Application of Probability Rough Set to Product Model Selection

Shouhua Yu, Tesheng Lin

College of Informatics South China Agricultural University

Guangzhou, China segrad@scau.edu.cn, ts_lin@163.com

Jingying Ou College of Public Management

South China Agricultural University Guangzhou, China

oujingying@scau.edu.cn

Abstract—Probability rough set is applied to product model selection, risk type decision making problem is converted to probability rough set model for analysis, further, minimum risk decision making rule is found. Take a furniture factory in Foshan City of Guangdong Province as an example, this model is applied to furniture color selection, as provides a scientific decision making method for this factory in furniture color selection.

Keywords-Probability rough set; Product model selection; Risk type decision making

I. INTRODUCTION Under current market economy conditions, the increasingly

intensive competition and continuously raised individualized customer demand for product urgently requires an enterprise to convert its marketing concept from traditional push type to new customer pull type. Meanwhile, an enterprise must realize small batch individualized production, and such mode of production requires an enterprise to master individualized customer demand during product model selection more accurately. In order to meet customer demand in time, an enterprise must instruct the formulation of production plans by more scientific and efficient market information analysis method.

Nowadays, more and more enterprises carry out informatization construction and raise matter management efficiency by management information system, but a lot of business data stored in management information system doesn’t provide useful market information for an enterprise and aren’t adequately used during production decision making of an enterprise. The factory in Foshan City of Guangdong Province to be analyzed in this paper is a typical example. This factory has realized the informatization of sales, inventory and other departments, however, when production plans are formulated, product type is still selected according to experience only, individualized customer demand isn’t dug from a lot of sales data, hence overstock is caused frequently, meanwhile, the products ordered by customers may be out of stock sometimes. Therefore, the main study purpose of this paper is to make full use of existing sales data of this factory in order to provide scientific decision making basis for product model selection of the enterprise and reduce production cost risk. Because customer demand determined through sales data analysis is of uncertainty and inconsistency, this paper applies

probability rough set model to data analysis. Probability rough set is a scientific tool that can effectively analyze inaccurate and inconsistent information, and it’s a new model that studies rough set theory from the view of probability theory. This paper applies probability rough set to analyzing existing sales data, further, finds out customer consumption behavior rule hidden in sales information, provides decision making basis for this factory in selecting marketable products and finding out minimum risk production decision making alternative.

II. BUSINESS ANALYSIS Presently the factory analyzed in this paper mainly

produces various teenager panel furniture, and opens speciality stores all over the country. In order to meet individualized service demand, this factory must provide every model of furniture in more than ten colors for customers to select; in order to guarantee timely supply of goods, this factory must produce and stock a certain amount of furniture in advance. Because customers’ furniture color selection is uncertain, the color of furniture produced in advance may be different from that of furniture ordered by customers. When ordered goods is out of stock, if supply time is enough, this factory can carry out additional production; otherwise, for punctual supply of goods, this factory may change the color of furniture that has been produced, in this way, it’s inevitable to increase production cost. In order to reduce color change operations, meanwhile, guarantee timely delivery of goods and reduce stock, when formulating production plans, it’s necessary for this factory to scientifically analyze the previous orders of customers, more accurately master the tendency of customers in selecting the color of every model of furniture, and reduce the risk of production cost rise caused by urgent work and rework as possible.

Furniture color selection is a risk type decision making problem, and this factory intends to select a decision making alternative with minimum risk, and make selected furniture color meet customer demand. For solving such problem, factors that influence customers’ selection of furniture color must be analyzed first. After analyzing the order records of this factory, this paper finds main influencing factors, i.e. location of customer, furniture type and furniture model. Therefore, this paper applies probability rough set model to analyzing the correlation between any of the three influencing factors and

Second International Symposium on Information Science and Engineering

978-0-7695-3991-1/09 $26.00 © 2009 IEEE

DOI 10.1109/ISISE.2009.48

583

furniture color, digs the rule of customers’ color preference, further, provides certain instruction for manufacturers in furniture color selection.

III. APPLY PROBABILITY ROUGH SET TO ANALYZING PRODUCT MODEL SELECTION

A) Fundamental principle of probability rough set Probability rough set model is a new model that studies

rough set theory from the view of probability theory, and it makes up and generalizes certain rough set model. Hereafter its fundamental principle is briefly described.

Assume U is a domain made up of limited objects and R is equivalence relation in U, and they form an equivalence class below.

U/R={X1, X2,… ,Xn} (1)

x is an object in U, record the equivalence class of x as [x], P is assumed as σ algebraic probability measure formed by subset classes defined in U, and then triple Ap=(U,R,P) is called probability approximation space. Every subset class in U is called concept that represents a random event. P(X|Y) represents the conditional probability of appearance of X when event Y occurs, and may be interpreted as the probability at which a randomly selected object belongs to X under concept Y.

Assume 0≤β<α≤1, for random X ⊆ U, we define that the probability (I ) lower approximation PI(X) and upper approximation ——

PI β (X) of X for probability approximation space Ap=(U,R,P) according to parameters α, β are as follows:

}])[|(|{)( αα ≥∈= xXPUxXPI (2)

}])[|(|{)( ββ >∈= xXPUxXPI (3)

Probability ( I ) positive domain, boundary domain and negative domain of X for Ap according to parameters α, β are as follows:

}])[|(|{)(),,( αβα α ≥∈== xXPUxXPIXpos (4)

}])[|(|{),,( αββα <<∈= xXPUxXbn (5)

}])[|(|{\),,( ββα β ≤∈== xXPUxPIUXneg (6)

When PIα(X)= ——PI β (X) or equivalently, bn(x,α,β)=φ, X is

definable in probability (I) for Ap according to parameters α, β, otherwise X is called probability (I ) rough set for Ap according to parameters α,β [1].

When probability rough set model is applied to analyzing an actual problem, α,β are determined first and P(X|[x]) is defined, so normally P(X|[x]) can be calculated according to data. α,β can be viewed as upper and lower limits of prior probability P(X) of object x in subset X, i.e. the range of

fluctuation of P(X). Parameter β is the lower limit of P(X) and meets constraint condition 0≤β<P(X)≤1. When P(X|[x]) is smaller than β, i.e. information acquired from data shows that the probability at which x belongs to subset X is small, so object x in equivalence class [x] shall be classified into negative domain of X. Parameter α is the upper limit of P(X) and meets constraint condition 0≤P(X)<α≤1. When P(X|[x]) is larger than α, i.e. information acquired from data shows that x is likely to belong to subset X, object x in equivalence class [x] may be classified into positive domain of X. When P(X|[x]) is between α and β, i.e. information acquired from data doesn’t show any further information than prior probability P(X), object x in equivalence class [x] is classified into boundary domain of X[2].

With the development of probability rough set theory, researchers have described various models, e.g. variable accuracy probability rough set model, i.e. define the confidence interval of parameter α as 1/2<α≤1, assume β =1-α, change two parameters to one parameter, so that variable accuracy probability rough set model is obtained[3]. Another researcher put forward variable accuracy Bayes rough set model which does not contain parameters α,β but directly substitute them by prior probability P(X) [4]. In combination with the practice, this paper selects and uses probability (I) rough set described above.

B) Modeling This paper analyzes product model selection during

production. This issue is a risk type decision making problem many production enterprises are faced with, and it can be converted to probability rough set model for analysis[5]. Take this factory as an example, hereafter the establishment of probability rough set model is analyzed for furniture color selection of this factory.

For this factory, the product model selection problem is to determine products in which colors shall be produced in advance for every model of furniture. This factory may analyze which color is popular according to previous purchase behaviors of customers, and formulate a production plan according to previous sales volume, meanwhile, this factory must consider that furniture in existing colors may be unmarketable if customer’s color preference changes, i.e. customers may purchase furniture in other colors, or loss may be caused due to color change of existing furniture for timely supply of goods. In order to better deal with such risk, it’s necessary to formulate a set of scientific decision making basis of color selection for furniture production.

Assume furniture color variable as ω, this factory, for the production decision making problem above, will conduct 3 decision making behaviors as follows: a) produce a model of furniture in color ω in advance; b) not produce a model of furniture in color ω; c) produce uncolored semi-finished product of a model of furniture only, i.e. spray base color only, and color the uncolored semi-finished product when the inventory of this model of furniture in color ω is smaller than the order quantity.

According to rough set theory, assume the entirety of furniture of this factory is domain U, represent certain furniture

584

by object x in U, and view furniture color ω as a concept. For every object in U, f(x) is called description of x, objects with the same description can’t be discriminated, and the entirety of objects with description the same as description of x is recorded as [x]. In this way, we often substitute [x] for f(x) under one to one correspondence on the premise of not causing any misunderstanding when expounding a problem. For the production decision making problem, this paper describes sold furniture x by three attributes order customer, product model and product type, i.e. classify the sold furniture by equivalence relation R made up of the three attributes, and the equivalence class generated by (U,R) is U/R={[x]|x ∈ U}. Through describing R, the set of sold furniture in color ω can be approximately portrayed.

For sold furniture in color ω, its positive domain is represented by pos(ω), if furniture x belongs to positive domain, decision r1 is adopted, i.e. all the furniture equivalent to furniture x ([x], equivalence class of x) shall be in color ω; if furniture x belongs to negative domain of ω (neg(ω)), decision r2 is adopted, i.e. not produce a model of furniture in color ω; if furniture x belongs to boundary domain of ω (bn(ω)), decision r3 is adopted, i.e. produce uncolored semi-finished product of this model of furniture. The domain of x can be determined according to the previous sales data and probability rough set theory. Therefore, the three decisions can be expressed below.

a) Decision r1: x∈pos(ω), i.e. r1: [x]→pos(ω);

b) Decision r2: x∈neg(ω), i.e. r2: [x]→neg(ω);

c) Decision r3: x∈bn(ω), i.e. r3: [x]→bn(ω).

At this point, decisions form a set A={r1,r2,r3}. Assume λ(ri|ω) is the risk of adopting decision ri when a customer purchases product in color ω ; λ(ri|~ω) is the risk of adopting decision ri when a customer purchases product not in color ω. P(ω|[x]) is the probability of furniture in color ω among the equivalence class [x] of sold furniture x, and P(~ω|[x]) is the probability of furniture not in color ω among the equivalence class [x] of sold furniture x. According to the acquired probability, conditional risk R(ri|[x]) of adopting decision ri can be calculated by total probability formula.

])[|(~])[|(])[|( 21 xPxPxrR iii ωλωλ += (7)

There of, λi1=λ(ri|ω), λi2=λ(ri|~ω), i=1,2,3

According to the formula above, the conditional risk value of every decision ri can be calculated, further, the decision with the minimum risk can be found out. The decision selection condition can be expressed below.

a) Adopt r1: [x]→pos(ω), if R(r1|[x])≤R(r2|[x]) and R(r1|[x])≤R(r3|[x]);

b) Adopt r2: [x]→neg(ω), if R(r2|[x])≤R(r1|[x]) and R(r2|[x])≤R(r3|[x]);

c) Adopt r3: [x]→bn(ω), if R(r3|[x])≤R(r1|[x]) and R(r3|[x])≤R(r2|[x]).

P(ω|[x]) + P(~ω|[x])=1, so P(~ω|[x])=1-P(ω|[x]), substitute the formula above in the formula of R(ri|[x]), sort the inequations above, and find out which decision is related to conditional probability P(ω|[x]) at which object x belongs to concept ω under [x] when risk loss λij (i=1,2,3; j=1,2) is known. Through simplification, the minimum risk decision making rule can be re-expressed below.

a) Adopt r1: [x]→pos(ω), if P(ω|[x]) ≥α;

b) Adopt r2: [x]→neg(ω), if P(ω|[x]) ≤β ;

c) Adopt r3: [x]→bn(ω), if β < P(ω|[x]) <α . There of,

)()( 32121131

3212

λλλλλλα

−+−−

=, )()( 22323121

2232

λλλλλλβ

−+−−

=, αβ ≤ .

So far, risk type decision making problem of this factory in product model selection has been converted to a probability rough set model.

In practice, if a customer purchases a model of furniture in color ω, the risk (λ11) of producing finished product of this model of furniture in color ω in advance isn’t larger than the risk (λ31) of producing uncolored semi-finished product of this model of furniture, and the risk (λ31) of producing uncolored semi-finished product of this model of furniture is smaller than the risk (λ21) of not producing finished product of furniture in color ω; if a customer doesn’t purchase the product of a model of furniture in color ω, the risk (λ12) of producing finished product of this model of furniture in color ω in advance is larger than the risk (λ32) of producing uncolored semi-finished product of this model of furniture, and the risk (λ22) of not producing finished product of this model of furniture in color ω isn’t larger than the risk (λ32) of producing uncolored semi-finished product of this model of furniture, namely λij(i=1,2,3; j=1,2) meets the relational expression:

λ11≤λ31< λ21, and λ12> λ32≥λ22. Therefore,α ∈(0,1], β ∈[0,1).

From the probability rough set model of the problem above, it can be seen that the decision making basis of producing a model of furniture in color ω is the value of P(ω|[x]), i.e. the probability at which an objects in x’s equivalence class [x] belongs to ω, and the value represents the probability at which the same type and model of furniture ordered by the same speciality store is in color ω for such decision making problem.

(8)

|*| represents the base number of the set ’*’, i.e. the number of elements in the set, this value can be conveniently calculated according to sales data. Next an example is used for further describing how to apply this model to production decision making.

|||][|])[|(

xxxP ∩= ωω

585

C) Case analysis Assume U={x1,x2,x3,…} is a domain made up of previous

furniture order records, and relevant knowledge expression system is shown in Table 1. e.g. the equivalence classes of objects x1,x2,x3 in the table are [x1]={a: Shenyang Speciality Store, b:3909, c: reading table}; [x2]={a: Liaoning Anshan Speciality Store, b:X6, c: three-piece door panel}; [x3]={a: Yangzhou Speciality Store,b:3609, c: night table} respectively. ω ={Light blue}, ~ω={Non light blue} . Hereafter probability rough set is used for analyzing whether to produce light blue products in next production plan of the three types of furniture.

First, a probability knowledge expression system is established, as shown in Table 1, assume K=(U, P, AT, V, f ) as furniture probability knowledge expression system. U is domain, i.e. a set made up of order records. P is σ algebraic probability measure made up of all subsets of U. AT is a set made up of limited attributes, AT=C ∪D,C∩ D=φ. C is conditional attribute set, including three attributes a) order customer, b) product model, c) product type; D is decision attribute, including one attribute only, i.e. d. product color. V is range of values of attributes, f: U→V is information function. Objects in the furniture probability knowledge expression system of this paper are records obtained through statistical analysis of orders of all speciality stores of this factory all over China, and several objects are listed in TABLE I.

TABLE I FURNITURE KNOWLEDGE EXPRESSION SYSTEM

U Conditional attribute C Decision attribute D

Object x

Order customer a

Product model

b

Product type c

Product color d

1 Shenyang Speciality Store 3909 Reading table Turquoise

blue

2 Anshan Speciality Store X6 Three-piece

door panel Pink

3 Yanghzou Speciality Store 3609 Night table Light blue

4 Shanghai Qingpu Speciality Store 3959 Reading table Brilliant red

5 Zhejiang Yiwu Speciality Store 3845

Bed curtain Pink

6 Yanghzou Speciality Store 3609 Night table Light yellow

7 Shenyang Speciality Store 3909 Reading table Pink

8 Yanghzou Speciality Store 3609 Night table Turquoise

blue … … … … …

Decision set is A={r1,r2,r3}, r1,r2,r3 represent three

decisions respectively, i.e. produce light blue furniture, not produce light blue furniture and produce uncolored semi-finished product of this model of furniture. According to production experience, risk parameters are λ11=50, λ21=2200, λ31=500, λ12=1000, λ22=50, λ32=600. According to the definition of model in the previous section, α=0.47, β=0.24. Through statistical analysis of objects in knowledge system, 50% of the first type of furniture in previous order records are

light blue, i.e. P(ω|[x1])=0.5. In a similar way, P(ω|[x2]) =0.25, P(ω|[x3])=0.05. Further,

}{}])[|(|{),,( 1xxPUxpos =≥∈= αωβαω ;

}{}])[|(|{),,( 2xxPUxbn =<<∈= αωββαω ;

}{}])[|(|{),,( 3xxPUxneg =≤∈= βωβαω .

Therefore, the minimum risk production decision of this problem is to produce a certain amount of light blue 3909 reading tables in advance according to the previous sales volume and inventory of 3909 reading tables in Shenyang Speciality Store; produce a certain amount of uncolored semi-finished products of X6 three-piece door panel according to the sales volume and inventory of Liaoning Anshan Speciality Store; not produce light blue 3609 night tables because the sales volume of such night tables in Yangzhou Speciality Table fails to meet related requirement.

IV. CONCLUSION According to rough set theory, knowledge can be acquired

from mass data without processing any information beyond data, then risk type decision making problem is converted to a probability rough set model. Furthermore, decision making basis can be directly obtained from the statistical information of knowledge expression system. This paper applies probability rough set model to analyzing the sales data of products in different colors of every model of furniture of a furniture factory in Foshan City of Guangdong Province, obtains the minimum risk decision making rule for production color selection, further, provides a production decision making method that not only meets individualized customers demand but also reduces production cost risk for this factory, and this method can aid this factory’s decision making during furniture production.

REFERENCES [1] Zhang Wenxiu, Wu Weizhi, Liang Jiye, and Li Deyu, Theroy and

Methods of Rough Sets, Beijing:Science Press, April 2006, pp.132-147. [2] Wojciech Ziarko, Rough Sets, Fuzzy Sets, Data Mining, and Granular

Computing, Berlin: Springer Berlin Heidelberg, 2005, pp.288. [3] Sun Bingzhen and Gong Zengtai, Variable precision probabilistic rough

set model, Journal of Northwest Normal University(Natural Science Edition), Vol.41, April 2005, pp.23-26.

[4] Dominik Slezak and Wojciech Ziarko, Variable Precision Bayesian Rough Set Model, Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing 9th International Conference[C].Berlin: Springer Berlin Heidelberg, 2003, pp.314.

[5] Zhang Huaizhong, Probabilistic Rough Set Model on Bayes Decision, Mini-micro Systems, Vol.25, March 2004, pp.407-409.

586