Joint optimal determination of process mean, production quantity, pricing, and market segmentation with demand leakage

Accepted Manuscript

Joint optimal determination of process mean, production quantity, pricing, and market segmentation with demand leakage Syed Asif Raza, Mihaela Turiac PII: DOI: Reference:

S0377-2217(15)00787-0 10.1016/j.ejor.2015.08.032 EOR 13183

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

16 July 2014 25 June 2015 19 August 2015

Please cite this article as: Syed Asif Raza, Mihaela Turiac, Joint optimal determination of process mean, production quantity, pricing, and market segmentation with demand leakage, European Journal of Operational Research (2015), doi: 10.1016/j.ejor.2015.08.032

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Highlights • Developed novel mathematical models for process targeting problem • Considered the demand leakage effect into market segmentation • Develop harmony search meta-heuristics to solve the problem

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• Integrated framework for process mean, pricing, production and market segmentation

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• Numerical experimentation to highlight the proposed integrated framework

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Joint optimal determination of process mean, production quantity, pricing, and market segmentation with demand leakage

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Syed Asif Razaa,∗, Mihaela Turiacb a College

of Business and Economics Qatar University, Doha, Qatar b The Bucharest University of Economic Studies Bucharest, Romania

Abstract

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The selection of an optimal process mean is an important problem in production planning and quality control research. Most of the previous studies in the field have analyzed the problem for a fixed exogenous price. However, in many realistic situations, besides product quality, product pricing is a paramount factor that determines the purchase behavior in the market. In the experts’ opinion an integrated framework that incorporates pricing as a decision tool could significantly improve a firm’s profitability. Most of the manufacturing firms yield products with distinguishable characteristics and therefore it is desired that these products to be sold in market at differentiated prices. Whereas the market segmentation achieved using differentiated prices is often imperfect, and therefore a firm may experience demand leakages. Thus, an optimal price decision must incorporate demand leakage effects for the firm to benefit from its differentiated pricing strategy. In this paper, these issues are addressed by proposing an optimal framework for joint determination of process mean, pricing, production quantity and market segmentation using differentiated pricing. This research discusses a production process that manufactures multi-class (grade) products based on their quality attribute. The products are sold in primary and secondary market at differentiated prices while experiencing demand leakages. The nonconforming items are reworked at an additional cost. Mathematical models are developed to address the problem under both price-dependent deterministic and stochastic demand situations. We propose a harmony search meta-heuristic for solving the models. A numerical experimentation is presented to study the significance of the proposed integrated decision framework.

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Keywords: Process Mean, Pricing, Market Segmentation, Demand Leakage, Harmony Search

∗ Corresponding

author Email addresses: [email protected] (Syed Asif Raza ), [email protected] (Mihaela Turiac)

Preprint submitted to *

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1. Introduction

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In this paper, we interface the contemporary Revenue Management (RM) tools for pricing on a process mean selection problem for a manufacturing firm and consider a consumer behavior for market segmentation. The process mean problem is among the topics of economics of quality control that has received much attention from researchers and industrial practitioners in the last 60 years. The problem, also known as targeting problem, mainly addresses the optimum selection of process mean which affects the expected profit/cost per item in production processes such as canning/filling, metal plating, grinding, glass and steel industries (Shao et al., 2000; Hariga and Al-Fawzan, 2005; Park et al., 2011; Duffuaa and El-Gaaly, 2013a,b). A manufacturer often follows in the production process the specification limits of a quality performance measure which may comprise weight, volume, concentration, thickness, length, etc. A product is therefore graded as conforming product if the quality performance measure is within the specification limits. Otherwise, the product is classified as nonconforming and may be sold at a reduced price, or reprocessed or scrapped (Roan et al., 2000; Darwish et al., 2013). Darwish et al. (2013) have reported one common industrial practice in manufacturing sector examined by USA federal agency, to set a higher mean of the production process in order to meet a given specification limit. This practice indeed leads to a ’giveaway’ cost (see Roan et al. (2000) for more details). Alternatively, if a tight production process mean is selected, then the manufacturing process may result a large proportion of nonconforming items, rework costs, increased scrapped volume, etc. Thus, an optimum selection of the process mean is highly desirable to improve the profitability of a manufacturer. Since Springer (1951) initiated the field of process targeting with cost minimization in a canning process, the problem evolved to integrate various decisions control. Bisgaard et al. (1984) integrated pricing decisions in the targeting problem, proposing that the nonconforming products to be sold at a price proportional to the amount of material consumed by each nonconforming product. Golhar (1987) considered the problem of optimum process mean in a canning process where the cans filled above a specification limit are sold at a fixed price while the under-filled cans are emptied and refilled at a reprocessing cost. Sampling inspection planning instead of full inspection was incorporated into the problem in Boucher and Jafari (1991). Optimal process mean with rectifying inspection was studied in Al-Sultan and Al-Fawzan (1997) for the targeting problem in a filling process with time dependent process mean. Shao et al. (2000) developed strategies for determining the optimal process mean of industrial processes when rejected goods can be held and sold to other customers in the same market at a later time. Bowling et al. (2004)’s work explored the targeting problem in the context of multi-stage serial production process. Darwish (2009) developed an integrated and hierarchical model of optimal process targeting in a single-vendor single-buyer supply chain. Based on a quality loss function, Chen and Kao (2009) determined the optimal process mean and screening limits. A reverse programming routine that identifies the relationship between the process mean and the settings within an experimental factor space was founded by Goethals and Cho (2011), and Goethals and Cho (2012). As identified here, researches have formulated different process targeting models for optimizing the firms’ profitability in terms of product uniformity or cost reduction, by approaching products’ characteristics in various production processes with multiple stages or inputs of production. The research field of process targeting evolved in many other directions, however, a detailed review of the specific literature is not in the scope of this paper. Besides setting an optimal process mean, a manufacturer’s objective is to maximize its profitability. The profitability of a firm depends on many factors and, as rightly described in Chen et al. (2004), pricing is a paramount factor that affects significantly the purchase behavior. In the filed of Revenue Management (RM), commonly defined as the science of profitability, price differentiation is among the most successful tactics for improving profitability of a firm (Talluri and Ryzin, 2004; Phillips, 2005). Price differentiation is applied by many businesses, for example, in the online stores vs. retail stores, where products might 3

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be offered by a firm at discounted prices for online sales without the option of touch and feel, whereas retail stores sales are higher priced and the customers can interact with the products or the sales staff. Another common example is the airline ticket sales, when the price differentiation is practiced by offering early sales with deeply discounted prices for advanced purchases of tickets with restrictions in changes or cancelations. This type of sale targets especially the leisure (economy) class passengers, whereas for late arriving business class passengers with higher willingness to pay, the airlines reserve some cabin capacity. Price differentiation results in market segmentation which in most of the related studies, is assumed to be perfect, and therefore demand cannibalization (demand leakage) is rather ignored in the models. While the evidence of increase profitability is proved under perfect segmentation, in most practical situations demand leakages are inevitable (Zhang et al., 2010). The essentiality of fencing improvement has been recently addressed to mitigate the customers’ spillover from one market segment to another, in Zhang et al. (2010). The authors assumed predetermined market share (maximum perceived demand) for each market segment with demand leakage from full price to discounted price market segment, depending on the difference of prices from the two market segments. Zhang and Bell (2012) presented an overview of price fencing in RM’s practice and its taxonomy. In a contrast to this, Phillips (2005) proposed a price differentiation with customer’s cannibalization to determine the optimal price and to segment a single market using differentiated pricing, for a price-dependent deterministic demand. The fundamental variant used in Phillips (2005) and in Zhang et al. (2010) is the notion of differentiation price. By using a differentiation price as in Phillips (2005), the market share can also be controlled unlike in Zhang et al. (2010). This important approach of differentiation price from Phillips (2005) is re-visited later in Raza (2015a,b). These studies generalized the modeling framework by considering an optimal market segmentation with demand leakage effect and in addition to Phillips (2005), the studies considered the price-dependent stochastic demands with known distribution, as well as with unknown distribution when the distribution-free approach (see Raza (2014) for details) was applied. Anderson and Xie (2014) integrated pricing and market segmentation decisions using opaque selling mechanisms. Xiaa et al. (2015) studied competition and market segmentation in a call center service supply chain. Integrating the targeting problem with production (inventory) decisions attracted for years the attention of many researchers. Gong et al. (1988) developed an integrated targeting-inventory model with constant process mean during the production cycle. This model was later generalized by Al-Fawzan and Hariga (2002) considering a time-dependent process mean. Roan et al. (2000) integrated the issues of production lot size and raw material procurement policy with the targeting problem. Hariga and Al-Fawzan (2005) determined simultaneously the optimal production lot size and the process mean for container-filling processes in multiple markets. Lee et al. (2007) considered a targeting-inventory problem for a production process where multiple products are processed. The targeting problem was integrated with single-vendor single-buyer problem in Darwish (2004), assuming an equal size shipment policy and salvage value for scrapping nonconforming items. Chen and Lai (2007) formulated a model that finds the optimal process mean, specification limits and manufacturing quantity under rectifying inspection plan. Chen and Khoo (2009) addressed the targetinginventory problem for serial production systems under quality loss and rectifying inspection plan. Darwish (2009) extended his earlier work and developed an integrated hierarchical targeting-inventory model for a two-layer supply chain. Park et al. (2011) established a profit model that determined the optimal common process mean and screening limits in a production process with multiple products. Recently, Darwish et al. (2013) developed a model that determines the optimal process mean for a stochastic inventory model under service level constraint. To the authors’ knowledge, the issue of optimal price setting and market segmentation using a differentiated pricing with demand leakage in the context of process mean (targeting) problem has not been addressed. This paper is expected to contribute towards this research avenue by developing novel mathematical models for joint optimal determination of process mean, pricing, production quantities and 4

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market segmentation using a differentiation price. Two distinct models are developed considering the pricedependent deterministic demand and then the price-dependent stochastic demand. A Harmony Search (HS) meta-heuristic is applied to the optimization problem. We present a numerical experimentation with both the models and study the model related parameters onto the profitability of a firm and its optimal joint decision on control parameters as discussed earlier. The remainder of this paper is organized as follows. In section 2, is defined the problem of a manufacturing firm that looks for a joint optimal determination of process mean, production quantity, pricing and market segmentation, under price-dependent deterministic demand situation first, and later considering the pricedependent stochastic demand. In Section 3, a harmony search based meta-heuristic is proposed to solve the two models. Section 4 studies the models using a numerical experimentation obtained by applying the harmony search algorithm with data from related literature. Lastly, in section 5 the conclusions of the results are discussed and the directions for the future research are proposed.

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2. Model Development

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In this section, we propose an integrated model of Revenue Management (RM) problem and process target problem for a monopolist manufacturing firm who is selling two products depending upon a quality characteristic, x, in a single selling period. The product quality characteristic, x, is acquired using a manufacturing process and is normally distributed with probability distribution function, φ(x), and cumulative probability distribution function, Φ(x). Figure 1 illustrates the problem of a target mean for normally distributed process with multi-classes of the products. Likewise several related studies (see Hariga and AlFawzan (2005), Lee et al. (2007), and Duffuaa and El-Gaaly (2013a,b) for details), it is assumed that normally distributed process has an unknown mean, µ, and a standard deviation, σ. The products manufactured are distinguished by the quality characteristic, x, based on which the products are classified into three classes. If the quality characteristic, x, is such that x ≥ l1 , then the product is classified as class (grade) 1 product and it is sold in a primary market at price p1 per unit. Else, if the product quality characteristic follows l2 ≤ x < l1 , the product is considered of class (grade) 2 and is sold in a secondary market at a discounted price p2 per unit. However, the products with quality characteristic, x < l2 are reworked at a rework cost, r, per unit. It assumed that, p1 ≥ p2 , and the firm’s expected cost per unit for a product from market segment, i, is ci , ∀ i = {1, 2}, which is dependent on the quality characteristic, x. The inspection process that classifies the products into three grades is assumed perfect and with negligible cost. This paper extends the research works on the process targeting problem with predetermined fixed prices, by incorporating pricing and market segmentation decisions with demand leakage effects. We propose a price-demand relationship model and we consider two types of demand situation commonly used in literature: (i) price-dependent deterministic demand; and (ii) price-dependent stochastic demand. 2.1. Deterministic Demand

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We start by developing the model for a firm’s profitability when it experiences price-dependent deterministic demand. We assume there are two market segments, the full (primary) market, and discounted (secondary) market. We refer the primary market as the market segment 1 which is designated for customers who are willing to pay the full price, p1 , and prefer class 1 products. Whereas, the (secondary) market segment 2 is reserved for customers willing to pay the discounted price, p2 , for a product class 2. It assumed that, p1 ≥ p2 , and the firm’s expected cost per unit for a product from market segment, i, is ci , ∀ i = {1, 2}, which depends on the quality characteristic, x. In market segment i, the price-dependent deterministic demand is assumed linear, which is widely used in the literature because of its simplicity and sufficiency to catch

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Figure 1: Multi-class process mean

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important managerial implications (see Al-Sultan (1994),Choi (1996),Chiang and Monahan (2005),Zhang et al. (2010) for details). Thus, the price-dependent deterministic demand ui (pi ) = [αi − βi pi ]+ is experienced by the firm in market segment i, and for brevity, ui = ui (pi ), ∀ i = {1, 2}. The price-dependent ∂ηi deterministic demand ui follows Increasing Price Elasticity (IPE) property in non-strict sense, thus ≥ 0, ∂pi ∂ui pi ∂ui ∂pi where ηi = − , from where ≤ 0. Further, it is assumed that the firm offers its products in yi ∂pi two market segments which may have distinct sensitivity to price variation. Nevertheless, the customers from market segment 1 have more willingness to pay and therefore, the firm is expected to have more price sensitivity in market segment 2, such that β1 ≤ β2 . In the price-dependent deterministic demand functions, ui , the maximum deterministic demand that a firm can experience in market segment i, is αi . This study extends the work of Phillips (2005) where the market segmentation strategy using a differentiation pricing is proposed, for a manufacturing firm experiencing price-dependent demand with leakage from full price market segment to discounted market segment. Some related studies on differentiated pricing similar with Phillips (2005) are Zhang and Bell (2007) and Zhang et al. (2010) which also considered the demand leakage effects. However, the major difference between these papers and Phillips (2005)’s work is the use of notation which we refer to as differentiation price. This differentiation price is the price which mainly segments the market so that a price higher than this price is regarded as full price, whereas, a price lower than this differentiation price is considered as discounted price. Manipulating the differentiation price, in addition to prices in each market segment, enables the firm a greater control of creating additional (or reducing) market share (also referred to as maximum allowable demand) from one market segment to another depending, upon other related parameters of the model. In contrast to this approach, Zhang and Bell (2007) and Zhang et al. (2010) assumed predetermined market shares, αi , ∀ i = {1, 2} that are fixed and there is no usage of differentiation price notation to optimally determine the market shares. Next, while segmenting the market using a price differentiation strategy, υ, such that p1 ≥ υ, and υ ≥ p2 , a firm must observe, p1 ≥ p2 . Thus, from the price differentiation strategy, υ, the maximum deterministic demand that can be experienced in discounted price market segment 2 would be α2 = υ β2 . Figure 2 presents a hypothetical model for market segmentation 6

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Figure 2: Conceptual model for price differentiation

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for a firm. Unlike some recent related studies Phillips (2005) and Raza (2015b) , the model presented here is a generalized representation where two distinct market segments that exist such that the market shares of these market segments are α1 ≤ α2 , and the price sensitivities are β1 ≤ β2 . Noticeably, when α1 = α2 , and β1 = β2 , then it is simplified to the model presented in Phillips (2005), and Raza (2015b). This mainly represents that a firm is segmenting its single market cumulative demand, i.e., α − β p into two segments using a differentiation price, υ. When a firm selects a differentiation price, υ, such that p1 ≥ υ ≥ p2 ≥ 0, then the maximum demand that can be experienced in market segment 2 would be, β2 υ. It is obvious to observe α1 here that, when υ ≥ 0, and p1 ≥ p2 , the maximum price in market segment 1 would be p1 = . Given a β1 differentiation price υ, the maximum price in market segment 2 would be υ, which also yields p1 ≥ υ ≥ 0. Referring again to Figure 2, we observe here that in the case of perfect segmentation, which means no demand leakage from full price market segment into discounted price market segment, if a firm offers a price, p1 , the expected profit would be represented by an area within the perimeter ”mnfh”. Naturally, the profit from discounted market segment is the area within the perimeter ”abdefh”, if the firm sets the discounted price, p2 . It is important to mention here that this profit representation shown in Figure 2 may change

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significantly depending upon the selection of problem related parameters, decision controls, p1 , p2 , and the differentiation price, υ.

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Table 1: Notations

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Process variability, i.e., standard deviation Upper specification limit for class (grade) 1 products Lower specification limit for class (grade) 2 products Fixed production cost per unit item Variable production cost per unit item Expected cost per unit for product class (grade) i, ∀ i = {1, 2, 3} Probability distribution of quality characteristic x Cumulative probability distribution of quality characteristic x Standard normal transformation of l1 Standard normal transformation of l2 Standard normal probability distribution of δi , ∀ i = {1, 2} Standard normal cumulative probability distribution of δi , ∀ i = {1, 2} Demand leakage factor Maximum perceived deterministic (riskless) demand in market segment i, ∀ i = {1, 2} Price sensitivity of deterministic demand in market segment i, ∀ i = {1, 2} Price dependent deterministic demand in market segment i, ∀ i = {1, 2} Adjusted price-dependent deterministic demand in market segment 1 with demand leakage Adjusted price-dependent deterministic demand for market segment 2 with leakage Stochastic demand factor for product class (grade) i, ∀ i = {1, 2} Total yield capacity Holding cost per unit of an excess inventory of product class i, ∀ i = {1, 2} Shortage cost per unit of an unmet inventory of product class i, ∀ i = {1, 2} Estimated rework cost per nonconforming item Stochastic price-dependent demand for product class i, ∀ i = {1, 2} Mean of the stochastic demand factor, ξi , i, ∀ i = {1, 2} Standard deviation of the stochastic demand factor, ξi , i, ∀ i = {1, 2} Probability distribution function for price-dependent stochastic demand of product class i, ∀ i = {1, 2} Cumulative probability distribution function for price-dependent stochastic demand in product class i, ∀ i = {1, 2} Total profit to the firm Expected value of a parameter

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Parameters: σ l1 l2 a b ci = ci (µ) φ(x) Φ(x) δ1 = (l1 − µ)/σ δ2 = (l2 − µ)/σ ∅(δi ) ϕ(δi ) θ αi βi ui = ui (pi ) y1 = y1 (p1 , θ) y2 = y2 (p1 , p2 , υ, θ) ξi C hi ki r Di µξi σ ξi fi (·)

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Fi (·)

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Π = Π(µ, p1 , p2 , υ, q1 , q2 , w) E(·) Decision variables: pi qi υ µ w

Price for product class i, ∀ i = {1, 2} Production quantity for product class i, ∀ i = {1, 2} Differentiation price Target process mean Number of nonconforming products reworked

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y1 = (1 − θ)u1 y2 = θ u1 + u2

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Following Phillips (2005) and Raza (2015a,b), we assume that the fences (segments) achieved using the differentiation price are imperfect and there is a θ proportion of customers’ demand, which belongs to the full price market segment and cannibalize (leak) to the discounted price market segment. In this paper, θ is assumed constant and independent of the prices p1 and p2 , but, later it will be identified in the model that any change in demand incurred due to price differentiation would impact the amount of demand that would leaked from full price market segment to discounted price segment. Next, in this paper, we follow a price differentiation and demand leakage model, suggested recently in Raza (2015b). The adjusted demand for each product class would be: (1) (2)

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In Equation 3, given that a firm manufactures products with a random quality characteristic, x, such that, x ≥ l1 , the firm will gain profit, (p1 − c(x)) y1 − b y1 (x − l1 ) from selling these products in a primary market as class 1 products. Likewise, the products with quality characteristics l2 ≤ x < l1 will be sold in a secondary market as class 2 at price p2 , resulting the profit (p2 − c(x)) y2 . Whereas the products with quality characteristic x < l2 would undergo a rework and the expected rework cost (or profit) is E(Π) − r w − w c(x). The quality characteristic, x, is a random variable and therefore, depending upon this quality characteristic, the firm’s manufacture would be three product classes and the total profit, Π, would be the sum of the individual expected profits from each product class (Al-Fawzan and Hariga, 2002; Duffuaa and El-Gaaly, 2013a,b). Moreover, the profit for each product class is perceived from the newsvendor problem with pricedependent deterministic demand (see Petruzzi and Dada (1999), Yao et al. (2006), Raza (2014) and Raza (2015a) for details), given a random quality characteristic, x. Also, since the demand is price-dependent deterministic, a firm’s decision would be to produce yi , ∀ i = {1, 2}, that can be easily determined once the prices, pi , ∀ i = {1, 2} are known (see Arrow et al. (1951), Zhang et al. (2010), Raza (2014), and Raza (2015a) for details). Indeed, by integrating pricing and market segmentation decisions, the problem’s complexity is increased, as the price-dependent deterministic demand and leakage information is also needed. Finally, the firm’s total profit, Π, is a random variable established below, which has µ, p1 , p2 and υ as decision controls.   (p1 − c(x)) y1 − b y1 (x − l1 ) x ≥ l1 ; (p2 − c(x)) y2 , l2 ≤ x < l1 ; Π= (3)  E(Π) − r w − w c(x), x < l2 .

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The total expected profit, E(Π) would be: Z ∞ Z E(Π) = ((p1 − c(x)) y1 − b y1 (x − l1 )) φ(x) dx + +

l1 Z l2

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−∞

l1

l2

(p2 − c(x)) y2 φ(x) dx

(E(Π) − r w − w c(x)) φ(x) dx

(4)

Where in Equation 4, the first term represents the expected profit obtained from selling class 1 products, the second term is the expected profit from selling products class 2 and the last term is the expected profit (or expected cost) from the reworked products. In the following, we simplify the expected profit Z function ∞

E(Π) established in Equation 4. In Equation 5, we use the following notations: 1 − ϕ(δ1 ) = φ(x)d x, Z l1 Z l2 Z ∞ Z l1 Zl1l2 ϕ(δ1 ) − ϕ(δ2 ) = φ(x)d x, ϕ(δ2 ) = φ(x)d x, c1 = c(x)d x, c2 = c(x)d x, c3 = c(x)d x, l2

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∞

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(x − l1 )φ(x)d x, and we simplify the profit function as below: E(Π)

= p1 y1 + p 2 y2

Z

∞

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φ(x)d x − y1 φ(x)d x − y2

(E(Π) − r w)

Z

l2

−∞

Z

∞

l1 Z l1

c(x) φ(x)d x − b y1

c(x)φ(x)d x

l2

φ(x)d x − w

Z

Z

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(x − l1 )φ(x)d x

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and g =

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c(x) φ(x)d x

−∞

(5)

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(a + bµ) (1 − ϕ(δ1 )) + σ b ∅(δ1 )

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We assume that the direct unit production cost for the finished product is a linear function of the quality characteristic, x. Previous studies have followed this framework are Bisgaard et al. (1984); Shao et al. (2000); Hariga and Al-Fawzan (2005). Given a quality characteristic, x, the production cost per unit is c(x) = a+bx, where a is the fixed production cost and b is the cost of obtaining a specific quality characteristic for one unit of finished product. Therefore, the expected unit production cost of class 1 products is given by c1 as follows: Z ∞ c1 = c(x)φ(x)d x l Z 1∞ = (a + b x)φ(x) dx (6)

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Where in Equation 6, ∅(δ1 ) = √12π e− 2 δ1 . In order to determine the giveaway cost of excess quality for class 1 products, we first determine the expected additional quality characteristic, g, that is given away when a process mean is set to a mean µ. Following the recent related studies on process mean (target) determination in Duffuaa and El-Gaaly (2013a,b), we can write: Z ∞ g = (x − l1 ) φ(x)d x l1

= =

=

E(x|x ≥ l1 ) (1 − ϕ(δ1 ))   ∅(δ1 ) (1 − ϕ(δ1 )) µ+σ 1 − ϕ(δ1 ) µ (1 − ϕ(δ1 )) + σ∅(δ1 )

(7)

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Notice here that Equation 7 gives the expected quality characteristic of giveaway. Since b is the cost per unit quality of characteristic (Hariga and Al-Fawzan, 2005), the expected giveaway cost per unit item of class 1 would be b g. Z l1 Next, the expected production cost for grade 2 item, c2 = c(x) φ(x)d x is determined similarly as l2

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c2

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l2 Z l1

(a + b x)φ(x) dx

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follows:

l2

(a + bµ) (ϕ(δ1 ) − ϕ(δ2 )) + σb (∅(δ2 ) − ∅(δ1 )) (8) Z l1 1 2 1 2 1 1 l1 − µ where in Equation 8, ϕ(δ1 ) − ϕ(δ2 ) = φ(x)d x, ∅(δ1 ) = √ e− 2 δ1 , ∅(δ2 ) = √ e− 2 δ2 , δ1 = and σ 2π 2π l2 l2 − µ δ2 = . σ Z =

l2

c(x)d x is determined in the

Lastly, the expected production cost of a non-conforming item, c3 =

following:

c3

= =

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−∞

l2

(a + b x)φ(x) dx

−∞

(a + bµ)ϕ(δ2 ) − σ b ∅(δ2 )

(9)

Given the firm’s capacity, C, if the process mean is set to µ, the total expected number of nonconforming Z l2 items would be, w = C φ(x)d x = C ϕ(δ2 ). −∞

E(Π)

=

p1 y1 (1 − ϕ(δ1 )) − c1 y1 − b g y1 p2 y2 (ϕ(δ1 ) − ϕ(δ2 )) − c2 y2

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Using the costs’ expressions derived earlier, the simplified expected profit function, E(Π), would be:

+

(E(Π) − r ϕ(δ2 ) C) ϕ(δ2 ) − c3 ϕ(δ2 ) C

(10)

Finally, after rearranging the terms in the profit function, would yield: =

1 {p1 y1 (1 − ϕ(δ1 )) − c1 y1 − b g y1 + p2 y2 (ϕ(δ1 ) − ϕ(δ2 )) − c2 y2 − (r ϕ(δ2 ) + c3 ) ϕ(δ2 ) C} 1 − ϕ(δ2 )

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E(Π)

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Thus, the firm’s optimization problem for the deterministic price-dependent deterministic demand in both market segments is formulated as follows: DP :

E(Π)

(11)

subject to: p1 − υ ≥ 0

(12)

(1 − ϕ(δ1 )) C − y1 ≥ 0

(14)

υ − p2 ≥ 0

(13)

(ϕ(δ1 ) − ϕ(δ2 )) C − y2 ≥ 0

(15)

(1 − ϕ(δ2 )) C −

(16)

µ − l1 ≥ 0 12

2 X i=1

yi ≥ 0

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In Equation 11, the total expected profit E(Π), as established in Equation 11, is a multiple decision variables function and these variables include µ, p1 , p2 and υ. In the deterministic constrained nonlinear optimization problem, DP, Equations 12 and 13 are the price differentiation constraints. These constraints imply that the price p1 , for product class 1, must not fall below the price differentiation policy, υ. Also, the price p2 , for the product class 2, must not exceed the price differentiation, υ. The constraints in Equations 14 to 16 are the capacity limitations. Equation 14 implies that the price-dependent deterministic demand, y1 , must not exceed the expected number of class 1 items, (1 − ϕ(δ1 )) C. Likewise, the constraint in Equation 15 states a similar limitation for class 2 products so that the price-dependent deterministic demand y2 must not exceed the allocated capacity, (ϕ(δ1 ) − ϕ(δ2 )) C. Similarly, the constraint in Equation 16 limits the total production of the firms to its capacity C. Since the products with quality characteristic x < l2 are regarded as nonconforming products, it is quite intuitive to realize that the firm must set the process mean as, µ ≥ l1 , to ensure the conformity of the products. In most practical situations the process mean must lie such that l2 ≤ µ ≤ µ, where µ is the maximum permissible process mean, often µ ≥ l1 (Duffuaa and El-Gaaly, 2013a,b). As discussed earlier for E(Π) in DP, once prices, pi , ∀ i = {1, 2} are determined, we can compute the production quantities, yi , ∀ i = {1, 2}. 2.2. Stochastic Demand

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In this section, the model developed earlier for price-dependent deterministic demands is extended to a situation when a firm experiences price-dependent stochastic demand Di , for products class i, ∀ i = {1, 2}. The price-dependent stochastic demand Di has components from the price-dependent deterministic demand yi and from the stochastic price independent demand ξi . The price independent stochastic demand, ξi , has a probability distribution function fi and a cumulative probability distribution function Fi , both continuous, twice differentiable, inverse-able and following an increasing failure rate (Petruzzi and Dada, 1999). These characteristics are often found in some commonly used distributions like Uniform, Normal or Lognormal (Petruzzi and Dada, 1999; Yao et al., 2006). Moreover, ξi is assumed to be bounded in [ξ i , ξ i ] and the expectation of ξi is µξi , with standard deviation σξi . In addition to this, it costs the firm a holding cost hi to stock an item of class i that is unsold, and similarly, a shortage penalty ki per unit for any unmet demand of product class i. To incorporate demand randomness ξi , two modeling approaches, the additive and the multiplicative are often utilized. Recent discussions on these modeling approaches can be found in Petruzzi and Dada (1999) and Yao et al. (2006). In this paper, the additive modeling approach is used, and thus the price-dependent stochastic demand, Di , for market segment i is: Di = yi + ξi ,

∀ i = {1, 2}

(18)

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Petruzzi and Dada (1999) suggested that for an additive approach, a more convenient demand model is the linear function, which has been already adopted in this research for the reasons highlighted earlier in this paper. The total profit, Π, of the firm in this scenario is expressed in Equation 19.   p1 min{q1 , D1 } − h1 [q1 − D1 ]+ − k1 [D1 − q1 ]+ − c(x) q1 − b q1 (x − l1 ) x ≥ l1 ; p2 min{q2 , D2 } − h2 [q2 − D2 ]+ − k2 [D2 − q2 ]+ − c(x) q2 , l2 ≤ x < l1 ; Π= (19)  E(Π) − r w − w c(x), x < l2 .

Similar to Equation 3, in Equation 19 the total profit for the firm is formulated, when the firm experiences price-dependent stochastic demand in each market segment. The formulation relies on the newsvendor problem modeling with pricing and shortage and holding costs in each of the two market segment (Raza, 2014). The first term of Equation 19 is the expected profit from grade 1 products, the second term is the

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expected profit from grade 2 products and the last term is the expected profit (often a loss) from nonconforming products after rework. Following some earlier studies (see Chen et al. (2004); Yao et al. (2006)), we can use the below relationships to simplify the profit function built in Equation 19:

+

Eξi [Di − qi ]

= qi − Eξi [qi − Di ]+ ,

∀ i = {1, 2}

+

= yi − qi + Eξi [qi − Di ] ,

(20)

∀ i = {1, 2}

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min{qi , Di }

(21)

The expression Eξi [qi − Di ]+ can be next deduced using integration by parts such that: Eξi [qi − Di ]+

Z

=

ξ

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i

qi −yi i

qi −yi

(qi − (yi + ξi ))fi (ξi )d ξi (qi − yi − ξi )fi (ξi )d ξi

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qi −yi

=

ξ

Fi (ξi )d ξi

(22)

i

(23)

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Using the simplification from Equation 22, the total profit function would be:  Z q1 −y1   (p + k − c(x)) q − k y − (p + k + h ) F1 (ξ1 )d ξ1 − b q1 (x − l1 ), x ≥ l1 ;  1 1 1 1 1 1 1 1   ξ  1 Z q2 −y2 Π=  (p + k − c(x)) q − k y − (p + k + h ) F2 (ξ2 )d ξ2 , l2 ≤ x < l1 ; 2 2 2 2 2 2 2 2    ξ  2  E(Π) − r w − w c(x), x < l2 . Z

E(Π) =

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From Equation 23 we obtain the mathematical expression for the total expected profit, E(Π) ∞

(p1 + k1 − c(x)) q1 − k1 y1 − (p1 + k1 + h1 )

l1

Z

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+

l1

(p2 + k2 − c(x)) q2 − k2 y2 − (p2 + k2 + h2 )

l2

Z

l2

−∞

q1 −y1

ξ

Z

1

q2 −y2

ξ

F1 (ξ1 )d ξ1 − b q1 (x − l1 ) F2 (ξ2 )d ξ2

2

!

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(p1 + k1 ) q1 − k1 y1 − (p1 + k1 + h1 )

− b q1 − q2

Z

Z

∞

l1

(x − l1 ) φ(x) dx +

q1 −y1

ξ

F1 (ξ1 )d ξ1

∞

φ(x) dx

l1

1

φ(x)d x − q1

(p2 + k2 ) q2 − k2 y2 − (p2 + k2 + h2 )

l1

l2

φ(x) dx

(24)

After rearranging the terms of Equation 24, the firms’ profit can be written as: !Z Z E(Π)

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Z

c(x)φ(x)d x + (E(Π) − r w)

Z

l2

−∞

φ(x) dx − w 14

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Z

ξ

q2 −y2 2

Z

∞

c(x)φ(x)d x

l1

F2 (ξ2 )d ξ2

!Z

l1

φ(x) dx

l2

l2

−∞

c(x)φ(x)

(25)

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Using again the simplification procedure outlined in the price-dependent deterministic demand case presented earlier in Section 2.1, we substitute the costs expressions in Equation 25 and we obtain following simplified formulation for profit, E(Π). ( ! Z q1 −y1 1 E(Π) = (p1 + k1 ) q1 − k1 y1 − (p1 + k1 + h1 ) F1 (ξ1 )d ξ1 (1 − ϕ(δ1 )) − c1 q1 − b g q1 1 − ϕ(δ2 ) ξ 1 ! ) Z q2 −y2 F2 (ξ2 )d ξ2 (ϕ(δ1 ) − ϕ(δ2 )) − c2 q2 − (r ϕ(δ2 ) + c3 ) ϕ(δ2 ) C + (p2 + k2 ) q2 − k2 y2 − (p2 + k2 + h2 ) 2

(26)

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Z l2 Z l1 Z ∞ φ(x) d x. φ(x) d x, ϕ(δ1 ) = φ(x) d x, ϕ(δ1 ) − ϕ(δ2 ) = where in Equation 26, we have 1 − ϕ(δ1 ) = −∞ l2 l1 Z l1 Z l2 Z ∞ Z ∞ Also, c1 = c(x)φ(x)d x, g = (x − l1 )φ(x)d x, c2 = c(x)φ(x)d x, and c3 = c(x)φ(x)d x, are l2

−∞

determined from Equation 6, Equation 7, Equation 8 and Equation 9, respectively. The firm’s stochastic problem (SP) which mainly considers the price-dependent stochastic demand situation, is formulated as follows: SP :

E(Π)

(27)

subject to:

(28)

(1 − ϕ(δ1 )) C − q1 ≥ 0

(30)

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υ − p2 ≥ 0

(29)

(ϕ(δ1 ) − ϕ(δ2 )) C − q2 ≥ 0 (1 − ϕ(δ2 )) C − µ − l1 ≥ 0

2 X i=1

qi ≥ 0

(31) (32) (33)

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In SP, the objective function E(Π) has several decision variables, and these variables include process mean, µ, prices pi , ∀ i = {1, 2}, production quantities qi , ∀ i = {1, 2}, and differentiation price, υ. The constraints in Equations 28- 33 are similar to DP’s constraints, however, the production quantities qi , ∀ i = {1, 2} are replacing, yi , ∀ i = {1, 2}, due to the stochastic price-dependent demand in SP. In the stochastic demand situation, the firm would hold safety stock, qi − yi , ∀ i = {1, 2} and depending on the problem related parameters, the safety stock can be positive or negative (see Arrow et al. (1951) and Petruzzi and Dada (1999) for details). 3. Harmony Search Meta-Heuristic As notified in Darwish et al. (2013), the process mean (targeting) problems are often very complex, being difficult or even infeasible to solve them with the existing numerical optimization methods. With no exception, the problems considered here, DP and SP, are complex constrained nonlinear optimization problem. Thus, the meta-heuristics based direct search methods are a better choice for solving both problems. 15

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The major advantage of the meta-heuristics would be that they can find near optimal solution without gradient information. It is important to notice here that, due to the problem nature, using a gradient based method such as Karush Kuhn Tucker (KKT) (see Bertsekas (1999)) can be computationally very inefficient. Moreover, the derivative analysis does not lead to important insights about the problems’ high number of parameters and decision variables. Other gradient or Hessian-based optimization methods applied on such complex problems are usually time consuming and may offer results which are sensitive to the starting point selection in the searching procedure, or which need complex derivative calculus of a large memory space. Since there is no proof on the uni-modality (convexity) of the profit functions, there is a possibility that the problems may have many local optima while KKT does not guarantee convergence to a global optimal solution Bertsekas (1999), and finding the global optimal solution could be a prohibitive task. A traditional optimization approach using KKT optimality conditions is also expected to fall short of any analytical insights as it would require solving several nonlinear equations numerically only with strong possibility of no closed-form solution. On the other hand, both practitioners and academic researchers have recognized that meta-heuristics are an efficient approach of hard optimization problems and they have contributed considerable to its development. The research field of meta-heuristics is growing rapidly and many efficient alternatives have been designed to find near-optimal solution for hard optimization problems. Due to their diverse features, is difficult to make a precise classification of the existing meta-heuristics, however, some classification criteria adopted by researchers are search type based on single solution vs. population (Boussa¨ıd et al., 2013), neighborhood structures or memory usage (Dorigo and St¨ utzle, 2004), source of inspiration (nature or non-nature inspired) end others. Among the popular meta-heuristics techniques are genetic algorithm (GA), scatter search (SS), tabu search (TS), simulated annealing (SA), ant colony optimization (ACO), harmony search (HS). The common evolutionary computation (EC) techniques from GA and SS use mainly the principle of evolution based on survival-of-the-fittest and replicate some natural phenomena like genetic inheritance. GA is a very popular EC technique based on the natural selection and the mechanisms of population genetics. Its theory was first proposed in Holland (1975) and assumes three main operators of reproduction, crossover, and mutation, where reproduction is the process of survival-of-the-fittest selection, crossover is a partial swapping between two parent strings to produce two offspring strings and mutation is the occasional random inversion of bit values that generates non-recursive offspring. The main characteristic of the GA is that it maintains and evaluates a set of solutions, providing an efficient exploration of the search space. SS differs from other EC techniques by providing unifying principles for joining solutions based on generalised path constructions in Euclidean space and by providing a search strategy rather than randomisation (Glover et al., 2003). Major contributions to SS were done by, but ont only, Laguna (2002). The use of adaptive memory incorporated in SS relates it with the later TS meta-heuristic formalized in Glover (1986). TS is based on a single neighbor-hood structure and simulates the natural phenomenon of human memory where the search memorizes a number of previously visited states or solution along with a number of states that might be considered unwanted. The candidate solutions are stored in a tabu list. The definition of tabu criterion, tabu list size, and information stored in the tabu list are often regarded very critical towards the performance of tabu search algorithm. The SA is a memoryless single-point search algorithm based on Monte Carlo strategy iterative method and it was proposed as an optimization framework for multivariate and combinatorial problems. Drawn from the analogy with annealing in solids, the algorithm applies to very large and complex systems, and was defined for the first time in Kirkpatrick et al. (1983) as an optimization technique. Another nature-inspired meta-heuristic is ACO, introduced by Dorigo (1992) inspired by the foraging behavior of ant colonies in search for food and in practice, it uses the learning principle to identify the good combinations of solution components by sampling the search space in every iteration. As described in Dorigo and St¨ utzle (2004) ACO follows the collective trail-laying and trail-following behavior whereby 16

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communications between individuals and environment rely on the use of chemicals deposited by the ants called pheromones, for marking paths on the ground while walking from food sources to the nest. The cooperative iteration leads the ant colony towards the shortest path which confers successful applicability of ACO in discrete optimization problems. HS is a relatively new population-based meta-heuristic inspired by musical improvisation process to seek a pleasant harmony, conceptualized by Geem et al. (2001) on the analogy that searching for the perfect harmony in music is similar with searching for the optimality in an optimization process. A performing music seeks the best harmony generated by an instrument, which is determined by an aesthetic estimation of the musician that plays the instrument. Similarly, an optimization algorithm seeks the global optimum which represents the values of the objective function. The aesthetic quality of the harmony is determined by a set of the sounds played different joined instruments, as the objective function value is determined by the values of its variables set. The musicians search for a better harmony by improvising the aesthetic valuation through practice, as the optimization algorithm search for a better value of the objective function by improving its values in an iterative process. Therefore, the algorithm’s parameters can be identified from the following analogies: each practice performed by the musician represent one iteration of the HS algorithm; the musicians represent the problem’s decisions variables; the sound pitches of each musician are the values of each decision variable; the esthetic standard of the harmony is the objective function of the problem; and the global optimum represents the perfect harmony. The parameters of HS, as will be described later, allows for a better control of diversification and intensification than other meta-heuristics in the search for a optimal solution. GA may have some drawbacks founded in convergence towards local optima rather than the global optimum, lack of good local search ability or low convergence speed. In this regard, Huang and Lin (2013) used HS as a fine tuning of the GA’s crossover mechanism to improve the original GA algorithm for solving the combined heat and power economic dispatch problem. While TS technique explores the search space beyond the local optimality to return the near-optimal solution, HS can be applied to determine the global optimal solution near the surrounding area of such near optimal solution (Mashinchi et al., 2011). One improvement of the slow convergence speed of SA was recently approached in (Askarzadeh, 2013) by integrating the adaptive search method of HS and chaotic search, in discrete SA algorithm for finding the optimum design of photovoltaic/wind hybrid system. Lee and Geem (2005) provide various examples of engineering optimization with HS algorithm and demonstrate the effectiveness and robustness of this metaheuristic. A more recent portfolio of HS applicability in both continuous and discrete optimization problems can be found in Manjarres et al. (2013). In practice, the possible choices a musician may have when improvises one pitch are: (i) playing any pitch exactly from memory; (ii) playing something similar - a neighbor pitch of one pitch from the memory; or (iii) playing a random pitch from the possible range of sounds. Likewise, when each decision variable chooses one value in the HS algorithm, it could have followed any one of the three rules: (i) choosing anyone value from the harmony memory (HM) which is defined as memory consideration; (ii) choosing an adjacent value of the one value from HM, referred here as pitch adjustments, and (iii) generating a totally random value from the possible value range, regarded as randomization. These three rules are formalized in the HS algorithm as two other control parameters: (i) Harmony Memory Considering Rate (HMCR); and (ii) Pitch Adjusting Rate (PAR). As can be observed from Figure 3 outlines the optimization procedure of HS meta-heuristic algorithm. It mainly contains following steps: • Step 1: Initialize the optimization problem and algorithm parameters • Step 2: Initialize HM • Step 3: Improvise a New Harmony (NH) from HM 17

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• Step 4: Update the HM

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• Step 5: Repeat Steps 3 and 4 until the termination criterion is satisfied

18

Step 1: Initialization

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START

Initialize the algorithm parameters HM, HMS, HMCR, PAR,ψ, Maximum number of Searches, Number of decision variables, Bounds of each decision variable

Step 2: Initialize the harmony memory

Evaluate f ( x )

Include harmony x j Yes in Initial HM Sort HM by f (x)

xi '{x , x ,...x ( 2) i

( HMS) i

d

xi ' Xi

f (x')

Satisfies feasibility? Yes

Generate Uniform Number for xi ' , i 1, N

M

xi ' such that xi ' m xi ' ψ

No

No

Generate a Random Number

Adjust

No

x ' { x1' , x1' ,... xN ' }

No

HMCR

}

j=HMS

Improvise NH

RAND Yes

RAND Yes

d

No

PAR

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Step 3: Improvise a new harmony

(1) i

Yes

f (x) Satisfies feasibility?

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Improvise NH

x ' { x1' , x1' ,... xN ' }

Step 5: Repeat Step 3 &4 until termination criterion satisfied

Satisfies feasibility?

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Step 4: Update HM

f (x')

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Generate Uniform Number for xi , i 1, N

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Generate a harmony

xj , j {1,2,...HMS}

Replace worst entry in HM with NH

No

Yes

Evaluate f (x' )

NH better than worst harmony in HM

Yes

No

Sort HM by f (x)

Reached maximum Searches?

No

Yes STOP

Figure 3: Harmony Search (HS) meta-heuristic flow chart

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Generate a Random Number

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3.1. Step 1: Initialize the optimization problem and algorithm parameters First, we consider an unconstrained optimization problem specified below: Maximize :

f (x)

(34) x∈X

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subject to: (35)

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where, f (x) is the objective function and x is the set of decision variables such that, xi ∈ Xi , ∀ i = {1, 2, · · · , N }. Xi is the set of possible values bounded of the decision variables xi such that xi ≤ xi ≤ xi ,∀ i = {1, 2, · · · , N }. N is the total number of decision variables. In DP, the objective function f (x) = E(Π) is established in Equation 11 and the decision variables set is x = {µ, p1 , p2 , υ}. For SP, we have f (x) = E(Π) established in Equation 26, and x = {µ, p1 , p2 , q1 , q2 , υ}. The bounds on the control variables α1 are l1 ≤ µ ≤ l1 + 6 σ, and for the controls p1 , p2 and υ the upper bound is , and lower bound at zero. β1 In addition to this, for problem, SP, the upper bound on qi , ∀i = {1, 2} is capacity, C, and lower bound at zero. The HS algorithm parameters required to solve the optimization problem are Harmony Memory Size (HMS), HMCR, PAR, and the termination criterion i.e., the maximum number of permissible searches. The purpose of HMCR and PAR is to improve the HM (Lee and Geem, 2005) which will be discussed in Step 3. 3.2. Step 2: Initialize HMS

(36)

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In this step, the HM outlined in matrix below, is created.  (1)  x  x(2)   HM =   ···  x(HM S)

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Equation 36 is filled randomly while following the bounding criterion on each decision variable, xi , and meeting the feasibility requirements. Once HM is filled, the solutions x(i) are sorted for the case maximization objective, in the descending order by their objective function value, f (x(i) ). 3.3. Step:3 Improvise an NH from HM

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In this step, an NH solution x0 = {x01 , x02 , · · · , x0N } is generated based on HMCR parameter. For instance, (1) (2) (HM S) a decision variable x0i , can be either chosen among the existing values in HM, {xi , xi , · · · , xi } or, a 0 0 New Harmony, xi is generated such that xi ∈ Xi and also satisfies the feasibility criterion. This selection process can be outlined as follows:  (1) (2) (HM S) x0i ∈ {xi , xi , · · · , xi } with probability HMCR 0 xi ← (37) 0 xi ∈ Xi with probabilitiy, 1- HMCR

Once x0i is determined, then it is checked for feasibility. In case of infeasibility, a New Harmony is improvised by repeating this step until feasibility is satisfied. The HMCR is the probability of choosing one value from the historic values stored in HM and (1- HMCR) is the probability of randomly choosing one feasible values not limited to values stored in HM. For instance, in this implementation we have assumed HMCR=0.85. This implies that HS algorithm will choose a feasible value from HM with 85% probability. Whereas, HS can 20

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also generated totally new but feasible value outside HM with 15% probability. It is not recommended to have HMCR=1. This criterion is used in pursuit of improving the HS process by considering value outside HM. This feature resembles mutation rate in GA. Every component of the NH vector, x0 = {x01 , x02 , · · · , x0N } is considered to determine whether it should be pitch adjusted. PAR parameter is used in this process to determine whether a decision variable, x0i as follows:  Yes, with probability PAR Pitch adjustment decision on x0i ← (38) No, with probabilitiy, 1- PAR The pitch adjustment process is only executed once a value in NH is chosen from the HM with a probability, PAR. We have selected PAR=0.35 in this implementation of HS. Naturally, 1- PAR is the probability of having no pitch adjustment on a variable x0i in NH. In case the pitch adjusting decision for a variable, x0i in NH is Yes, then the variable x0i is adjusted as follows:

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(39)

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where ψ is the bandwidth with for a continuous decision variable used for adjustment. ψ is selected randomly such that ψ ∈ U [ψ, ψ]. As earlier suggested in Lee and Geem (2005), ψ = −1, and ψ = 1 which has been adapted in this implementation of HS as well. A detailed illustration of the NH continuous search strategy based on the HS meta-heuristic algorithm can be found in Lee and Geem (2005). The HMCR and PAR parameters introduced in the harmony search help the algorithm find globally and locally improved solutions, respectively. 3.4. Step 4: Update the HM

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In Step 4, if the NH vector is better than the worst harmony in HM in terms of the objective function value, the New Harmony is included in the HM and the existing worst harmony is excluded from the HM. The HM is then sorted by the objective function value. 3.5. Step 5: Repeat Steps 3 and 4 until the termination criterion is satisfied

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In Step 5, the computations are terminated when the termination criterion is satisfied. If not, Steps 3 and 4 are repeated. In Table 2, we have selected the best observed values for the parameters for HS meta-heuristic after conducting a series of numerical experimentation with these parameters. Table 2: HS meta-heuristic parameters

Features

Parameters

Harmony Memory Generation New Harmony improvisation Bandwidth adjustment Stopping criterion

HMS=10 HMCR=0.85, PAR =0.35 ψ = −1, ψ = 1 5000 searches

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4. Numerical Example

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In this section, a numerical study is presented for DP and SP. In DP, a firm experiences price-dependent deterministic demand and in SP, the demand experienced is price-dependent stochastic. The purpose of the numerical illustration is to examine the impact of demand leakage onto a firm’s strategy for an integrated optimal control of price differentiation, pricing, capacity allocation and process mean decisions. This research problem can be regarded as an interface between production system and pricing theory from economics. We have selected the problem related parameters from published related research in both pricing and process targeting. In Table 3 we present the data selected for the parameters of the integrated problem of RM and process targeting. The pseudocode of the HS flowchart from Figure 3 was implemented in MATLAB Release 2013a and tested on Intel Core i7-3520M processor 2.9 GHz and 8 GB RAM, with Windows 7, 64-bit operating system. Table 3: Parameters selection for numerical experimentation 200 11 9 4 1 500 7 9 15 0 {3,5} {2,3,4,5}

Best guess of the authors

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C l1 l2 a b α1 β1 β2 r µξi σξi σ

Source

Best guess of the authors perceived from process targeting literature (Duffuaa and El-Gaaly, 2013a,b) Best guess of the authors perceived from process targeting literature (Hariga and Al-Fawzan, 2005) Best guess of the authors perceived from process targeting literature (Phillips, 2005; Zhang et al., 2010) Best guess of the authors Perceived from pricing literature (Zhang et al., 2010)

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Authors’ selection for numerical experimentation, ∀ i = {1, 2}

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4.1. Deterministic Demand In the deterministic case it is obvious to notice that σξi = 0, ∀ i = {1, 2}, and therefore, demand information is perfectly known to a firm. In Table 4 is reported the impact of demand leakage (θ) and process variability (σ) onto the firm’s profitability. An increase of both parameters values, demand leakage, θ, and process variability, σ, tends to diminish the optimal expected profit, E(Π∗ ) of the firm. Thus, for a process variability σ = 2, an increase in demand leakage from 0 to 15% causes a reduction around 15.2% of the optimal profit gains from $2522.035 to $2139.07. As the process variability, σ, increases, the firm’s profit will have a more diminishing trend. Thus, when the process variability is high, in this example σ = 5, the optimal expected profit of $1419.52 is diminished at 15% demand leakage with almost 23.5% to $1065.90. Furthermore, an increase in demand leakage, θ, causes a firm to lower the process target mean, µ, as with a high demand leakage level, the expected demand for product class 2 is likely to increase. It can be noticed from Table 4 that, when σ = 2 and there is no demand leakage, the optimal process target mean is 13.62, which decreases to 13.25 when demand leakage increases to 15%. Similarly, at high process variability σ = 5, with no demand leakage, the optimal process target mean is 16.63 which is also reduced to 15.94 when the firm experiences a demand leakage of 15%. Noticeably here that a firm’s optimal strategy is directed towards minimizing the price difference p1 − p2 as the demand leakage factor, θ, increases in response to high proportion of the demand leakage from product class 1 to the product class 2. Figure 4 illustrates the impact of the exogenous factors of demand leakage and process variability onto the firm’s profit under deterministic demand assumption. 22

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Table 4: Numerical experimentation with deterministic demand

θ

µ∗

p∗1

p∗2

υ∗

E(Π∗ )

CPU(sec.)

2

0 0.05 0.1 0.15

13.62 13.52 13.67 13.25

51.52 51.35 51.45 51.21

49.56 49.63 40.73 49.60

51.44 50.89 40.97 49.76

2522.03 2396.37 2255.65 2139.07

11.19 12.78 16.24 16.73

3

0 0.05 0.1 0.15

14.64 14.77 14.70 14.30

52.68 52.63 52.45 53.01

46.00 42.39 51.50 34.57

47.78 43.36 51.80 34.57

2122.84 2004.89 1903.60 1745.99

11.32 12.21 17.59 19.89

4

0 0.05 0.1 0.15

15.66 15.75 15.90 15.18

54.46 53.60 53.92 55.05

52.80 52.25 40.80 55.03

54.44 53.15 40.95 55.04

1757.67 1659.09 1545.74 1422.08

14.15 14.45 20.30 20.50

5

0 0.05 0.1 0.15

16.62 16.78 17.00 15.79

54.61 54.66 54.83 55.86

48.91 53.56 52.21 55.70

50.39 54.33 52.25 55.70

1415.49 1327.95 1235.71 1083.16

12.95 15.79 28.09 25.03

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σ

2600

σ=2 σ=3 σ=4 σ=5

2400

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2200

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E(Π∗)

2000 1800 1600 1400 1200 1000

0

0.02

0.04

0.06

0.08 θ

0.1

0.12

0.14

0.16

Figure 4: Impact of demand leakage (θ) and process variability, σ for deterministic demand situation

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4.2. Stochastic Demand In the stochastic demand situation the price-dependent deterministic demand parameters values are set as in the previous subsection and, as we have assumed, for each product class the stochastic demand mean is µξi = 0, ∀ i = {1, 2}. For simplicity σξi are assumed equal for each product class segment, thus, σξ = σξi , ∀ i = {1, 2}, and for brevity purposes only two values of process variability are used in the numerical experimentation, such √ σξ√= {3,  5}. Consistent with Mostard et al. (2005), the stochastic factor of  that, demand is assumed ξi ∈ − 3σξ , 3σξ . In a complex problem like this one, the numerical experimentation is conducted with uniform and normal distributions only. Tables 5 and 6 report the numerical experimentation results when a firm faces stochastic price-dependent demand distributed uniformly and normally, respectively. Also, we assumed the holding per unit of excess inventory and the shortage cost per unit of unmet inventory such that, hi = ki = 1, ∀ i = {1, 2}. We can remark here that depending on the market demand situation, a firm may hold a positive (or negative) safety stock which is actually a differential of capacity allocation for a product class and price-dependent deterministic demand, at a given price for that product class. In both uniform and normal distribution scenarios, the findings on the process variability influence, σ, onto the firm’s profit and the production process mean are similar to deterministic demand situation. Therefore, the firm’s profit will follow a descending trend as long as process standard deviation is intensifying. This is intuitive since a high value of standard deviation will scatter the quality characteristic of the products from the target mean and will generate more non-conforming products in production output, thus, the profit decrease as more non-conforming products are produced. Also, the optimum process mean setting is higher if the process standard deviation increases which is obvious since a high process variability indicate a high probability of violating the specification limits resulting more products of class 2 and 3. These results are consistent with the findings from process targeting literature (Hariga and Al-Fawzan, 2005; Duffuaa and El-Gaaly, 2013a,b). Similarly, demand leakage, θ, has a diminishing influence onto the firm’s profit, as it intensity increase, contributing beside the process variability, σ at the profitability’s descending trend. For instance, at any given demand variability, σξ , an increase in demand leakage, θ, drives the firm to increase more of its capacity for class 2 products. Naturally, the firm increases prices in both product classes, while reducing the price differential, p∗1 − p∗2 , to mitigate the demand leakage effect. For example, in the uniform demand distribution case, when σξ = 3, at a process variability of σ = 2, the firm will increase its capacity for class 2 products with 42.4% when demand leakage increase from 0% to 15% and the price differential p∗1 − p∗2 drops from 17 units to less than 1, causing a profit decrease of 14.2%. Likewise, the example for normal distribution demand gives the same response of the firm’s decisions on prices and quantities allocation which will generate a decrease on the optimal profit slightly moderate than in the case of uniform demand distribution. Interesting to notice here is that, unlike in the deterministic case, demand leakage becomes less influential onto the optimal process mean due to the cumulative stochastic effect from both market demand and process variability. Thus, the firm is lowering the process target mean, µ, as demand leakage, θ, increases, only if the process variability, σ, and demand variability, σξ , are low. In the example with uniform distribution, when σ = 2 and demand variability σξ = 3, the firm is lowering the process mean, µ, with 1.6%, as demand leakage increase to 5%. For the same σξ = 3, if the process variability increases beyond 2, then the firm will increase the process mean while demand leakage intensifies. While studying the effect of demand variability, σξ , as its level increases the firm tends to increase the process target, i.e., the mean quality characteristics, and obviously the firm’s profitability diminishes as well. Similar to the numerical study with deterministic demand, Table 5 and Table 6 reports a numerical experimentation study under stochastic demand with uniform and normal distribution, respectively. The comparative overall results from the two table shows that a firm experiences better profitability when demand is observed with normal distribution. This finding is due to the structure of normal distribution, while the rest of the problem related parameters are the same. The reported CPU time differs considerably from one case to another, due to the 24

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different structures of the uniform and normal distributions and the MATLAB implementation of demands’ probability distribution functions.

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Table 5: Numerical experimentation with uniformly distributed stochastic demand

θ

µ∗

p∗1

p∗2

υ∗

q1∗

q2∗

E(Π∗ )

CPU(sec.)

2

0 0.05 0.1 0.15

13.87 13.74 13.69 13.64

53.02 51.86 52.60 51.76

35.79 50.37 51.86 50.97

37.88 51.84 52.59 51.01

127.52 128.98 117.03 115.80

13.63 15.24 15.88 16.21

2425.58 2331.14 2199.83 2080.27

256.83 256.27 295.78 259.47

3

0 0.05 0.1 0.15

14.72 14.80 14.73 14.72

52.06 53.47 52.95 52.84

44.12 51.97 52.11 52.70

46.60 53.42 52.94 52.81

133.87 117.65 114.97 109.22

15.81 15.12 15.72 15.80

2059.09 1949.85 1842.49 1730.46

239.03 264.43 309.22 299.43

4

0 0.05 0.1 0.15

15.83 15.63 15.84 15.93

54.42 53.38 53.66 53.46

51.93 48.42 49.30 53.03

5

0 0.05 0.1 0.15

16.78 16.85 17.11 17.31

55.09 55.03 55.34 55.31

53.04 52.93 45.98 49.16

2

0 0.05 0.1 0.15

13.40 13.42 13.72 13.73

51.66 51.72 51.12 51.37

3

0 0.05 0.1 0.15

14.89 14.69 14.85 14.76

4

0 0.05 0.1 0.15

5

0 0.05 0.1 0.15

3

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117.50 118.32 110.18 105.02

13.93 14.83 13.88 13.46

1695.01 1594.03 1494.17 1396.49

276.09 280.20 230.17 307.33

55.00 54.20 47.11 49.18

112.40 107.18 98.79 93.73

12.80 12.52 11.68 11.02

1357.51 1266.83 1168.80 1083.64

294.72 283.97 249.16 259.40

48.52 49.43 48.32 48.89

51.65 51.71 49.21 49.23

136.73 129.32 125.93 117.37

20.19 19.87 15.56 15.40

2418.73 2293.59 2162.37 2036.60

285.42 285.36 211.99 254.18

54.26 52.34 52.56 52.78

51.89 50.44 51.44 52.35

54.09 52.33 52.43 52.77

117.11 124.48 116.19 108.48

14.51 16.08 14.82 15.54

2003.95 1915.45 1800.63 1689.51

376.54 265.10 206.82 292.93

15.79 15.69 15.92 15.90

53.61 53.52 53.75 53.44

50.85 51.77 52.57 52.50

53.48 53.51 53.61 52.69

121.75 116.28 108.38 103.88

14.12 14.62 13.50 13.61

1658.86 1558.71 1456.82 1356.27

275.75 302.69 229.56 311.21

16.83 16.81 17.10 17.25

54.79 54.68 55.08 55.12

45.35 53.07 53.82 53.05

47.77 54.68 54.98 53.61

113.33 108.06 99.55 93.64

12.54 12.65 11.70 11.15

1312.27 1228.71 1137.11 1047.57

240.65 256.04 314.51 266.19

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σξ

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Table 6: Numerical experimentation with normally distributed stochastic demand

θ

µ∗

p∗1

p∗2

υ∗

q1∗

q2∗

E(Π∗ )

CPU(sec.)

2

0 0.05 0.1 0.15

13.71 13.76 13.64 13.62

51.12 52.03 51.46 51.44

45.74 48.28 50.71 50.98

47.94 49.75 51.45 50.98

141.29 127.98 125.00 118.16

15.62 15.01 16.66 16.93

2466.06 2340.20 2221.75 2095.71

1328.39 1316.07 1649.51 1331.56

3

0 0.05 0.1 0.15

14.68 14.87 14.70 15.05

54.00 53.30 52.30 53.60

51.61 43.15 49.27 39.02

53.84 44.42 50.73 39.08

121.10 119.65 119.61 105.05

16.09 14.63 15.97 13.32

2068.86 1953.97 1852.17 1721.35

1722.42 1433.52 1470.33 1452.89

4

0 0.05 0.1 0.15

15.56 15.93 15.71 16.09

52.65 54.45 53.22 53.95

50.47 52.39 47.76 50.29

5

0 0.05 0.1 0.15

16.89 16.65 16.92 17.13

55.61 54.28 54.35 55.22

50.77 52.54 53.68 54.99

2

0 0.05 0.1 0.15

13.73 13.64 13.57 13.64

51.92 52.32 51.23 51.87

3

0 0.05 0.1 0.15

14.63 14.63 14.63 14.89

4

0 0.05 0.1 0.15

5

0 0.05 0.1 0.15

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5

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15.30 13.14 14.54 12.67

1708.49 1605.19 1503.87 1404.97

1839.81 1514.64 1603.22 1545.32

53.28 53.94 54.20 55.03

109.10 112.55 106.20 95.07

12.34 13.23 12.31 11.62

1361.54 1279.34 1189.24 1100.70

1362.08 1580.55 1637.33 1738.16

49.42 50.49 50.07 51.46

51.66 52.32 51.20 51.86

135.27 125.90 125.93 115.13

15.24 16.62 17.62 16.56

2436.42 2310.56 2190.22 2062.36

1278.09 1733.00 1666.71 1571.16

52.17 52.38 51.90 52.84

49.58 49.97 50.74 49.86

52.16 51.89 51.85 50.50

133.15 125.03 121.31 108.92

16.58 16.56 16.55 14.48

2049.25 1936.17 1822.16 1705.99

1603.89 1460.05 1454.12 1480.27

15.55 15.85 15.81 15.90

52.98 53.58 53.41 53.83

50.31 50.49 52.28 53.48

52.85 52.37 53.16 53.78

127.22 116.72 111.47 102.75

15.33 13.84 14.04 13.60

1679.47 1576.91 1477.67 1377.38

1592.71 1471.24 1663.11 1614.23

16.75 16.93 16.78 17.24

54.63 55.08 54.17 55.15

50.25 53.10 53.18 54.98

52.64 54.71 54.11 55.12

115.42 106.40 106.44 94.54

12.91 11.94 12.81 11.27

1337.46 1245.90 1157.40 1068.72

1771.12 1566.77 2010.81 1770.73

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52.63 53.79 48.40 50.41

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σξ

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4.3. Impact of optimal price differentiation

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In this section, a comparative analysis is presented where the impact of optimal segmentation is studied numerically for price-dependent deterministic demand situation. Earlier, Phillips (2005) presented a similar comparative analysis of how does an optimal price differentiation strategy can improve profitability to a firm in contrast to an arbitrary price differentiation strategy. Figure 5 presents a comparison of the fixed price differentiation strategies, υ = {30, 40, 50} and the optimal price differentiation, υ ∗ . Typically, when the price differentiation is fixed, for instance at υ = 30, then the firm will optimize the prices, p1 , and p2 , along with the capacity allocation decisions, q1 , q2 , and w for class 1 products, class 2 products and reworked items, respectively. From Figure 5 it can be clearly observed that a firm can achieve superior profitability while segmenting the market using the proposed optimal price differentiation strategy, υ ∗ . The numerical experimentation results reported in Figure 5 are generated for a low process variability σ = 2. A similar analysis with a high process variability, σ = 5, is reported in Figure 6. Form Figure 6 it is more evidently remarked that the optimal price differentiation is more attractive when profitability is a performance measure and process variability is high. The figure provides a graphical support to the fact that, an increase of process variability is influential towards the performance of the proposed integrated framework, undoubtedly, the leakage also is found to be influential. The findings on optimal price differentiation strategy influence onto the firm’s profitability when it experience stochastic demand behavior are obviously similar to the deterministic scenario, thus, we resume to expose the results for only one demand situation. 2550

υ=30 υ=40 υ=50

2500 2450

∗

2350 2300

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E(Π∗)

υ

M

2400

2250 2200 2150

PT

2100

0

0.02

0.04

0.06

0.08 θ

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2050

0.1

0.12

0.14

0.16

AC

Figure 5: Impact of optimal segmentation at process variability, σ = 2 for deterministic demand

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1450

υ=30 υ=40 υ=50

1400

∗

υ

1350

1250 1200

M

E(Π∗)

1300

1150 1100

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1050 1000

0

0.02

0.04

0.06

0.08 θ

0.1

0.12

0.14

0.16

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Figure 6: Impact of optimal segmentation at process variability, σ = 5 for deterministic demand

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5. Conclusions, limitations and future research suggestions:

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In this paper, we proposed mathematical models for an optimal joint determination of process mean, pricing, production quantity and market segmentation for a manufacturing firm. The market segmentation is achieved using a differentiation price, however, it is assumed imperfect and therefore the demand leakage is experienced. We proposed two models, the first model considered the price-dependent deterministic demand and the later model considered the price-dependent stochastic demand. The models developed are solved using a meta-heuristic and we have implemented a harmony search algorithm. A numerical illustration is presented to explore the effects of the model related parameters on the joint optimal decisions of the firm. This research has highlighted following findings:

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1. When a firm adopts an integrated optimal strategy of process targeting, pricing and price differentiation decisions, it yields significantly superior profits compared to the situation when the firm selects an arbitrary price differentiation to segment the market. From the numerical study, this finding is consistently observed in both deterministic and stochastic demand situations. 2. As the process variability increases, the firm response in a deterministic demand scenario, is to increase the process mean to maintain the desired productivity of the conforming items for each product class. 3. As response to an increase in demand leakage, the firm raises the prices in both market segments, however, it also skims the price differential between the two product classes to mitigate the demand leakage.

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Although this is the first study that suggests a mathematical model to outline an integrated optimization approach of pricing and process targeting problem with the demand leakage effect, yet, there are many avenues still unexplored. At present, the proposed model assumes that the distinct market segments are determined and maintained without any fencing cost incurred. As rightly identified in Zhang et al. (2010), a fencing cost may be observed by the firm, once it enforces a control mechanism to minimize the demand leakage between market segments. Also, there is only a single process considered in this study, whose mean needs to be determined by the firm along with other decision controls to optimize its profit. Nevertheless, in many real life situations it is not uncommon that the products are manufactured through a number of processes. Furthermore, this study uses additive model for price-dependent stochastic demand with linear deterministic price-dependent demand curve, which is the most widely used in literature (Zhang et al., 2010; Raza, 2015b). However, depending upon the modeling framework of stochastic price-dependent demand and the type of deterministic price-dependent demand curve, the model can be substantially different in performance (Smith et al., 2007). Nevertheless, a detailed comparative analysis of modeling frameworks and price-dependent deterministic demand curve for this problem is not presented in this paper. The future work directions may include the use of the proposed integrated approach to a firm that manufactures products in more than one production process, therefore, an extension of this study could be to consider optimal means for a combination of processes. The present analysis has considered the firm in monopoly only, but an interesting avenue would be to consider a game theoretic approach to the problem of the firm in a duopoly or oligopoly. We have formalized our problem based on the risk-neutral newsvendor problem under a given demand distribution, however, the risk-averse newsvendor with pricing and quantity decisions can take an new interesting joint approach with the targeting problem when a risk-averse decision criteria as expected utility (EU) maximization, mean-variance (MV) analysis, or conditional value-at-risk (CVaR) minimization (see Katariya et al. (2014) for more details) is considered into the the problem. In this study, we have presented a meta-heuristic using harmony search, indeed a comparative analysis of the proposed meta-heuristic is needed to benchmark its performance. To this need, an interesting research direction would be to consider developing tighter bounds on the expected revenue functions for the models 30

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developed here. One such possibility would be develop a lower bound on the expected total revenue using distribution-free approach (see Raza (2014), and Raza (2015b) for details). This performance of harmony search may also be tested by comparing it with other meta-heuristics which were discussed earlier in this paper. Acknowledgement

This publication was made possible by the support of an NPRP Grant # 4-173-5-025 from the Qatar National Research Fund. The statements made herein are solely the responsibility of the authors. References

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