Capacity Planning under Postponement Strategies. DECISION SCIENCES INSTITUTE Effects of Price and Production Postponement on Capacity Planning

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1 DECISION SCIENCES INSTITUTE Effects of Price and Production Postponement on Capacity Planning (Full Paper Submission) Dipankar Bose XLRI Xavier School of anagement, India Samir Barman University of Oklahoma, USA Ashis K Chatterjee II Calcutta, India ac@iimcal.ac.in ABSTRACT We examine the effects of price and production postponement strategies in a multi-product environment Using an additive demand uncertainty,we develop an expected profit maximization model similar to a two-stage stochastic program. A computational study is undertaken to observe the impacts of intercept variance, intercept correlation and cross-price responsiveness on the capacity investment and optimal expected profit. For no production postponement, the effect of intercept uncertainty on capacity investment is critical and depends on whether the products are substitute or complementary. The results also show that capacity and postponement decisions are not strategic complements. KEYWORDS: Capacity planning, Demand uncertainty, Production postponement, Price postponement INTRODUCTION Globalization and rapid technological advances have increased the business complexity in terms of both product variety as well as uncertainty in product demands. In a single period environment, in the context of uncertainty a manufacturer needs to set the plant capacity based on the estimates of the demand curve parameters. However, for deterministic demand scenarios, the manufacturer invests in capacity based on the knowledge of the demand curve function with certainty. Along with the capacity, the manufacturer sets either the price or quantity for the product and the market determines the other based on the demand curve. This allows the manufacturer to determine the price-quantity combination that maximizes the expected profit. This optimal quantity so determined also defines the optimal capacity of the plant. In real life, however, due to the uncertainty in demand, the decision making process becomes complex. Specifically, situation may arise where the manufacturer may need to set the price along with the capacity level and start production before knowing the actual demand data. This is referred to as no postponement" strategy (ieghem and Dada, 1999). There may also be other situations where capacity is decided a priori, but the manufacturer decides on production or price or both after the demand curve is realized. These cases are referred to as production 1

2 postponement", price postponement" and price and production postponement", respectively (ieghem and Dada, 1999). This paper builds upon the work by (ieghem and Dada, 1999) which compares price versus production postponement in the context of capacity decision in a single product setting. It assumes additive demand uncertainty", where the uncertainty is price independent and present in the intercept of the demand curve. We extend their work into a multi-product setting under various postponement strategies. We interpret demand intercept as demand opportunity for a product" and slope as "price responsiveness". While understanding the effects of postponement strategies on optimal profit has attracted the attention of many researchers, most of the literature has come out with characteristics of optimal solutions and multiple dominant conditions. oreover, the literature lacks studies that compare various postponement strategies by examining the combined effects of variance and correlation in demand intercepts along with cross-price responsiveness on plant capacity and profit. In this paper, we develop models similar to two-stage stochastic optimization models(choi and Ruszczynski, 2008), (Francas et al., 2009) and (uriel et al., 2006) for various strategies in a multi-product setting, with the assumption of additive demand uncertainty. The objective is to determine the capacity that maximizes the expected profit using the Sample Based Optimization procedure,similar to solving the Newsvendor type stochastic optimization model (Choi and Ruszczynski, 2008; Francas et al., 2009; uriel et al., 2006). To gain further insights to the problem, the model has been solved multiple times with different parameter values. The resulting experimental study helps us develop some guidelines for comparisons among the strategies. Although some of these guidelines may be found in the literature in specific contexts, we generalize them over the range of intercept variance, correlation or cross-price responsiveness in a multi-product setting. The rest of the paper is organized as follows. The relevant literature is presented in Section 2, followed by the derivation of optimal capacity, price, demand and profit for the deterministic setting in Section 3. In Section 4, we develop the profit models on various strategies for multi-product setting under uncertainty in price dependent demand. We solve the models using a wide range of intercept variance, correlation or cross-price responsiveness values. Discussion on the findings and intuitive justifications for the results are presented in Section 5. Section 6 concludes the paper. LITERATURE REVIEW Two types of uncertainty assumptions in price dependent demand are common in the literature. The concept of additive uncertainty is introduced in (ieghem and Dada, 1999), (ills, 1959),(Zabel, 1972) and (Thowsen, 1975), while that of multiplicative uncertainty is provided in(karlin and Carr, 1962), (Nevins, 1966) and (Zabel, 1970). The additive uncertainty comes with a linear demand function, while the multiplicative uncertainty with a iso-elastic demand function. Both the additive and multiplicative effects are combined in the models presented in (Young, 1978),(Young, 1979) and (Zabel, 1988). A comparison between additive and multiplicative demand uncertainties is shown in (Petruzzi and Dada, 1999) and (Xu et al., 2010). The effects of demand uncertainty on capacity decision have attracted the attention of many researchers (see (ieghem, 2003) for a detailed review). (Eppen, 1979) considers a multi-location newsboy problem with an opportunity for centralization. It shows that the initial inventory increases with the increase in both demand variance and correlation, if and only if, the 2

3 initial inventory is higher than the expected demand; whereas expected profit always decreases with the increase in demand variance. (ieghem and Dada, 1999) develop capacity planning models using the newsboy framework in a single product setting under various postponement strategies. They consider additive demand with two types of price postponement strategies, clearance and holdback. They argue that the production postponement strategy always dominates the no-production postponement strategy. However, for both the postponement strategies, the capacity investment level decreases as variability increases with moderate and high capacity cost, whereas, the converse is true for low capacity costs. Furthermore, price postponement strategies make the capacity investment and production relatively insensitive to uncertainty. (Bish and Wang, 2004) also develop a two-stage stochastic program for price postponement strategy considering additive demand uncertainty in a two-product environment. They provide the necessary and sufficient conditions for the optimal investment strategy. With a two-product scenario, (Goyal and Netessine, 2007) study the strategies related to the choice between dedicated and product flexible technologies with competition. They conclude that product flexibility is more effective for substitute products clubbed with low correlation and low capacity cost. Recently, (Bish and Suwandechochai, 2010) analyzed two postponement strategies, price and production postponement and price postponement only, and compare the effects of these strategies and product substitutability on the flexible plant optimal capacity. For both the postponement strategies, they show that the optimal expected profit and optimal capacity decrease as the degree of product substitutability increases. This research extends the previous work (ieghem and Dada, 1999) in a multi-product environment with a view to examine the effect of intercept variance, intercept correlation, cross-price responsiveness and various postponement strategies on optimal capacity and expected profit. While both (Chod and Rudi, 2005) and (Bish and Wang, 2004) address the issue,the former study focuses on flexible plant with both the postponement strategies and the latter deals with the combination of dedicated and flexible plants. Similarly, (Bish and Suwandechochai, 2010) consider only the effect of cross-price responsiveness for a two-product flexible plant. Contrary to these studies, we consider multiple dedcated plans. However, none of the studies show explicitly the combined effect of the different levels of above-mentioned parameters on the capacity as well as profit. Furthemore, unlike(goyal and Netessine, 2007), this paper uses holdback" as a price postponement strategy. There exist several other studies related to the capacity decision under postponement strategies and product flexibility. For example, (Anupindi and Jiang, 2008) extend the work of (ieghem and Dada, 1999) with a duopoly model and compare the solutions with monopoly optimal solutions. (Goyal and Netessine, 2011) discuss the effect of product substitutability and demand correlation on the choice between either volume or product flexibility,or both.. OPTIAL CAPACITY AND PROFIT FOR DETERINISTIC SETTING For products, we represent additive demand uncertainty function as, D = ξ BP, where, D = demand vector [ 1], ξ = demand intercept vector [ 1], B = Price responsiveness matrix [ ] and P = price vector [ 1]. With no uncertainty in the demand curve intercept, ξ is replaced by A, where A is the vector of deterministic demand curve intercept [ 1]. The deterministic demand curve is expressed as D=A BP. For example, in a three product setting the demand curve functions for the products are: 3

4 D 1 =A 1 b 11 P 1 b 12 P 2 b 13 P 3 D 2 =A 2 b 21 P 1 b 22 P 2 b 23 P 3 D 3 =A 3 b 31 P 1 b 32 P 2 b 33 P 3 Note, for I j, b ij > 0 denotes complementary products, b ij < 0 denotes substitute products and b ij = 0 denotes that products have no cross-price responsiveness. Now consider, C = Cost vector [ 1] and C K = arginal cost of capacity (equal for each of the dedicated plants). Under deterministic assumption, a manufacturer decides on capacity, price and production quantity simultaneously. The demanded quantity and the capacity are same as there is no needto build idle capacity. Then, total profit = (P C C K ) T D 3 In three product setting, total profit = Π = (P i C i C K )D i Now, Π P = 0 i i Hence, A B(2P C C K ) = 0 Finally, P * = [B 1 A + C + C K ] (1) D * = A BP * Π * = D * (P * C C K ) The superscript * denotes optimality. ODEL FORULATION Assumptions and ethodology Consider a manufacturer producing products, facing a problem of deciding on the capacity before realizing the demand. Unlike the classic newsboy problem, in this research we do not consider salvage value for unsold quantities. Instead, we assume remaining inventory is disposed of at a given cost, called dispose of cost. Note that such cost will arise only in situations where production postponement is not allowed. Furthermore, we assume that capacity investment cost C(K) is linear in capacity K; i.e., C(K) = C Strategy:S K, where C Strategy:S is the marginal cost of product capacity when operated with Strategy S. We also assume that one unit of capacity is required to produce one unit of the product, so capacity is expressed as the number of product units that can be produced, and there is no penalty for producing below the optimal capacity level. We consider demand intercept for each of the products i, ξ i R +, is a random draw from a multivariate normal distribution function. 4

5 The mean of the marginal distribution function is μ i, the variance is σ 2 i, and the covariance of the joint distribution is σ ik = ρ ik σ i σ k, where ρ ik is the correlation coefficient between products i and k. Note, 1 ρ ik 1 for product I k. In case of the price postponement models,due to simultaneous selection of price and demand quantity the models become quadratic in nature. In literature, the models under uncertainty are formulated as a two-stage stochastic program (see (Dantzig, 1955; Elmaghraby, 1959; Agizy, 1967; Wagner, 1999)) and are solved by the sample based optimization procedure. As discussed before, we discretize the demand intercept space to generate a large number of intercept scenarios. The scenarios then become equally likely, which allows us to express the expected profit as an average of the profits over all scenarios. The two-stage stochastic problem over multiple scenarios is then linked together in a single model and determines the capacity that maximizes the expected profit. A similar procedureis used in a computational study (Biller et al., 2006) for a two-product scenario in order to compute profit for fixed versus postponed pricing. Herein, we obtain the discretized random data using the Cholesky Decomposition" of the variance-covariance matrix of a multivariate normal distribution. For each demand intercept parameter set, we generate a large number of equally likely scenarios. The problems are solved using the CPLEX 10.2 software. Notations We use following notations: i = Product 1 to j = Scenario generated, 1 to N D i = Demand of product i P i = Unit price of product i, in case of no price postponement C i = Unit cost of product i O i = Unit dispose of cost of product i K i = Capacity of plant dedicated for producing product i K = Capacity of product flexible plant Π j = Profit at scenario j ij = Production quantity of product i at scenario j = Sales of product i at scenario j P d ij = Per unit price of product i at scenario j for demand d ij ξ ij = Realized demand curve intercept for product i in scenario j d ij = Realized demand of product i at scenario j The cross-price responsiveness values between products i and k may be expressed as b ik i k. Similarly, own-price responsiveness values for product i can be expressed as b ii. Then, d ij = ξ ij b ik P kj k=1 5

6 Table 1 presents the profit functions under various postponement strategies for the single product scenario. The profit maximization models are provided in Table 2. Table 1: Profit expressions under various postponement strategies PRICE POSTPONEENT NO YES PRODUCTION POSTPONEENT SITUATION NO YES D > K PK (C + C K )K (P C C K )K D K PD O(K D) (C + C K )K (P C)D C K K D > K P K K (C + C K )K P K K (C + C K )K D K P D D O(K D) (C + C K )K (P D C)D C K K Table 2: Profit maximization models under various strategies STRATEGY NO POSTPONEENT PRICE POSTPONEENT OPTIIZATION ODEL N aximize [(P i +O i ) ] [(C Strategy:1 +C i +O i )K i j=1i=1 i=1 Subject to: G ij K i, K i 0 N aximize [(P j=1i=1 Subject to: = ξ ij b ik P Z kj k=1 K i +O i ) ] [(C Strategy:2 +C i +O i )K i i=1, P, K i 0 PRODUCTION POSTPONEENT N aximize [(P i - C i ) ] C Strategy:3 j=1i=1 i=1 Subject to: G ij K i, K i 0 K i 6

7 PRICE AND PRODUCTION POSTPONEENT N aximize [( P j=1i=1 Subject to: = ξ ij b ik P Z kj k=1 K i - C i ) ] C Strategy:4 i=1 K i, P, K i 0 odels No Price and Production Postponement In this case, the capacity for each dedicated plant as well as the price and production quantity for each product is set before the uncertainty is realized. While capacity higher than the production level always remains unutilized, production above the capacity creates infeasibility to the model. Hence, the production quantity is made equal to the capacity. On the other hand, if the demand is less than the production, extra quantity is disposed of at a predetermined cost O i, which implies production and sales could be different. Under this strategy, although price is not allowed to be postponed to influence the demanded quantity, the manufacturer sets the price at the time of capacity decision. However, from a computational viewpoint, demand d ij becomes a second stage variable, as it is generated using the first stage variable P i and random intercept value ξ ij. This changes the nature of the problem from linear to quadratic. However, the Hessian matrix may not remain positive semi-definite for the problem. Consequently, one cannot obtain the single optimal solution. The same is true for all the strategies, where there is no price postponement. To make the problem solvable, we assume that the price under the no price postponement situation as the optimal price at the deterministic setting (Equation 1). From Table 1, the profit expression becomes: Π Strategy:1 = [P i in(d i,k i ) O i ax(k i D i,0) C i K i ] C Strategy:1 K i Now, sales quantity = Z i = in(d i, K i ) Then, quantity overproduced = ax(k i D i,0) = Z i + K i Hence the expression can be rewritten as: Π Strategy:1 = [(P i + O i )Z i ] [(C Strategy:1 + C i + O i )K i 7

8 For a given price vector, depending on the realization of ξ ij values, realized demand may become negative. For our purpose, we introduce a parameter, G ij as non-negative demand realization. In case of no price postponement, G ij is a function of ξ ij only and is expressed as G ij = ax(d ij,0). Also, = in(g ij,k i ) implies G ij and K i, :i,j. Price Postponement Only In this strategy, capacity for each plant is decided before the demand curve is realized and the production again equals the capacity. Here we consider the price postponement with holdback strategy, instead of the price postponement with clearance strategy. The upper bound of the sales quantity is the production or capacity, andif the sales quantity is less than the production, the extra quantity is disposed of. Here, the price is scenario dependent and varies with the sales quantity. Note that P D and D exhibit the same point on the curve. Similarly, P Z and Z, and P K and K also represent the two points on the same demand curve, where, Z K. From Table 1, the profit expression becomes: Hence, Π Strategy:2 = [P Z i Z i O i (K i Z i ) C i K i ] C Strategy:2 K i where, Z i =ξ i b ik P Z k i k=1 Production Postponement Only In this strategy, again capacity and price are decided a priori, but the production is based on the realized demand. This eliminates the chance of overproduction and capacity acts as an upper bound on production. Here, sales quantities are equal to the production quantities. Like no postponement strategy, the price is again considered as the optimal price at the deterministic setting. From Table 1, the profit expression becomes: Π Strategy:3 = [(P i C i )in(d i,k i ) C Strategy:3 K i Both Price and Production Postponement Under price and production postponement, the situation is similar to only price postponement with holdback. However, in this case the production quantity is decided after the realization of the optimal demand quantity". The optimal demand is based on the optimal price, for which the cost of production acts as a lower bound. However, it is not possible to impose a constraint where production cost acts as a lower bound for the optimal price (P i C i, i). With such a constraint, realized demand may become negative with a high intercept variance. As a result, production could become negative and the model could become infeasible. Therefore, we impose a non-negativity constraint on production quantity. Under this modification, when price becomes less than the cost, production adjusts itself to zero in order to achieve optimality. Thus, even if optimal price becomes less than production cost, such scenarios do not have any impact on the optimal profits as they become zero. On the other hand, capacity, which is decided a priori acts as an upper bound on the production quantity and sales equal to production quantity. From Table 1, the profit expression becomes: 8

9 Π Strategy:4 = [(P i C i )in(d i,k i ) C Strategy:4 D i = ξ i b ik P k i k =1 K i COPUTATIONAL RESULTS Experimental Details The primary objective of this experiment,performed in a three product environment ( = 3), is to compare the effects of various factors on the optimal expected profit and corresponding capacity investment forfour strategies discussed earlier. For all of the experimantal conditions, we consider the following parameter values: mean of demand opportunity = 500, own-price responsiveness = 4, cost = 20, dispose of cost = 10 (for each product) and capacity cost = 10 (for each of the strategies). The experiment included the following factors and their respective levels: Factor: Cross-price responsiveness;levels: 1.5, 1, 0.5, 0, 0.5, 1 and 1.5 Factor: Intercept correlation; levels: 0.99, 0.5, 0.25, 0, 0.25 and 0.5 Factor: Intercept variance;levels: 100, 200, 300, 400, 2000, 4000, 6000, 8000 and Factor: Postponement strategy: levels: four different strategies (mentioend earlier) Therefore, the experiment contains 1,512 (= ) experimental conditions. For each experimental condition, we randomly generate 10,000 demand intercept scenarios (N = 10,000). Note that ρ= 0.99 represents perfectly positive correlation among the intercepts, whereas ρ = 0.5 represents perfectly negative correlation among the intercepts. The major findings from the experiment are presented in Table 3. Table 3: Observations on different strategies STRATEGY NO POSTPONEENT (NP) PRICE POSTPONEENT ONLY FINDINGS Complementary Products o Optimal investment in capacity is less than that of deterministic setting o Increase in intercept variance reduces optimal investment in capacity Substitute Products or No Cross-price responsiveness o Optimal investment in capacity is higher than that of deterministic setting o Increase in intercept variance increases optimal investment in capacity Change in intercept correlation has no effect on capacity and expected profit Complementary products or products with no cross-price effect 9

10 PRODUCTION POSTPONEENT ONLY (PP) PRICE AND PRODUCTION POSTPONEENT o Change in intercept variance has no effect on optimal investment or expected profit o Both optimal capacity and expected profit remain almost same as corresponding deterministic values Highly substitute products o Both optimal capacity and expected profit are higher than corresponding deterministic values o Increase in intercept variance increases optimal investment or expected profit When optimal investment in capacity exceeds that of deterministic setting o Increase in intercept variance increases optimal investment in capacity Both optimal investment and profit in PP are higher than that of NP Change in intercept correlation has no effect on optimal investment or expected profit Increase in intercept variance increases optimal expected profit Additional production postponement adds no value in case of low intercept variance No Postponement For no postponement, the deterministic optimal price equals the ex ante price, and it increases as the cross-price responsiveness decreases. The deterministic optimal price is low for the complentary products and high for the substitute products. For low prices, the marginal opportunity loss for producing less than the demand (underage loss) is less for producting more than the demand (overage loss). Hence, for plants producing the complementary products the optimal production quantity is less than those producing the substitute products. Since capacity equals production quantity, for complementary products the optimal capacity with demand uncertainty is less than the corresponding deterministic optimal capacity. oreover, the optimal capacity decreases with the increase in intercept variance. The converese is true for the substitute products. In case of products with no cross-price responsiveness, the optimal capacity is not affected by any changes in the intercept variance. These findings are consistent with (Eppen, 1979). Finally, the optimal capacity remains unaffected by any changes in intercept correlation. Price Postponement Only Under the price postponement with the holdback option, the manufacturer is able to decide the price after having full information on the demand curve and therefore can control the demand. Thus, in favorable situations, the manufacturer gains from high demand and manages loss in adverse situations by not bringing the entire production in the market. We found, for substitute products, actual price remains below corresponding deterministic price for more than 50 percent of the cases. Therefore,in those casesthe optimal demandis higher than the deterministic production. As the intercept variance increases, this percentage also increases, thus raising the optimal capacity level for substitute products. This observation is in agreement with (ieghem 10

11 and Dada, 1999),whichanalyzes a single product environment. However, for complementary products or those with no cross-price responsiveness, the cost of overproduction is negligible. oreover, in exactly halfof the cases the optimal price is found to be less than the corresponding deterministic price regardless of the intercept variance level. As a result, both the optimal capacity and expected profit become equal to the corresponding values under the deterministic situations.. Production Postponement Only In case of production postponement, the production decision is made only after the demand curve is realized. Thus, there is only overcapacity loss, but no overproduction loss. As a result, production postponement reduces average overage loss, with no effect on the average underage loss. In order to minimize the total average loss, the required optimal capacity level becomes higher compared to the corresponding capacity for the no postponement strategy or that for the deterministic scenario. oreover, the optimal capacity increases as the intercept variance increases. Due to the higher optimal capacity, production postponement allows higher expected sales andexpected profitthan the no postponement strategy. As the gain from additional sales exceeds the cost of additional capacity, profit increases under production postponement. Asthe intercept variance increases, the gain from additional sales decreases, thus lowering the profit level. These observations appear to be true irrespective of the cross-price responsiveness level. Both Price and Production Postponement Under the price and production postponement, our results show that profit increases with the increase in variance. In other words, any increase in intercept variance provides an opportunity for the manufacturer to increase profitability. The observation may be explained by looking into the corresponding opportunity loss values. For complementary products with positive correlation, the price postponement effect on capacity is higher than the flexibility effect on price and production postponement strategy. Hence, capacity increases with the increase in variance.. While most of our findings pertaining to this strategy support (Bish and Suwandechochai, 2010), our results also show that for a dedicated plant price and production postponement always requires higher optimal capacity investment than that for only price postponement, irrespective of the cross-price responsiveness level. This finding is consistent with (ieghem and Dada, 1999), which concludes that postponement and capacity are strategic complements" in a single product setting. With price postponement, production postponement adds little value to profit when the demand intercept variance is low. Similar observation is also found in a single product environment (ieghem and Dada, 1999). For low intercept variance, the situation is very similar to the deterministicone. Thus, in such cases, the manufacturer need not postpone production and wait for the demand information. As a consequence, the situation with both price and production postponement becomes similar to that with only price postponement. 11

12 CONCLUSION We investigate the optimal capacity choice under additive demand uncertainty where uncertainty is introduced in the intercept of the demand curve. Assuming demand intercepts follow a multivariate normal distribution, we examine in a three product environment the effects of intercept variance and intercept correlation along with the cross-price responsiveness on the capacity and expected profit level for various postponement strategies. We considered four postponement strategies: (a) no postponement, (b) production postponement, (c) price postponement with holdback and (d) price and production postponement. The major findings of the paper, summarized below, can act as guidelines for a manufacturer to decide on postponement policies and capacity investment. For complementary products with no postponement, dedicated plants require less capacity investment than that for the deterministic situation, and the required investment decreases with the increase in intercept variance. The converse is true for the substitute products. The production postponement causes profit to increase by requiring higher capacityinvestment. However, any increase in the intercept variance causes an increase in investment but a reduction in profit. The intercept correlation level has no effect on the capacity investment and profit level for the dedicated strategies. With only price postponement, highly substitute products generate higher expected profit compared to the deterministic setting and it increases as degree of intercept variance increases. For complementary products and those with no cross-price responsiveness, the extent of intercept variance and intercept correlation has negligible effect on the optimal capacity and expected profit. Hence, it can be argued that price postponement acts as a better hedge against uncertainty than production postponement. In case of low variance, production postponement adds little value to expected profit when coupled with price postponement. REFERENCES Agizy,. E. (1967). Two-stage programming under uncertainty with discrete distribution function. Operations Research, 15(1): Anupindi, R. and Jiang, L. (2008). Capacity investment under postponement strategies, market competition, and demand uncertainty. anagement Science, 54(11): Biller, S., uriel, A., and Zhang, Y. (2006). Impact of price postponement on capacity and flexibility investment decisions. Production and Operations anagement, 15: Bish, E. K. and Suwandechochai, R. (2010). Optimal capacity for substitutable products under operational postponement. European Journal of Operational Research, 207(2): Bish, E. K. and Wang, Q. (2004). Optimal investment strategies for flexible resources, considering pricing and correlated demands. Operations Research, 52: Chod, J. and Rudi, N. (2005). Resource flexibility with responsive pricing. Operations research, 53: Choi, S. and Ruszczynski, A. (2008). A risk-averse newsvendor with law invariant coherent measures of risk. Operations Research Letters, 36(1):

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