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A game theory-based model for product portfolio management in a competitive market

A. Sadeghi, M. Zandieh Expert Systems with Applications,2010

Structure
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Introduction Literature review Description of the PPM problem Problem formulation Example Conclusions and future work

1. Introduction
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Consumers, industrial managers, and sales and marketing people, all demand products that improve their lifestyles or to gain an edge over the competition. So, product portfolios are interesting for many people. But unlimited product variety is not a way to be successful; there has to be an optimum. It is true for most companies that the Pareto rule applies:

80% of the sales and/or pro?ts come from 20% of the products . It is evident that a single product cannot

ful?ll the manufacturer needs and on the other hand, for diversity there exists limitation .
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In today’s highly competitive environment, determining an optimal product portfolio is very important for the survival of a ?rm. Optimal product portfolio has received considerable attention,because the rates of failure of new product portfolio and their associated losses are very high . The whole product component information(产品构建信息), engineering

portfolio decision (工程组合决策)is very crucial for the progress of a ?rm ,because it is very costly and difficult to change .The key questions are, what the best product portfolio is, and how manufacturer can ?nd it.
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Product portfolio management (PPM) is a general business concept that analyze the production ability (生产能力) and market potential, simultaneously, and then determine the best set of products to offer. PPM is developed to direct a product and its diversity

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including not only attributes(属性), levels, and price’s, but also analysis results,environmental requirements(环保需求), manufacturing procedures (生产流程), product performance information(产品 性能信息), and etc.Therefore PPM has been classified as a combinatorial optimization problem. Each company strives for the optimality of its product offerings through various combinations of products . ? The PPM problem may develop from two perspectives:(I) For attract the opinion of customers

in target markets. (II) For reduce the manufacture engineering costs. First is the problem of marketing managers, and second is the problem of producer. When both of them compose with each other as reflect to utility of costumers and engineering costs, this problem becomes to miss link between sale and production chain. Jiao and Zhang(2005) consider the customer–engineering interaction in product portfolio planning, which aims to create product family

speci?cations(产品族/系列规格) for a target market segment, and proposed a maximizing surplus share model(最大剩余份额模型).
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In competitive environment, we determine our product portfolio with regard to products that offer by competitors, while the competitors manage their product portfolios in regard to our products. Game theory can be used to model this problem. The proposed model constructs product portfolio

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based on customer–engineering interaction model in product portfolio planning which is developed by Jiao and Zhang .Present paper extends previous works in PPM with regard to customer–engineering concerns and competitive environment. It is not for any specific product, and it can be applied to a diversity of products or services.
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objective: develop a game theory-based model as a procedure of ?nding optimal product portfolio.

2. Literature review
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A PPM is de?ned as a decision making that optimizes some criteria, such as market share. The main contribution of the most researches in PPM is summarized in following issues: 1) Generating design alternatives via multi-objective optimization(通过多目标优化生成设计方案). 2) Accounting for uncertainty and competition when estimating the achievement of business goals. 3) Applying meta-heuristic algorithms(元启发式算法)

to solve a combinatorial problem during the product line design.
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The development of algorithms Heuristic ( identify product pro?le product line design) algorithms improved heuristic algorithms genetic algorithms.

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The development of models 1) Jiao and Zhang proposed a model to address the product portfolio planning problem,it considers customer preferences, choice probabilities and

platform based product costing. Also, a genetic algorithm procedure is applied. 2) Aiyoshi and Maki proposed a game problem under the constraints of allocation of product and market share simultaneously. Their research is considered several manufacturers in oligopoly market(寡头垄断市 场). This proposed model, on the one hand had the competitive circumstance, but on the other hand, did not has details such as large variety of customers' preferences, customer–engineering concerns, etc.

3) model in this paper considers both details and competitive circumstance.

3. Description of the PPM problem
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Considering the ?rm capabilities to produce products, a set of product portfolios have been identi?ed. Each product has certain desirability between customers. More speci?cally, we consider a scenario in which a set of products, have been identi?ed, given that the manufacturer (m) has the capabilities (both design and production) to produce all these products, . . A product portfolio, ,is a set consisting of some selected product. Combined with the products, a set of product portfolios are created, . 相关参数

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For example, if manufacturer m can produce 3 product, 7 product portfolio are available:( 2 N ?1 =7)
m

m ? Every product, pn , is associated with certain m engineering costs,denoted as n . There are multiple market segments,S ={s1, ... , sg , ... , sG}, each containing homogeneous customers,with a de?nite

c

size, Qg. The customer–engineering interaction is

embodied in the decisions associated with customers’ choices of different products. Various customer preferences on diverse products are represented by respective utilities, U m (utility of the gn gth segment for the nth product of mth manufacturer). Product demands or market shares, Dm gn (market share of the gth segment for the nth product of mth manufacturer), are described by the probabilities of customers’ choosing products. Customers choose a product based on the surplus
g

buyer rule. They have the option of not buying any products or buying competitors’ products.
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We assume that competitors respond to the manufacturer’s moves, meaning that, the competition react by introducing new products.

4. Problem formulation
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The present paper considers a market with G segments, S ={s1, ... , sg , ... , sG}, and 2 manufacturers that each of them can offer Nm products, and Jm product portfolios, . This gives the bimatrix-game(双矩阵对策) problem with 2 players and Jm strategy for each, (m = 1 or 2).

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The payoff for each player will of course depend on the combined actions of both players. A payoff matrix shows what payoff each player will receive at

the outcome of the game. For player m (m = 1 or 2), the payoff matrix, Fm, is as follows:

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In summary, a J1 × J2 – bimatrix game is played by two players,player 1 and player 2. Player 1 has a 1 finite set Z ? {z1 ,..., z1 } and player 2 has a ?nite J
1 1

2 set Z ? {z12 ,..., z J } of pure strategies. The payoff 2 2 matrixes [f1( z 1 , zb )], z1 ? Z 1 , zb ? Z 2 of player 1 a a and [ f 2 ( z1 , zb2 )], z1 ? Z 1 , zb2 ? Z 2 of player 2 are denoted a a by F1 and F2 respectively. This game is denoted by (F1, F2).
2 2

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Now the game (F1, F2) is played as follows. Players 1 and 2 choose, independent of each other, a strategy

2 and x 2 ? ?J 2 respectively. Here x11 ( x z 2 ) can be x ? ?J1 za b seen as the probability that player 1 (2) chooses his 2 z 1– th row ( z b – th column). The (expected) payoff for a player 1 is x1F1x2 and the expected payoff to player 2 is x1F2x2.
1

? A strategy pair ( ( x 1 , x 2 ) ? ?J ? ?J ) is an 1 2 equilibrium for the game (F1, F2) if

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The set of all equilibria for the game (F1, F2) is denoted by E(F1, F2). By a theorem of Nash this set is non-empty for all bimatrix-games (Nash, 1950). Some methods for calculating payoff matrix
1 2 arrays, f m ( z a , zb ) ,are there (see Section 2). We used the function that proposed by Jiao and Zhang (2005). This function is based on customerengineering interaction model in PPM. This is as follows:

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Eq. (3) is the expected shared surplus by offering a ' ' N m products ( N m ? N m ) product portfolio, consisting of ,to customer segments,sg, each with size Qg. The market potentials, Qg, can be given exogenously at the outset or estimated through a variety of techniques based on historical data or test markets. The utility of the gth segment for the nth product of mth U mdenoted as manufacturer isgn . This model assumes that customers only choose a

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product with a positive surplus. The choice probability, , m Dgn that a customer or a segment, sg, chooses a m product, pn , with Ncom competing products, is defined as follows:

? ? ?

where u is a scaling parameter(尺度参数). According to matrix (1) and Eq. (3), let the
1 2 function f m ( z a , zb ) be defined by

5. Example
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In this section, a simple example to use the proposed model is presented. For simplicity, we consider a market with two competitor (M = 2), and four different products (Nm = 4) for each. Feasible strategies, is defined as follows:

产品组合数=24-1=15

???

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Three segments are identified, i.e., s1, s2, and s3. Q1, Q2, and Q3 are assumed 0.2, 0.3 and 0.5, respectively. Table 1 shows the utilities of three m segments to every product ( U gn ) and cost of each ( Cnm ). Also, scaling parameter (u) is supposed 0.8. Therefore, 2 payoff matrixes F1 and F2 formed for manufacturer 1 and 2, separately. This game and obtained data from expected shared surplus values (Eq. (5)) are summarized in Fig. 1.

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The optimal result for each manufacturer is derived from the Nash equilibrium point of the game. A 1 2 strategy pair ( z8 , z 7 ) is an alone equilibrium for the game . The related payoff pair is (0.74, 0.83).

6. Conclusions and future work
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This paper proposed a game theory-based model that is used to maximize the expected shared surplus for a product portfolio managed. The product portfolio management (PPM) is an important optimization problem that includes the large set of constraints and characteristics. Therefore, it is very helpful for a manager to use a marketing decision support system which provides him the acceptable solutions with considering more terms. According to this goal, a

game theory-based model is proposed and applied to solve the problems involved in PPM. ? There are potentially unlimited opportunities for research in PPM.Future studies can focus on other characteristics to achieve more ideal results. Other notable directions for future researches include allowing for sequential entry strategies, time varying utilities and changing customer behaviors.

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相关概念
? 纳什定理:在一个有n个博弈方的博弈G=﹛S1,…,Sn: u1,…,un}中,如果n是有限的,且Si 都是有限集(对i=1 ,…,n),则该博弈至少存在一个纳什均衡,但可能包含 混合策略。 ? 产品组合:由不同的产品线构成,而产品线又是由不同的 产品项目构成。 ? 产品组合策略:在产品组合的深度、广度和相关性方面做 的筹划和安排。 ? 产品组合的广度:产品线的数量。 ? 产品组合的广度:产品项目(规格或品种)的数量

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遗传算法(Genetic Algorithm)是一类借鉴生物界的进化 规律(适者生存,优胜劣汰遗传机制)演化而来的随机化 搜索方法。其主要特点是直接对结构对象进行操作,不存 在求导和函数连续性的限定;具有内在的隐并行性和更好 的全局寻优能力;采用概率化的寻优方法,能自动获取和 指导优化的搜索空间,自适应地调整搜索方向,不需要确 定的规则。

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