A fuzzy multi-objective model for capacity allocation ...

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A fuzzy multi-objective model for capacity allocation and pricing policy of provider in data communication service with different QoS levels a

a

b

c

d

e

Wei Pan , Xianjia Wang , Yong-guang Zhong , Lean Yu , Cao Jie , Lun Ran , Han Qiao c f a

c

, Shouyang Wang & Xianhao Xu

g

Economics and Management School, Wuhan University, Wuhan 430072, China

b

Department of Management Science and Engineering, University of Qingdao, Qingdao 266071, China c

Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China d

School of Economics, Management Nanjing University of Information Science Technology, NanJing 210044, China e

School of Management and Economics, Beijing Institute of Technology, Beijing 100081, China f

School of Economics, Qingdao University, Qingdao 266071, China

g

School of Management, Huazhong University of Science and Technology, Wuhan 430074, China Available online: 21 Jul 2011

To cite this article: Wei Pan, Xianjia Wang, Yong-guang Zhong, Lean Yu, Cao Jie, Lun Ran, Han Qiao, Shouyang Wang & Xianhao Xu (2012): A fuzzy multi-objective model for capacity allocation and pricing policy of provider in data communication service with different QoS levels, International Journal of Systems Science, 43:6, 1054-1063 To link to this article: http://dx.doi.org/10.1080/00207721.2010.549581

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International Journal of Systems Science Vol. 43, No. 6, June 2012, 1054–1063

A fuzzy multi-objective model for capacity allocation and pricing policy of provider in data communication service with different QoS levels Wei Pana, Xianjia Wanga, Yong-guang Zhongb, Lean Yuc, Cao Jied, Lun Rane, Han Qiaocf, Shouyang Wangc* and Xianhao Xug

Downloaded by [Academy of Mathematics and System Sciences] at 21:13 21 April 2012

a

Economics and Management School, Wuhan University, Wuhan 430072, China; bDepartment of Management Science and Engineering, University of Qingdao, Qingdao 266071, China; cInstitute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; dSchool of Economics, Management Nanjing University of Information Science Technology, NanJing 210044, China; eSchool of Management and Economics, Beijing Institute of Technology, Beijing 100081, China; fSchool of Economics, Qingdao University, Qingdao 266071, China; g School of Management, Huazhong University of Science and Technology, Wuhan 430074, China (Received 7 December 2009; final version received 4 November 2010) Data communication service has an important influence on e-commerce. The key challenge for the users is, ultimately, to select a suitable provider. However, in this article, we do not focus on this aspect but the viewpoint and decision-making of providers for order allocation and pricing policy when orders exceed service capacity. It is a multiple criteria decision-making problem such as profit and cancellation ratio. Meanwhile, we know realistic situations in which much of the input information is uncertain. Thus, it becomes very complex in a real-life environment. In this situation, fuzzy sets theory is the best tool for solving this problem. Our fuzzy model is formulated in such a way as to simultaneously consider the imprecision of information, price sensitive demand, stochastic variables, cancellation fee and the general membership function. For solving the problem, a new fuzzy programming is developed. Finally, a numerical example is presented to illustrate the proposed method. The results show that it is effective for determining the suitable order set and pricing policy of provider in data communication service with different quality of service (QoS) levels. Keywords: pricing policy; quality of service (QoS); order allocation; fuzzy set theory

1. Introduction With the development of e-commerce, more and more companies realise the importance of data communication service in today’s fiercely competitive market environment. In data communication domain, one of the most challenging issues is provider selection. It has received considerable attention. Many papers and books have been published from the viewpoint and decision-making of customers. A key assumption of these models is that providers can meet different customers demand in data communication service with multiple quality of service (QoS) requirements. Armony and Haviv (2003) studied provider selection problems in which customer faces two firms which offered identical services for possibly different prices and response times. Kasap, Aytug, and Erengu (2007) investigated an optimisation problem when a buyer faces acquiring network capacity from multiple providers. Raghuram and Munindar (2004) reformulated two traditionally recommended approaches for provider selection and proposed a new agent-based approach in which agents cooperate to evaluate *Corresponding author. Email: [email protected] ISSN 0020–7721 print/ISSN 1464–5319 online ß 2012 Taylor & Francis http://dx.doi.org/10.1080/00207721.2010.549581 http://www.tandfonline.com

service providers. Kumara, Vratb, and Shankar (2006) proposed fuzzy goal programming for a provider selection problem with multiple sources that included three primary goals: minimising the net cost, minimising the net rejections and minimising the net late deliveries subject to realistic constraints regarding the buyer’s demands and the vendor’s capacity. In this proposed model, weightless technique is used in which there is no difference between objective functions. Amida, Ghodsypour, and O’Brien (2006) provided a fuzzy multi-objective provider selection, where the objective of the model is fuzzy when constraints and weights are deterministic. QoS is the collective summation of service measures, which determines the degree of user satisfaction of the service (Ahsan 2006). Common measures of QoS are delay, reliability and missing data probability in data communication networks. Delay specifies how long it takes for data to travel across the network from source to destination (Comer 2001). Reliability represents the variance in data transmission. Audio and video applications are quite sensitive


International Journal of Systems Science to delay and reliability, whereas common data services are insensitive to either (Ragsdale, Lynch, and Raschke 2000). Missing data represents information dropped or irrecoverably damaged by the network. It typically stems from data collision and buffer overflows. Sudden changes in transmission may also cause data loss (Teitelbaum and Sadagic 2002). In some important tasks, missing data sometimes leads to heavy loss in data transmission. Meanwhile, in real situations, multiple customers’ demands sometimes exceed provider service capacity. With respect to this condition, a few researchers studied order allocation and policy pricing problems of provider. Choi, Kim, Park, Park, and Whinston (2004) investigated an agent for selecting optimal order set in EC marketplace. In this model, the decision to accept an order, or the selection of optimal order set, critically depends on the production schedule when orders exceed production capacity. Korpela, KylaKheiko, Lehmusvaara, and Tuominen (2002) studied an analytic approach to production capacity allocation. They offered a sale plan where the limited production capacity is allocated to the customers based on their strategic importance and the risk involved. In studies of existing models and their relative pricing policy, many researchers have observed that demand for an item may be affected by its price. Polatoglu (1991) examined an inventory model for developing pricing and procurement decisions simultaneously. Kim, Hong, and Kim (2010) investigated pricing and ordering policies for price-dependent demand in a supply chain consisting of a single retailer and a single manufacturer. Pan, Wang, Zhang, Hua, and Xie (2008) constructed a mathematical model in which demand was price-dependent to analyse the affect of pricing and order size for a service product when customers are segmented into two types. Moreover, when providers decide order allocation and pricing policy they not only expect high profit, but also maintain high quality customer such as lower cancellation ratio and higher loyalty degree. Thus, it is a multi-criteria decision making problem involving several conflicting factors. Consequently a manager must analyse the tradeoff among several criteria. At the same time, for this problem, crisp data are inadequate to model real world situations, since human judgments are often vague and cannot be estimated by an exact numerical value. At the time of making decisions, the value of many criteria and constraints have been expressed in vague terms such as ‘very high in loyalty’ or ‘low in price’. To deal with this vagueness, Zadeh (1965) first introduced the fuzzy logic theory, which was oriented to the rationality of uncertainty due to imprecision or vagueness. In this

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situation, the theory of fuzzy sets is the best tool for handling vague information. Based on fuzzy logic approaches, Amid, Ghodsypour, and O’Brien (2007) constructed a fuzzy multi-objective supplier model. In this model, constraint and weight are also deterministic. Combining these features, we present a new fuzzy multi-objective model in this article, which reflects both subjective judgement and objective information in real-life circumstances. The proposed method in our research incorporates the concepts of stochastic theory, fuzzy sets and scenario analysis to conduct order allocation and pricing policy of provider. Therefore, this method will efficiently manage the vagueness and ambiguity existing in the available information as well as the essential fuzziness in human judgement and preference. In the following, we assume that a market includes a great number of consumers (when they request a service) and a provider (when he offers a service implementation). For simplicity, a provider offers the service in data communication service with different QoS levels to meet multiple consumers demand. Now the main problem is how to make a suitable order allocation and pricing policy when customers demand exceed provider service capacity in a fuzzy environment. The rest of this article is organised as follows. In Section 2, we construct a new multi-objective model for capacity allocation and pricing policy of provider in data communication services with different QoS levels. In Section 3, under considering information uncertainty, we establish a new fuzzy multi-objective order allocation model. In this model, not only the objectives are fuzzy, but also constraints include static variable when service demands are dependent on sale price. In Section 4, we offer a new algorithm for solving this problem. In Section 5, we give a numerical example. From data analysis, we obtain some useful results to support the order allocation and pricing policy decision of provider. Finally, the concluding remarks are obtained in Section 6.

2. A multi-objective model for capacity allocation and pricing policy of provider in data communication services with different QoS levels The provider receives demand information from customers and allocates the corresponding service order in multiple customer environments. The problem here is, when customers’ demands exceed provider capacity, how to allocate order and determinate sale price in its list of customers. Notably, order allocation is a multiple criteria decision-making problem and


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customer demand is price-sensitive. Considering these factors, the multi-objective decision model needs be built to allocate the order and determinate sale price among multiple customers. Meanwhile, in the existing related models, researchers rarely have simultaneously considered price-sensitive demand, multiple QoS levels, cancellation fee functions in different scenarios and static (stochastic) constraints when customer demand exceeds provider capacity. Our model recognises that these conditions must be considered in order to solve order allocation and pricing policy of provider. The following content discusses our model in detail. We first make the following assumptions: (1) (2) (3) (4)

Customer demand is price-dependent. Provider capacity cannot meet all customers. Profit consider missing penalty. Provider offers multiple services to meet customer demand. (5) Definition of cancellation fee function is only influenced by scenario. (6) The objective membership function is general. Moreover, the following parameters and decision variables are used to present formulations and solution procedures throughout this article. n k m Di

d e q pij

Vj Vj cj xij

yij

ij

Number of customers Number of constraints The number of QoS level Demand at the ith customer where i is a customer index, where Di ¼ {Di1, Di2, . . . , Dim}, i ¼ 1, 2, . . . , n The number of negative objectives The number of positive objectives The number of objectives, q ¼ d þ e Price of the ith customer at the jth QoS level where j is a QoS level index, pij 2 pi, j ¼ 1, 2, . . . , m Maximum supply quantity of the provider at the jth QoS level The minimal integer value of the provider is bigger than Vj, Vj ¼ Vj þ 1 Cost of the provider at the jth QoS level Transmission data quantity of the ith customer at the jth QoS level, xij 2 x, where xi ¼ {xi1, xi2, . . . , xim} and x ¼ {x1, x2, . . . , xn} The decision variable represents if data of the ith customer at the jth QoS level is transmitted or not, yij 2 yi, where yi ¼ {yi1, yi2, . . . , yim}, if data is transmitted, where yij ¼ 1; otherwise yij ¼ 0 Missing rate of transmission data for the ith customer at the jth QoS level

F(xij) Lij Rij V p

" ,

i i

fh fl

fp fh fl

Penalty function of xij for order cancellation Cancelation rate for the ith customer at the jth QoS level, Lij 2 Li Reliability degree for the ith customer at the jth QoS level, Rij 2 Ri Capacity for the provider A parameter for the pth objective membership function where p is a objective membership index, p ¼ 1, 2, . . . , q, p40 A coefficient for linear penalty function, "40 Coefficients for nonlinear penalty function, and are, respectively, exponential parameter and product parameter, 41, 40 Scale parameter of demand of the ith customer, where i ¼ {i1, i2, . . . , im} A coefficient for reflecting the relationship between price and demand of the ith customer, where i ¼ {i1, i2, . . . , im} The hth negative objective function where h is a negative objective index, h ¼ 1, 2, . . . , d The lth positive objective function where p is a positive objective index, l ¼ d þ 1, 2, . . . , d þ e The pth objective function where p is a objective index, p ¼ 1, 2, . . . , q, q ¼ d þ e The minimal value of the hth negative objective function The maximal value of the lth positive objective function

Then, a new general multi-objective model can be stated as follows: min fh ¼ f f1 , f2 , . . . , fd g,

ð2:1Þ

max fl ¼ f fdþ1 , fdþ2 , . . . , fdþe g,

ð2:2Þ

with the following constraint: ( ( ) ) n X m X gt ¼ P aðtÞij xij bt Bt ,

ð2:3Þ

i¼1 j¼1

where fh are the negative objectives or criteria-like costs, delay, etc., h ¼ 1, 2, . . . , d. fl the positive objectives or criteria such as reliability, bandwidth and so on, l ¼ d þ 1, d þ 2, . . . , d þ e, bt the tth independent variable, while a(t)ij represents the coefficient of xij in the tth constraint. Bt is the tth pre-assigned probability level and P{} represents the probability. However, to have a new specific multi-objective model for the provider selection problem, we assume that the criteria include profit f1, loyalty degree f2 and cancellation degree f3 and a major constraint is

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International Journal of Systems Science selecting a service that can satisfy a demand. Each customer has his own cancellation history and reliability record. A provider allocates the service to customers in his list. We let di and B, respectively, represent the coefficient of the ith decision variable and the preassigned probability level in the demand constraint. Meanwhile, the provider can provide up to V units of Pn the Pm service over the planning period, i.e. i¼1 j¼1 xij V. This is due to service capacity. Then, we have ! n X m X di yij xij D B: ð2:4Þ P i¼1 j¼1

Since information dropped or irrecoverably damaged in data communication service is unavoidable, a penalty function should be considered and defined. Notably, missing data usually causes loss to customers. Moreover, the more important the task is, the higher the penalty will be for not achieving desired targets. According to this relationship, we assume that scenarios are divided into three conditions and accordingly different penalty functions are defined as follows: (1) Without penalty function when the missing data xij is unimportant as Fðxij Þ ¼ 0:

ð2:5Þ

which should minimise the number of cancellation services. We let f1 , f2 and f3 represent the minimal cost, the maximal reliability and the minimal delivery cancellation, respectively. Then we have the final form of the integer multi-objective model for purchasing data communication services in multiple source networks is as follows: (

f1

) n X m X ¼ min fð pij cj Þ yij xij Fðxij Þg , i¼1 j¼1

( f2

¼ max (

f3

¼ min

n X m X

) Rij yij xij ,

i¼1 j¼1 n X m X

P

(reliability)

) Lij yij xij ,

i¼1 j¼1 n X m X

(cost)

(cancellation)

!

di yij xij D B,

(demand )

i¼1 j¼1 n X m X

yij ¼ 1,

(single customer selected )

yij 4 1,

(multi-customer selected )

yij 1,

(random customer selected )

i¼1 j¼1 n X m X i¼1 j¼1 n X m X i¼1 j¼1

(2) With linear penalty function when the missing data xij is common as Fðxij Þ ¼ ij xij ":

ð2:6Þ

(3) With nonlinear penalty function when the missing data xij is important as ij xij :

m X

yij 1,

Vj ¼ Vj þ 1, yij Vj1 xij yij Vj ,

ð2:7Þ

n X m X

(4) The objective function for cost can be stated as

i¼1 j¼1

Fðxij Þ ¼

f1 ¼

n X m X ð pij cj Þ yij xij Fðxij Þ ,

ð2:8Þ

i¼1 j¼1

which should minimise the cost of service. The objective function for reliability is defined as f2 ¼

n X m X

Rij yij xij ,

yij 2 f0, 1g,

j¼1

xij 0,

xij V:

Generally, users do not have exact and complete information related to decision criteria and constraints. For provider selection problems, the collected data does not behave crisply, some are typically fuzzy, and others are stochastic in nature. The following section discusses our model in a fuzzy environment.

ð2:9Þ

t¼1 j¼1

which should maximise the number of reliable units. The aggregate performance measure for delivery cancellation objective function is defined as f3 ¼

n X m X i¼1 j¼1

Lij yij xij ,

ð2:10Þ

3. A fuzzy multi-objective decision-making model for capacity allocation and pricing policy of service provider in web networks In this section, we construct a new order allocation model to maximise provider profit. The following content discusses our model in detail.

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3.1. Assumptions The provider receives demand information from customers and allocates the corresponding order in multiple customers environment. The problem here is, when the provider capacity is limited, how to allocate an order set in its list of customers for maximising the provider profit. Notably, order allocation is a multiple criteria decision-making problem. According to this situation, we construct a new multi-objective decision model to allocate the suitable order among multiple customers with QoS requirements. Meanwhile, in the existing related models concentrating on order allocation, researchers rarely have considered pricing policy, different QoS requirements, fuzzy information and limited capacity. Our model recognises that these phenomena simultaneously exist. The following content discusses our model in detail. We first make the following assumptions. According to the real situation, we will develop a new model presented in this section for order allocation and pricing policy of provider. The following section discusses this model in detail.

3.2. Mathematical model To have a new order allocation model, we assume that the demand for a web service depends on the selling price. The demand function is given by the following equation. D i ¼ i i pi

ð3:1Þ

The objective function for profit can be stated as f~1 ¼

m X ð pi ci Þ yi xi Fðxij Þ ,

ð3:2Þ

i¼1

which maximises the profit of provider. The objective function for loyalty degree is defined as f~2 ¼

m X

Ri yi xi

ð3:3Þ

i¼1

which should maximise the loyalty degree. The aggregate performance measure for cancellation rate objective function is defined as f~3 ¼

m X

Li yi xi ,

ð3:4Þ

i¼1

which minimises cancellation rate. Generally, decision makers do not have exact and complete information related to decision criteria. For order allocation problems, the collected data does not behave crisply, some are typically fuzzy in nature.

Our multi-objective model is developed to deal with these problems. In the new multi-objective order allocation model presented in this article, the sign ‘ e’ indicates the fuzzy environment. The symbol ‘0’ in the objectives indicates fuzziness of ‘ 5 ’, i.e. approximately greater than or equal to. In contrast, ‘9’ has a linguistic interpretation ‘essentially smaller than or equal to’. The following content discusses our model in a fuzzy environment. Meanwhile, some basic definitions of fuzzy sets, fuzzy numbers and linguistic variables from Kaufmann and Gupta (1991) and Negi (1989) are reviewed. Using the Bellman–Zadeh approach (Zadeh 1975), the fuzzy set F(objective or constraint) functions are defined by e ¼ fx, ~ g, F F

e p ¼ 1, 2, . . . , q f~p 2 F,

ð3:5Þ

where f~p is the functions of objective, f~p jx ! ½0, 1 the degree of membership to which x belongs to objectives and constraints, where x ¼ {x1, x2, . . . , xn}. The fuzzy objectives are thus uniquely determined by its membership functions f~p . The range of membership functions f~p is a subset of the nonnegative real numbers whose value is finite and usually finds a place in the interval [0, 1]. Let D represent the membership function of the solution, using Equaion (3.5), it is possible to obtain the solution proving the maximum degree as follows: ð3:6Þ max D ¼ max min f~p , 1pq

0 x ¼ arg max min f~p : 1pq

ð3:7Þ

Finally, to obtain Equaions (3.6) and (3.7), it is necessary to build the membership functions f~p by the corresponding f~p . This is satisfied by the use of the membership functions as follows: 8 > 1, f~p max f~p , > > )p >( < f~p min f~p f~p ¼ , min f~p 5 f~p 5 max f~p , > > max f~p min f~p > > : 0, f~p min f~p ð3:8Þ for maximised objective functions or by the use of the membership functions 8 > 1, fp min f~p , > > )p > > max f~p min f~p > > : 0, f~p max f~p ð3:9Þ

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Figure 1. The different membership functions.

for minimised objective functions: where min f~p and max f~p are, respectively, the minimum and maximum values of the objective function f~p (by considering single objective, we can obtain the corresponding minimum/maximum value). The construction of Equation (3.8) or (3.9) aims to solve the following problems: f~p ! min f~p , f~p ! max f~p ,

ð3:10Þ D ¼

p¼1

¼ maxfD g:

q X

wp ,

ð3:14Þ

wf~p ¼ 1,

wf~p 0,

ð3:15Þ

p¼1

where wf~p are the weighting coefficients that present the relative importance among the fuzzy goals. is the fuzzy goal. The suggested model can be represented in the form of a weighted max–min deterministic-crisp linear programming model as follows: ( ) q X wp p , ð3:16Þ max p¼1

s.t. p f~p , q X

wp ¼ 1,

ð3:17Þ

wf~p 0,

ð3:18Þ

ytj 2 f0, 1g,

ð3:19Þ

p¼1 n X

yi 1,

i¼1 n X i¼1

where D and f~p are defined in previous content. The optimal solution D for all fuzzy objectives and fuzzy constraints is given as follows:

q X p¼1

ð3:11Þ

where min f~p and max f~p are obtained through solving the multi-objective problem as a single objective. Figure 1 shows the different membership functions of the objective. Since for every objective function f~p , its value changes from min f~p to max f~p , it may be considered as a fuzzy number with the membership function f~p as presented in Equaion (3.8) or (3.9) (see Figure 1). Meanwhile, we present a decision making process. Firstly, we discuss the max-min operator, which was used by Zimmermann for fuzzy multi-objective problems (Zimmermann 1978, 1987, 1993; Sakawa 1993). Then, we show the convex (weighted additive) operator, which enables the decision maker to assign different weights to various criteria. In fuzzy programming modelling, using Zimmermann’s approach, a fuzzy solution is given by the intersection of all the fuzzy sets representing either fuzzy objectives or fuzzy constraints. The fuzzy solution for all fuzzy objectives and fuzzy constraints may be given as follows: ( ) q \ f~p , ð3:12Þ D ¼

D

The fuzzy weighted additive model can handle this problem, which is described as follows: the weighted additive model is widely used in vector-objective optimisation problems. The basic concept is to use a single utility function to express the overall preference of the decision maker to draw out the relative importance of the criteria (Hwang and Masud 1979; Lia, Xua, and Genb 2006). In this case, multiplying each membership function of fuzzy goals by their corresponding weights and then adding the results together produces a linear weighted utility function. Let wp denote the fuzzy weight of the kth objective or constraint, where p ¼ 1, 2, . . . , q. Based on Zadeh (1975), Sakawa (1993) and Tiwari, Dharmahr, and Rao (1987) and the new weighted additive model are as follows:

xi yi ¼

n X

Di yi V,

ð3:20Þ

i¼1

ci 5 pi

ð3:21Þ

pi 5 piþ1

ð3:22Þ

xi 0:

ð3:23Þ

ð3:13Þ

In a real situation, the confluence of different objectives and constraints has unequal importance to the decision maker. Weight should also be considered.

It is noticed that, pi becomes a decision variable, in addition to p, xi and yi.


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4. An algorithm

Table 1. Collected data for numerical example.

Because we constructed a fuzzy multi-objective model, a major difficulty is the uncertainty of data. To solve this problem, we reconstructed the new algorithm, it could overcome this difficulty. Our new algorithm was stated with 6 steps as follows:

i

ci

i

i

Li

Ri

1 2 3 4 5 6

2 3 4 5 6 7

100 150 120 130 160 150

7 6 5 4 3 2

0.1 0.15 0.12 0.1 0.2 0.14

0.8 0.7 0.6 0.8 0.9 0.7

(1) Construct model according to the criteria and the constraints of the customer and provider. (2) Let the objective membership function be constructed according to Equation (3.8) and Equation (3.9). (3) According to the actual situation, give the distribution of demand variables Di. (4) Construct an initial vector i, i, wk of the important factors. (5) Using the membership function, construct the equivalent crisp model of the multi-objective fuzzy model according to Equations (3.16)– (3.23). (6) Find the optimal solution vector x and p by lingo software, where x and p are the efficient solution of the multi-objective fuzzy model. The algorithm of the model is illustrated by a numerical example in the next section.

5. A numerical example In this section, we present a numerical example to illustrate the proposed method presented in this article and show that this method is an effective method for determining order allocation in multiple customers environment. We first make the following assumptions: (1) The provider has six customers with different QoS requirements. (2) Objective weight is w ¼ (0.6, 0.1, 0.3). Three provider capacities are limited. (3) Important parameters of objectives includes profit f~1 , loyalty degree f~2 and cancellation rate f~3 , data is provided in Table 1. (4) Using Table 2, the lower bounds and upper bounds value for three objectives are provided by which the total profit and loyalty degree are maximised, cancellation rate is minimised. This example consists of three fuzzy objectives as follows: Case 1: p ¼ 1, the values for the decision variables in the numerical example are shown in Table 3. ( ) q X wp p ¼ 0:8144: max p¼1

Table 2. The lower bounds and upper bounds value of the objective functions. Objective e f1 e f2 e f3

e fmax

e fmin

3765.55 90 19.4

0 60 10

Table 3. Decision variable values under p ¼ 1. i

yi

xi

pi

1 2 3 4 5 6

0 0 1 1 0 1

86 24 15 35 58 50

2 20.9 21 23.75 34 50


yi

xi

pi

1 2 3 4 5 6

0 0 1 1 1 1

0 9 2 20 31 47

14.29 23.5 23.6 27.5 43 51

Case 2: p ¼ 2, the values for the decision variables in the numerical example are shown in Table 4. ( ) q X wp p ¼ 0:7531: max p¼1

Case 3: p ¼ 3, the values for the decision variables in the numerical example are shown in Table 5. ( ) q X wp p ¼ 0:4055: max p¼1

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yi

xi

pi

1 2 3 4 5 6

1 0 1 1 0 0

66 11 0 34 86 78

4.86 23.17 23.9 24 24.67 36

The linear (nonlinear) programming software LINDO/LINGO is used to solve this problem. We can then get the following objective and constraints: ( ) 3 X w p p , max p¼1

s.t.

p f~p ðxÞ,

w1 þ w2 þ w3 ¼ 1, 6 X

yi 1,

yi 2 f0, 1g,

i¼1

xi 0, 6 X

yi xi ¼

i¼1

6 X

yi Di ¼ V,

i¼1

c i 5 pi , p1 5 p2 5 p3 5 p4 5 p5 5 p6 : In each of the three situations, the membership function is different, we analyse order allocation and pricing results that are shown in Tables 3–5. They disclose that variation in p will cause changes to order allocation as well as pricing policy in web networks. Moreover, our numerical example has also shown an interesting result about order allocation. In general, a firm should select the suitable customer by balancing profit and other factors. It means multi-criteria requirements should be simultaneously met. Notably, from our numerical example, we find that our model enables the managers to select the most suitable order set for provider taking into consideration multiple factors in a fuzzy environment.

not precisely known. According to this situation, in this article, a new fuzzy multi-objective order allocation model was developed under considering multiple class services, price-dependent demand and nonlinear membership function. This formulation can effectively handle the vagueness and imprecision of input data for solving order allocation problems. This article differs from past studies in that it includes the following six features: (1) Uncertain information includes objectives and customer demand. (2) Unit price is aligned with QoS level change. (3) Providers offer different QoS levels to customers. (4) Different penalty functions are defined in scenario analysis. (5) The objective membership functions consist of both linear and non-linear functions. (6) This article focuses on the decision-making of providers for order allocation and pricing policy when orders exceed service capacity. Thus, the proposed model in this article can help the decision makers to choose an appropriate order allocation scheme. Moreover, through translation, we transform the fuzzy multi-objective order allocation problem into a weighted max–min deterministic-crisp nonlinear programming model. This transformation simplifies the solution process, giving less computational complexity, and makes the application of fuzzy methodology more understandable. Finally, from an application point of view, it is also worth further investigating order allocation in different networks.

Acknowledgements This work was supported by the Fundamental Research Funds for the Central Universities (No.1101012), National Natural Science Foundation of China (NSFC No. 70221001, 70731001, 71071599, 71003057 and 60979010), the Technology Research and Development Program of Shandong Province (No. 2010GGX10128) and the Promotive research fund for excellent young and middleaged scientists of Shandong Province (No. BS2010SF008).

Notes on contributors 6. Conclusions When customer demand exceeds provider capacity, order allocation is one of the most important decision in e-commerce. In real situations, it is a multiple criteria decision-making problem in which the objectives are not equally important. At the same time, during the course of decision making, input data are

Wei Pan is currently a teacher in economics and management school at Wuhan University, China. He received his BS in urban construction from China University of Geosciences, Guangzhou, China, his MS in Geologic oceanography from Chinese Academy of Sciences and his PhD in Management Science at

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Department of Management, Huazhong University of Science and Technology in 2000, 2005 and 2008, respectively. He was a government official at Committee of Municipal and Rural Construction of Dalian City, China, 2000–2002. His research interests include Optimization, EoQ Model, Pricing, Internet services and distributed systems.


Yong-guang Zhong Associate Professor, key area of researching: Resources and Environmental Policy and Management.

Cao Jie received his PhD in management science and engineering, from Southeast University, China, in 2005 and MS in systems engineering 1999, respectively. He is currently a Professor at School of Economics and Management Nanjing University of Information Science and Technology. His research interests include Information Management and Emergency Management, Risk Management, Decision Support Systems, Knowledge Management and Financial Engineering. Lun Ran born in 1977, is currently an associate professor and PhD supervisor in the School of Management and Economics, Beijing Institute of Technology. He got all his degrees from BS to PhD from Beijing Institute of Technology. His research interests are revenue management, supply chain modelling, financial engineering and risk management. He is a program director of the Department of Management Science, National Natural Science Foundation of China. Han Qiao is currently a teacher in economics school at Qingdao University and also a PhD candidate at Academy of Mathematics and Systems Science Chinese Academy of Sciences (CAS), China. Her research interests include Energy, Economics, Network Optimisation and Game Theory. Shouyang Wang received his BS in autocontrol from Sun Yat-Sen University in 1982 and PhD degree in operation and control from Chinese Academy of Sciences in 1986. He is currently an Professor of Computer Science at Academy of Mathematics and Systems Science Chinese Academy of Sciences (CAS), China. His current research interests include Energy, Economics, Network Optimization and Decision Support. Prof Wang is the guest editor of international multijournal such as European Journal Operation Research, IIE Transactions, Annals of Operations Research, Energy Economics and so on. He is a director of the National Natural Science Foundation of China.

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