Rtðg1;UCF ; k; lÞ þmaxf�cD;t þ V�t;2ðg1 þ 1;UCF ; k; lÞ � Rtðg1 þ 1;UCF ; k; lÞ; dE½V�tþ1ðg1;UCF ; k0; l0Þ�g ð1Þ

By the induction hypothesis, V�t;1ði; j; k; lÞ ¼ V�t;2ði; j; k; lÞ at i = g1 + 1, and j = UCF. It is straight forward that

V�t;2ðg1 þ 1;UCF ; k; lÞ ¼ Rtðg1 þ 1;UCF ; k; lÞ þ max06a6UCD�g1�1

f�cD � a0 þ dE½V�tþ1ðg1 þ 1þ a0;UCF ; k0; l0Þ�g:

Substitute V�t;2ðg1 þ 1;UCF ; k; lÞ in the previous equation into (1), we then have

V�t;2ðg1;UCF ; k; lÞ ¼ Rtðg1;UCF ; k; lÞ þmax �cD;t þmaxaf�cD � aþ dE½V�tþ1ðg1 þ 1þ a;UCF ; k

0; l0Þ�g; dE½V�tþ1ðg1;UCF ; k

0; l0Þ�

n o¼ Rtðg1;UCF ; k; lÞ þmax max

06a06UCD�g1�1f�cD;t � ð1þ a0Þ þ dE½V�tþ1ðg1 þ 1þ a0;UCF ; k

0; l0Þ�g; dE½V�tþ1ðg1;UCF ; k

0; l0Þ�

� �¼ Rtðg1;UCF ; k; lÞ þ max

06a006UCD�g1

f�cD;t � a00 þ dE½V�tþ1ðg1 þ a00;UCF ; k0; l0Þ�g ¼ V�t;1ðg1;UCF ; k; lÞ

By mathematical induction, V�t;1ði; j; k; lÞ ¼ V�t;2ði; j; k; lÞ holds for all states 1 6 i 6 UCD and j = UCF. Moreover, the optimal action is also iden-tical.Similar to the proof in induction 1, we shown the other boundary condition in induction 2.

Induction 2:In Induction 2, we show V�t;1ði; j; k; lÞ ¼ V�t;2ði; j; k; lÞ holds for all states i = UCD, 1 6 j 6 UCD and the optimal actions are identical. Since the proofis almost identical to the proof of Induction 1, we omit the details for brevity.

Induction 3:1. Induction 1 and induction 2 serve as the boundary and initial conditions in the induction 3.2. We make an induction hypothesis that V�t;1ði; j; k; lÞ ¼ V�t;2ði; j; k; lÞ and optimal actions are identical in both algorithms for all states

(0 6 i 6 UCD, j = g2 + 1) and (0 6 i = g1 + 1 6 UCD, j = g2).3. We complete the proof by showing the desired properties at i = g1, j = g2.

By the optimality equation of both algorithms:

V�t;1ðg1; g2; k; lÞ ¼ Rtðg1; g2; k; lÞ þmaxa;b

�cD;t � a� cF;t � bþ dXk0 ;l0

Ptðg1 þ a; g2 þ b; k0; l0jg1; g2; k; lÞ � V�tþ1ðg1 þ a; g2 þ b; k0; l0Þ

8<:

9=;

¼ Rtðg1; g2; k; lÞ þmaxa;bf�cD;t � a� cF;t � bþ dE½V�tþ1ðg1 þ a; g2 þ b; k0; l0Þ�g;

where 0 6 a 6 UCD � g1, 0 6 b 6 UCF � g2.And,

V�t;2ðg1; g2; k; lÞ ¼ Rtðg1; g2; k; lÞ þmaxf�cD;t þ V�t;2ðg1 þ 1; g2; k; lÞ � rtðg1 þ 1; g2; k; lÞ;�cF;t þ V�t;2ðg1; g2 þ 1; k; lÞ � rtðg1; g2

þ 1; k; lÞ; dE½V�tþ1ðg1; g2; k0; l0Þ�g: ð2Þ

From the induction hypothesis,

V�t;2ðg1 þ 1; g2; k; lÞ ¼ V�t;1ðg1 þ 1; g2; k; lÞ ¼ Rtðg1 þ 1; g2; k; lÞ þmaxa;bf�cD;t � a� cF;t � bþ dE½V�tþ1ðg1 þ 1þ a; g2 þ b; k0; l0Þ�g;

where 0 6 a 6 UCD � g1 � 1, 0 6 b 6 UCF � g2.Also from the induction hypothesis,

V�t;2ðg1; g2 þ 1; k; lÞ ¼ V�t;1ðg1; g2 þ 1; k; lÞ ¼ Rtðg1; g2 þ 1; k; lÞ þmaxa;bf�cD;t � a� cF;t � bþ dE½V�tþ1ðg1 þ a; g2 þ 1þ b; k0; l0Þ�g;

where 0 6 a 6 UCD � g1, 0 6 b 6 UCF � g2 � 1.Substituting V�t;2ðg1 þ 1; g2; k; lÞ and V�t;2ðg1; g2 þ 1; k; lÞ in (2) with equations above. We then have


V�t;2ðg1; g2; k; lÞ ¼ Rtðg1; g2; k; lÞ þmax �cD;t þmaxa;bf�cD;t � a� cF;t � bþ dE½V�tþ1ðg1 þ 1þ a; g2 þ b; k0; l0Þ�g;�cF;t

�

þmaxa;bf�cD;t � a� cF;t � bþ dE½V�tþ1ðg1 þ a; g2 þ 1þ b; k0; l0Þ�g; dE½V�tþ1ðg1; g2; k

0; l0Þ�

�

¼ Rtðg1; g2; k; lÞ þmax maxa;bf�cD;t � ð1þ aÞ � cF;t � bþ dE½V�tþ1ðg1 þ 1þ a; g2 þ b; k0; l0Þ�g;

�

�maxa;bf�cD;t � a� cF;t � ð1þ bÞ þ dE½V�tþ1ðg1 þ a; g2 þ 1þ b; k0; l0Þ�g; dE½V�tþ1ðg1; g2; k

0; l0Þ�

�¼ Rtðg1; g2; k; lÞ þmax

a;bf�cD;t � a� cF;t � bþ dE½V�tþ1ðg1 þ a; g2 þ b; k0; l0Þ�g:

Obviously, the results of value iteration algorithm and revised algorithm are the same.Therefore, V�t;1ði; j; k; lÞ ¼ V�t;2ði; j; k; lÞ and optimal actions are identical under both algorithms. This completes the proof of Lemma 3.1 and therevised algorithm generates optimal policy and value function. h

In Lemma 3.2, we show an interesting and important structure of optimal capacity expansion policy. This structure can further reducecomputational complexity of portfolio optimization problem.

Lemma 3.2. If (a*,b*) is the optimal action for some state (m,n,k, l) in period t. Then, the optimal action is (a* � i + m,b* � j + n) for all states(i, j, k, l) where m 6 i 6m + a* and n 6 j 6 n + b*.

Proof. Let a*, b* be the optimal action of state (m,n,k, l) and m* = m + a*, n* = n + b*. Then,

V�t ðm;n; k; lÞ ¼maxa;b

Rtðm; n; k; lÞ � cD;tðaÞ � cF;tðbÞ þ dX

i0 ;j0 ;k0 ;l0Ptðmþ a;nþ b; k0; l0jm;n; k; l;a;bÞ � V�tþ1ðmþ a;nþ b; k0; l0Þ

8<:

9=;

¼ Rtðm;n; k; lÞ � cD;tðm� �mÞ � cF;tðn� � nÞ þ dX

i0 ;j0 ;k0 ;l0Ptðm�;n�; k0; l0jm;n; k; l;a; bÞ � V�tþ1ðm�;n�; k

0; l0Þ;

ða�;b�Þ ¼ arg maxa;b

Rtðm;n; k; lÞ � cD;tðaÞ � cF;tðbÞ þ dX

i0 ;j0 ;k0 ;l0Ptðmþ a;nþ b; k0; l0jm;n; k; l;a; bÞ � V�tþ1ðmþ a;nþ b; k0; l0Þ

8<:

9=;:

As a result, the following inequality holds for all i0 P m, j0 P n.

dE½V�tþ1ðm�;n�; k0; l0Þ� � cD;tðm� �mÞ � cF;tðn� � nÞP dE½V�tþ1ði

0; j0;k0; l0Þ� � cD;tði0 �mÞ � cF;tðj0 � nÞ; ð3Þ

where E½V�tþ1ði0; j0; k0; l0ji; j; k; l� ¼

Pk0 ;l0Ptði0; j0; k0; l0ji; j; k; lÞ � V�tþ1ði

0; j0; k0; l0Þ.

For any m 6 i 6m* and n 6 j 6 n*, add (4) to both side of inequality (3) and get inequality (5)

cD;t � ði�mÞ þ cF;t � ðj� nÞ; ð4Þ

dE½V�tþ1ðm�;n�; k0; l0Þ� � cD;tðm� � iÞ � cF;tðn� � jÞP dE½V�tþ1ði

0; j0; k0; l0Þ� � cD;tði0 � iÞ � cF;tðj0 � jÞ ð5Þ

By (5) and the optimality equation, the optimal action is therefore (a* � i + m,b* � j + n) for all states (i, j,k, l), where m 6 i 6m + a* andn 6 j 6 n + b*. This completes the proof of Lemma 3.2. h

Lemma 3.2 states that the optimal policy is monotone to the number of dedicated and flexible capacity on hand, i.e. the optimal policytends to invest less in capacity while more is currently on hand.

In Lemma 3.3, we show the computation complexity under the revised algorithm is less than the original algorithm. Since we only con-sider three actions for each state in our revised algorithm, the revised algorithm considered a significantly smaller action set for each state.

Lemma 3.3. Computation complexity of the original value iteration is always greater then that of the revised algorithm.

Proof. According to the definition and optimality equation of both algorithms, same amount of computational effort is required for eval-uating the cost of each expansion action. Since the number of actions evaluated by the revised algorithm is always less than or equal to theoriginal algorithm, solving the optimality equation using the revised algorithm required less computational time. h

In the next example, we compare the computational complexity of original value iteration algorithm with the revised algorithm. Thecomputational complexity of the revised algorithm is linear to the number of capacity states.

Example 3.1 (Computational complexity of the revised algorithm). Let T be the length of planning horizon. And, let K and L be the number ofpossible market states for product one and product two, respectively.

If we let fOVIA be the total number of actions evaluated by the original value iteration algorithm, fOVIA could be defined by:

fOVIA ¼ T � K � L�XUCD

i¼0

XUCF

j¼0

ðiþ 1Þ � ðjþ 1Þ !

¼ T � K � L� ðUCD þ 2ÞðUCD þ 1ÞðUCF þ 2ÞðUCF þ 1Þ4

� �: ð6Þ

On the other hand, let fRA be the total number of actions evaluated by the revised algorithm


fRA ¼ T � K � L� ½1þ 2ðUCD þ UCFÞ þ 3ðUCD � UCFÞ�: ð7Þ

Let n be the number of distinct capacity states, where n = (UCD + 1) � (UCF + 1). Obviously, while fOVIA is a O(n2) function, fRA is only a O(n)function. Therefore, the required computational time of the revised algorithm grows linearly with the number of capacity states n. Thismakes the revised algorithm very efficient for larger problems.

In the next section, we provide computation results under various types of problem setting and discussion major characteristic of multi-generation capacity planning problem.

4. Numerical study and comparative analysis

The main purpose of this section is to verify the performance of our algorithm and find the importance of optimal dynamic decisions inrapid changing environments. In Section 4.1, we illustrate the implementation procedure of our revised algorithm in a numerical example.Under the same set of data, we solve the capacity planning problem with and without considering throughput, price and demand uncer-tainties. Differences between deterministic and stochastic models are compared with Monte Carlo simulation in Section 4.2. To verify theperformance of our algorithm in general cases, we study the performance of our algorithm under 16 more sets of system parameters inSection 4.3. Experimental design techniques are used to construct the 16 parameter sets from an eight factors 28�4 fractional factorialdesign.

4.1. Optimal policy structure and implementation of dynamic programming algorithm

This research implements a software tool using Microsoft Visual Basic 2003 to solve the capacity expansion problem. We implementboth algorithms and compare the computational times. To facilitate the understanding, we use the following example to demonstratethe stochastic capacity planning results.

Example 4.1. Let d = 0.99, UCD = 20, UCF = 20, cD,t = 250, cF,t = 500, sD = 10, sF = 20, hD = 1, hF = 2, b1 = 3, b2 = 6, xD = 5, xF = 5. (To reflect actualsemiconductor manufacturing environments, these parameters are selected according to Chou et al. (2007).) Assuming a planning horizonof two years and time bucket size of one month, i.e. T = 24. Let market states of products be a Markov chains. The market state transitionprobability matrices are listed in Fig. 4.1.

The demand pattern and product life cycle settings are referred to Li and Graves (2007). Li and Graves (2007) consider a downwarddemand pattern for the old product demand pattern and an upward pattern for the new product, as shown in Fig. 4.2.

We assume that the actual demand of products follows Poisson distributions. (Song and Zipkin (1993) assume demand is Poissonprocess and affected by market state.) The mean of product one (two) demand is k1,k,t (k2,l,t) at time t under market state k (l). Since productlife cycles are considered, demands are non-stationary and the means change with time and market state. Yield/throughput rate followsuniform distribution (Wang and Gerchak, 2000).

Fig. 4.3 shows the optimal decisions rule if the initial market states are low for both products. The optimal capacity expansion decisionsat each state are plotted in Fig. 4.3. The horizontal axis is the initial number of dedicated equipments; vertical axis is the initial number offlexible equipments. In area I, when i 6 2 and j 6 18, the optimal purchasing action is to buy both types of equipments until i = 2, j* = 18. Inarea II and area III, the optimal action is to buy only dedicated or flexible equipments until the boundary of area IV. Lemma 3.2 shows theexisting of area I, area II, and area III. In the upper-right area IV, the optimal action is ‘‘do not buy any additional equipments”. Althoughwithout a proof, in all our numerical examples, a unique switching curve separate area IV. We verify this optimal policy structure and theexisting of switching curve in more than 100 set of system parameters that have been studied.

Compared to the original value iteration algorithm, the computational time is improved from 106.4 to 17.5 s by using the proposedrevised algorithm. In all our numerical study, the improvement in computational time is similar.

Fig. 4.1. Probability of market states transition.

Period

mea

n of

dem

and

Product 1

Product 2

Fig. 4.2. An example of product life cycle patterns.

Fig. 4.3. Optimal investment action of stochastic model.

Fig. 4.4. Optimal actions of deterministic model.


Example 4.2 (Deterministic decision model and current industry practice). In this example, we study capacity expansion plans that do notconsider uncertainties. We let system parameters be identical to that of Example 4.1, except that the demand/price/yield uncertaintiesare ignored and planning results are based only on mean value. To date, this deterministic approach is the most common industry practice.Fig. 4.4 shows the optimal action under this deterministic model, where i* = 2, j* = 15 for initial states in area I. We find that when uncer-tainties are ignored, capacity expansions are less aggressive in this example.

We verify the performance of both deterministic and stochastic models by Monte Carlo simulation in the next section.

4.2. Monte Carlo simulation and the importance of uncertainty considerations

To demonstrate the importance of considering uncertain factors, Monte Carlo simulation of 10,000 randomly generated samples areused to evaluate the performance of both deterministic and stochastic models in Examples 4.1 and 4.2. Since both demands and pricesdepend on market states, we first generate 100 randomly generated market state chains. These market state chains are generated accordingto the market state transition probability matrix defined in Example 4.1. As a result, these market state chains are random realization of the

Table 4.1Simulation result under deterministic and stochastic models.

Performance in Monte Carlo simulation of 10,000 samples Model

Deterministic Stochastic

Mean profit 20963.18 21642.39Idling of capacity D 9.7174 9.7174Idling of capacity F 295.162 489.3746Type 1 demand shortage 217.624 143.8434Type 2 demand shortage 87.6319 45.1436

Table 4.2High/low levels of factors.

Factor Level: low Level: high

cD 250 375cF 500 750hD 1 2hF 2 4b1 3 6b2 6 12xD 5 8xF 5 8

Table 4.3Factor setting in the 28�4 design.

Run cD cF hD hF b1 b2 xD xF

1 � + � � + � + +2 + + + + + + + +3 + � + + � + � �4 + + � � + + � �5 � + � + � + + �6 � � � � � � � �7 + + + � � � + �8 � � + + � � + +9 � � + � + + + �10 � � � + + + � +11 � + + � � + � +12 + � + � + � � +13 � + + + + � � �14 + + � + � � � +15 + � � + + � + �16 + � � � � + + +

Table 4.4Simulation result.

Run Model Mean profit Improvement in mean (%) VaR90 Improvement in VaR90 (%)

1 Deterministic 21011.0 4.4 18580.38 1.7Stochastic 21952.4 18898.43

2 Deterministic 13199.1 11 11804.02 10.5Stochastic 14729.8 13053.53





7 Deterministic 17292.5 1.4 14919.4 �0.2Stochastic 17548.4 14875.65







14 Deterministic 20896.3 1.9 18259.9 �0.2Stochastic 21293.4 18221.25





market state Markov chains. Each of market state chains is then used to generate 100 sets of demand/price/yield data for each time period.The 10,000 samples Monte Carlo simulation results are listed in Table 4.1.

In Table 4.1, mean profit is improved by 3.2% under the stochastic model. In this example, when uncertainties are considered, more flex-ible equipments are acquired to avoid capacity shortage and improve robustness.

4.3. Experimental design and result analysis

In the previous section, the effectiveness and robustness of our stochastic model is showed in one example. To verify the effectiveness ofstochastic model under different cost parameters and probability distributions, we use the Monte Carlo simulation method proposed inSection 4.2 to construct an eight factors bi-level 28�4 fractional factorial experiment. Table 4.2 shows the high/low level of each cost param-eter. The design of 28�4 fractional factorial experiment is given in Table 4.3.

Again, for each experiment run, we evaluate the performance of each model using Monte Carlo simulation of 10,000 samples. Table 4.4summarized the mean profit of both stochastic and deterministic model. In the 16 experiment runs, mean profits under the stochastic mod-el are all greater than that of the deterministic model.

In addition to mean profits, Value-at-risk (VaR) values under the 90% confidence interval are listed in Table 4.4. In economics, VaR is ameasure of decision risks, especially in investment problem (McNeil et al., 2005). For example, VaR90 represents a value that the probabilityof observing a loss exceeded this value will not be greater than 0.1. Therefore, there is a 90% probability that the value of an investmentportfolio is greater than VaR90. Table 4.4 shows that the VaR90 is improved by the stochastic model in 14 out of the 16 runs, except in run 7and 14. In run 7 and 14, the VaR90 value of deterministic model is about 0.2% better than that of the stochastic model.

5. Conclusion

This paper investigates the capacity portfolio planning problem under uncertainties. We showed the significance of considering uncer-tainties in capacity portfolio planning. Another significant point is we developed a new algorithm that decreases computational time effec-tively. The computational time of the revised algorithm grows linearly with the number of capacity states. In the future, the concept of thisalgorithm might also be applied to others dynamic decision problems.

In conclusion, we believe that a stochastic capacity portfolio planning model can improve not only mean profits but also decisionrobustness. The risk of capacity expansion decision is also significantly reduced by including price, demand, and yield uncertainties. In ra-pid changing manufacture environment, the dynamic capacity expansion policy provides better capacity expansion decision.

Acknowledgements

This research was supported in part by the National Science Council of Taiwan under Grant NSC98-2221-E-002-039 and NSC96-3114-P-002-012-Y.

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an innovative approach for strategic capacity portfolio planning under uncertainties

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