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A decentralized two-level transportation problem in a housing material manufacturer: Interactive fuzzy programming approach
European Journal of Operational Research 141 (2002) 167–185 www.elsevier.com/locate/dsw
Production, Manufacturing and Logistics
A decentralized two-level transportation problem in a housing material manufacturer: Interactive fuzzy programming approach Masatoshi Sakawa
a,*
, Ichiro Nishizaki a, Yoshio Uemura
b
a
b
Department of Artificial Complex Systems Engineering, Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima 739-8527, Japan System Analysis EDP Department, Juken Sangyou Co., Ltd., 1-1 Mokuzaikou-minami, Hatsukaichi, Hiroshima 738-8502, Japan Received 21 November 2000; accepted 23 July 2001
Abstract In this paper, we deal with a transportation problem in a housing material manufacturer and derive a satisfactory solution to the problem by taking into account not only the degree of satisfaction with respect to objectives of the housing material manufacturer but also those of two forwarding agents to which the housing material manufacturer entrusts transportation of products. The two forwarding agents handle the regular transportation and the small lot transportation, and each of them assigns work force, the transformation tasks properly. There are two objectives in the housing material manufacturer: one is to minimize the transportation cost and the other is to minimize the opportunity loss with respect to transportation time. The objectives of the two forwarding agents are to maximize their profits. We formulate the transportation planning and work force assignment problem as a decentralized two-level integer programming problem, and derive a satisfactory solution by applying an interactive fuzzy programming method. Ó 2002 Published by Elsevier Science B.V. Keywords: Optimization; Transportation planning; Work force assignment; Two-level programming; Interactive fuzzy programming
1. Introduction In this paper, we treat a transportation problem in a housing material manufacturer. The housing material manufacturer does not transport its
* Corresponding author. Tel.: +81-82-424-7695; fax: +81-82424-7695. E-mail address: [email protected] (M. Sakawa).
products from its factory or warehouse to customers on its own account but entrusts the transportation to forwarding agents. There are two kinds of forwarding agents: one handles the regular transportation and the other handles the small lot transportation. Minimizing the transportation cost and the opportunity loss with respect to transportation time, the housing material manufacturer gives the two forwarding agents orders for the transportation of the products to the customers. The two forwarding agents assign work force
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so as to maximize their profits, taking ability of drivers into account. Such a transportation planning and work force assignment problem can be formulated as a decentralized two-level integer programming problem. To make rational and efficient plan for transportation of the products and to take into account cooperative relation between the housing material manufacturer and the two forwarding agents, by applying interactive fuzzy programming, we try to derive a satisfactory solution. In a simple two-level mathematical programming problem, there are two decision makers at the upper and the lower levels, and the upper level decision maker determines a decision and thereafter the lower level decision maker chooses a decision. Moreover, assuming that each decision maker completely knows the objective functions of the upper and the lower level decision makers and the constraints and, for a decision of the upper level decision maker, the lower level decision maker responds rationally, the upper level decision maker specifies a decision so as to optimize the objective. Then a solution defined as the above mentioned procedure is called the Stackelberg (equilibrium) solution. The Stackelberg solution has been employed as a solution concept when decision making problems are formulated as twolevel mathematical programming problems [1,2,5, 15,18], whether there is a cooperative relationship between the decision makers or not. Furthermore, it should be noted that the Stackelberg solution does not always satisfy Pareto optimality because of its noncooperative nature. In situations where the Stackelberg solution is employed, it is supposed that the two decision makers do not have motivation to cooperate mutually. However, if there are commercial transactions between two firms in the long term, or a parent firm orders a subsidiary firm some tasks, the two firms may have motivation to cooperate each other and be willing to derive a solution satisfactory to both firms. In the case study dealt with in this paper, the housing material manufacturer has been connected in business with each of the forwarding agents over a long period of time and there exists cooperative relationship between them. For this cooperative relationship, it is nat-
ural that the housing material manufacturer does not optimize only its own objectives but makes decisions cooperatively by balancing its own satisfaction with those of the two forwarding agents. Thus, requiring a cooperative decision making method different from the Stackelberg solution models, we employ interactive fuzzy programming approach to the transportation planning and work force assignment problem. Methods for obtaining solutions to two-level programming problems where there is cooperative relationship between decision makers have been presented within interactive fuzzy programming by Lai [7], Shih et al. [14], and Sakawa et al. [9–13]. Especially, the interactive fuzzy programming method for a decentralized two-level linear programming problem developed by Sakawa et al. [13] is applicable to the transportation problem in the housing material manufacturer by revising it partly because the transportation problem in the housing material manufacturer can be formulated as a decentralized two-level integer programming problem with the two objectives of the upper level decision maker. In Section 2, we formulate the transportation problem in the housing material manufacturer and solve four individual programming problems with a single objective function which is each of the two objectives of the housing material manufacturer and the two objectives of the two forwarding agents in order to understand characteristics of the transportation problem and to specify parameters of membership functions of fuzzy goals. In Section 3, the fuzzy goals for the objective functions are identified and the two fuzzy goals of the housing material manufacturer are aggregated by three methods. Then, we derive a satisfactory solution to the transportation problem by applying the interactive fuzzy programming method composed of two interactive phases.
2. Formulation of the transportation problem The transportation problem in the housing material manufacturer is formulated as a decentralized two-level integer programming problem. The upper level decision maker is the housing
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Table 1 Data on transportation of Firm U Customer
Xi
ai (U/product)
bi (U/product)
ci (days)
di (days)
1 2 3 4 5
300 150 250 100 100
1000 2000 2200 800 2000
2000 4000 3000 1400 3400
10 2 2 7 2
5 1 1 3 1
material manufacturer and there are two decision makers at the lower level who are the two forwarding agents handling the regular transportation and the small lot transportation. The housing material manufacturer determines allocation of the transportation tasks to five customers between the two forwarding agents, and they assign work force, taking three levels of ability of drivers into account. The housing material manufacturer and the two forwarding agents have the following objectives. 2.1. Objective functions To describe concisely, let Firm U, Firm L1 and Firm L2 denote the housing material manufacturer, the forwarding agent handling the regular transportation and the forwarding agent handling the small lot transportation, respectively. The first objective function of Firm U. Coordinating orders to Firm L1 and Firm L2, Firm U minimizes the transportation cost, z11 ðy; zÞ ¼
5 X
ðai yi þ bi zi Þ;
ð1Þ
i¼1
where ai and bi are unit costs of the regular and the small lot transportation to customer i, respectively, and are shown in Table 1; yi and zi denote decision variables which are the numbers of the products to be sent to customer i by Firm L1 and Firm L2, respectively. Because the decision variables y ¼ ðy1 ; . . . ; y5 Þ and z ¼ ðz1 ; . . . ; z5 Þ of Firm U are the numbers of the products to be sent to the five customers, they are integer variables. The second objective function of Firm U. Firm U also minimizes the opportunity loss with respect to the transportation time which is a delay from the earliest delivery time,
z12 ðu; vÞ ¼
5 X 3 X i¼1
ðeij uij þ fij vij Þ;
ð2Þ
j¼1
where eij and fij are, respectively, the values of the opportunity loss when drivers with ability level j in Firms L1 and L2 transport the products to customer i, and they are defined by eij ¼ oij oi3 and fij ¼ pij pi3 ; oij and pij are transportation times when the drivers with ability level j of Firms L1 and L2 transport the products to customer i. Because we use a larger index j for drivers with higher ability, oi3 and pi3 are the smallest values. oij and pij are shown in Tables 2 and 3, respectively. uij and vij denote decision variables which are the numbers of the assigned drivers with ability level j in Firms L1 and L2 who transform the products to customer i. Because the decision variables u ¼ ðu11 ; . . . ; u53 Þ and v ¼ ðv11 ; . . . ; v53 Þ of Firms L1 and L2
Table 2 Drivers data of Firm L1a qj o1j , e1j (days) o2j , e2j (days) o3j , e3j (days) o4j , e4j (days) o5j , e5j (days) s1j s2j s3j s4j s5j a
(U) (U) (U) (U) (U)
Level 1
Level 2
Level 3
100 10, 2
120 9, 1
150 8, 0
3* , –
2, 1
1, 0
3*, –
2, 0
2, 0
7, 2
6, 1
5, 0
3*, –
2, 1
1, 0
40,000 – – 34,000 –
57,000 36,000 36,000 48,000 36,000
72,000 44,000 44,000 60,000 44,000
The symbol * denotes drivers who cannot be assigned.
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Table 3 Drivers data of Firm L2a rj p1j , e1j (days) p2j , e2j (days) p3j , e3j (days) p4j , e4j (days) p5j , e5j (days) t1j t2j t3j t4j t5j a
(U) (U) (U) (U) (U)
2.2. Constraints
Level 1
Level 2
Level 3
60 5, 2
72 4, 1
90 3, 0
1, 0
1, 0
1, 0
3*, –
2, 0
2, 0
Constraints on the amount of transportation. Because the amount of transportation to customer i is fixed at Xi , the sum yi þ zi of the numbers of the products to be sent to customer i by Firms L1 and L2 equals Xi and the constraints on the amount of transportation are represented as
2*, –
1, 0
1, 0
Xi ¼ yi þ zi ;
1, 0
1, 0
1, 0
40,000 32,000 – – 32,000
52,000 43,000 43,000 49,000 43,000
62,000 54,000 54,000 58,000 54,000
The values of Xi are shown in Table 1. Constraints on the minimal orders. Firm U must order Firms L1 and L2 to transport more than 30 and 20 products to each customer, respectively. The constraints on the minimal orders are represented as
The symbol * denotes drivers who cannot be assigned.
are the numbers of drivers, they are integer variables. The objective function of Firm L1. By assigning drivers efficiently, Firm L1 maximizes the following profit defined by taking necessary expenses and costs on drivers from the income from Firm U: z21 ðy; uÞ ¼
5 X i¼1
( ai yi
0:3ai yi þ
3 X
!) sij uij
:
j¼1
i¼1
j¼1
where tij denotes the cost on a driver with ability level j who transports the products to customer i, and values of tij are shown in Table 3.
ð5Þ
yi P 30;
i ¼ 1; . . . ; 5;
ð6Þ
zi P 20;
i ¼ 1; . . . ; 5:
ð7Þ
Constraints on assignment of work force . Firms L1 and L2 must transport the products to the customers before the specified date of delivery and they cannot assign transportation of the products to drivers who take longer time than the specified date of delivery. The constraints on assignment of work force are represented as X yi 6 qj uij ; i ¼ 1; . . . ; 5; ð8Þ
ð3Þ The necessary expense is defined as 30% of the income from Firm U. In the objective function (3), sij denotes the cost on a driver with ability level j who transports the products to customer i, and values of sij are shown in Table 2. The objective function of Firm L2. Similarly to Firm L1, Firm L2 also maximizes the following profit defined by taking necessary expenses and costs on drivers from the income from Firm U: ( !) 5 3 X X z22 ðz; vÞ ¼ bi zi 0:3bi zi þ tij vij ; ð4Þ
i ¼ 1; . . . ; 5:
j2Ki
zi 6
X
rj vij ;
i ¼ 1; . . . ; 5;
ð9Þ
j2Li
where qj and rj are capacities of transportation of a driver with ability level j in Firms L1 and L2, respectively. Let Ki ¼ fj j oij 6 ci g and Li ¼ fj j pij 6 di g, where ci and di denote the specified dates of delivery to customer i for Firms L1 and L2, respectively, and they are shown in Table 1. Constraints on the numbers of drivers. For each ability level, Firms L1 and L2 must assign from one to three drivers and the constraints on the numbers of drivers are represented as 16
5 X i¼1
uij 6 3;
j ¼ 1; 2; 3;
ð10Þ
M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
16
5 X
vij 6 3;
j ¼ 1; 2; 3:
ð11Þ
i¼1
Constraints on redundant assignment of work force. To prevent redundant assignment of work force, the following constraints, meaning that actual transportation to each customer must be smaller than the sum of the volume of the order and the capacity of transportation of a driver with the lowest ability level, are imposed: X qj uij 6 yi þ q1 1; i ¼ 1; . . . ; 5; ð12Þ j2Ki
X
rj vij 6 zi þ r1 1;
i ¼ 1; . . . ; 5:
ð13Þ
j2Li
3. Interactive fuzzy programming We apply the interactive fuzzy programming method to the transportation planning and work force assignment problem, which is formulated as the decentralized two-level integer programming problem, because the housing material manufacturer denoted by Firm U makes decisions by taking into account not only its own satisfaction but also the satisfaction of the two forwarding agents denoted by Firm L1 and L2. To effectively support cooperative decision making in decentralized organizations, the interactive fuzzy programming methods have been developed. In the methods, it is assumed that multiple decision makers can coordinate satisfaction of them and arrive at an agreement, and interactive process is performed by a delegate of them or an analyst who makes contact with the decision makers. Consequently, the delegate or the analyst controls all of the decision variables, paying more attention or respect to the upper level decision maker and considering satisfactory balance between the two levels. Before examining the application, we show an outline of the interactive fuzzy programming method. The interactive fuzzy programming method by Sakawa et al. [13] was developed for obtaining a satisfactory solution to a decentralized
171
two-level linear programming problem in which there are a single decision maker at the upper level and two or more decision makers at the lower level and each decision maker has a single objective function. The algorithm is composed of two phases. In the first phase, the decision makers at both levels identify membership functions of their fuzzy goals for the objective functions. Let l0 denote the membership function of the upper level decision maker and li , i ¼ 1; . . . ; k, the membership functions of the k lower level decision makers. Taking into consideration overall satisfactory balance between the two levels by consulting the ratio of satisfactory degrees between the two levels, D¼
mini¼1;...;k li ðzi Þ ; l0 ðz0 Þ
ð14Þ
the upper level decision maker specifies the minimal satisfactory level d^ and updates it if necessary, and then a tentative solution is obtained. In this phase, the lower level decision makers are treated impartially, and therefore they can be regarded as a group. In the second phase, consulting the ratios of satisfaction between the upper level decision maker and each of the lower level decision makers, the upper level decision maker specifies minimal and/or maximal satisfactory levels to some of the lower level decision makers and updates them if necessary. By coordinating the satisfactory degrees of the decision makers individually, the final satisfactory solution can be derived. The interactive procedure is depicted by a flowchart in Fig. 1. Through this interactive process, although more attention or respect to the upper level decision maker is paid, we can obtain the final satisfactory solution well-balanced between the both levels, and the satisfactory degree of the upper level decision maker and that of each lower level decision maker are coordinated appropriately. First, to incorporate fuzziness of human judgment of the decision makers, we identify membership functions of fuzzy goals for their objective functions. To do so, we solve four singleobjective programming problems in which four objective functions shown in the previous section
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Fig. 1. Interactive procedures in the fuzzy programming.
are optimized separately, and determine parameters of the membership functions by consulting the optimal values and solutions of the four problems.
minimize z11 ðy; zÞ subject to ðy; z; u; vÞ 2 S;
3.1. Individual optimization problems
where S denotes the feasible region satisfying the constraints (5)–(13). Problem (15) is solved by using ‘lp_solve’ developed by Berkelaar [4], which is a software for solving mixed integer program-
We first solve the problem minimizing the transportation cost z11 of Firm U:
ð15Þ
M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
ming problems and is based on the branch-andbound techniques, and subsequent problems formulated in this paper are also solved by using lp_solve. The obtained optimal solution to Problem (15) is shown in Table 4. For the obtained solution, the transportation planning is one-sided to the regular transportation of Firm L1, and Firm L2 receives only the minimum orders. Therefore, the profit of Firm L2, z22 , is extremely small. Similarly, the results of solving the rest of the problems, which minimize or maximize the opportunity loss z12 of Firm U and the profits of Firms L1 and L2, are shown in Tables 5–7. For the result of minimization of the opportunity loss z12 of Firm U shown in Table 5, the opportunity loss of the small lot transportation in Firm L2 is zero while that of the regular transportation in Firm L1 becomes three days because of the constraints (12). Moreover, the objective function (2) does not include the decision variables y and z and it is found that the transportation planning is not one-sided between the regular transportation and the small lot transportation in the planning shown in Table 5. Therefore, this planning produces impartial profits to Firms L1 and L2. For the result of maximization of the profit z21 of Firm L1 shown in Table 6, because the P transportation is planned such that the income 5i¼1 ai yi from Firm U becomes larger, the solution shown
173
in Table 6 is one-sided to the regular transportation. On the other hand, to keep expenses to the drivers as low as possible, Firm L1 assigns the transportation to drivers with relatively lower ability, and consequently the transportation time and the opportunity loss of Firm U become large. Similarly, for maximization of the profit z22 of Firm L2 shown in Table 7, the solution is onesided to the small lot transportation. Because the capacity of transportation of Firm L2 is not so large, the transportation to customer 1 by Firm L1, y1 ¼ 120, is more than its minimal order. We identify the membership functions of the fuzzy goals for the objective functions, consulting the obtained solutions and related information of the four individual programming problems with a single objective function. In this paper, we use linear membership functions such as (16) which is for objective functions to be minimized, and employ the Zimmermann method for determining parameters of the membership functions [19]. 8 0; z > z0 ; > > < 0 zz 1 0 lðzÞ ¼ ð16Þ > z1 z0 ; z < z 6 z ; > : 1; z 6 z1 : In the Zimmermann method, parameter z1 is specified as the optimal value of the corresponding individual programming problem and parameter z0 is specified as the worst objective function value among the optimal solutions to the other
Table 4 An optimal solution to the problem minimizing z11 Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
280 20
130 20
229 21
70 30
80 20
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 0 2
0 0 1
0 2 0
1 0 0
0 1 0
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
1 0 0
1 0 0
0 1 0
0 0 1
1 0 0
Objective function value
z11
z12
z21
z22
(U) 1,552,800
7 (days)
(U) 551,860
(U) 100
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Table 5 An optimal solution to the problem minimizing z12 Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
150 150
130 20
150 100
70 30
80 20
Assigned drivers in Firm L1. (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 0 1
0 0 1
0 0 1
1 0 0
0 1 0
Assigned drivers in Firm L2. (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 2
1 0 0
0 2 0
0 0 1
1 0 0
Objective function value
z11
z12
z21
z22
(U) 1,746,000
3 (days)
(U) 439,260
(U) 221,000
Table 6 An optimal solution to the problem minimizing z21 Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
250 50
130 20
230 20
80 20
80 20
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
1 0 1
0 0 1
0 2 0
1 0 0
0 1 0
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 1
1 0 0
0 1 0
0 1 0
1 0 0
Objective function value
z11
z12
z21
z22
(U) 1,576,000
8 (days)
(U) 570,000
(U) 17,200
individual programming problems. The obtained parameters are shown in Table 8. Next, by using the interactive fuzzy programming method [13], we try to derive a satisfactory solution such that the satisfactory degree of the housing material manufacturer, Firm U, is regarded as important and the satisfaction of Firm U and those of the forwarding agents, Firms L1 and L2, are well-balanced. 3.2. Aggregation of the fuzzy goals of Firm U In our problem, the upper level decision maker, Firm U, has the two objective function while there
are a single decision maker at the upper level and two decision makers at the lower level. Therefore, to apply the interactive fuzzy programming method, we aggregate the fuzzy goals for the two objective functions of Firm U. In this paper, we examine three aggregation methods: the aggregation by a minimal component, the aggregation by weighting coefficients, and the aggregation by a distance from aspiration levels. The three aggregation methods are known as methods for scalarizing multiobjective mathematical programming problems [6,8,16,17]. The characteristics of the three methods are summarized as follows.
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Table 7 An optimal solution to the problem minimizing z22 Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
120 180
30 120
30 220
30 70
40 60
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 0 1
0 1 0
0 1 0
1 0 0
0 1 0
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 2
2 0 0
0 2 1
0 1 0
1 0 0
Objective function value
z11
z12
z21
z22
(U) 2,152,000
6 (days)
(U) 31,000
(U) 852,400
Table 8 Parameters z1 and z0 of the linear membership functions 1
z z0
z11
z12
z21
z22
1,552,800 2,152,000
3 8
570,000 31,000
852,400 100
In the aggregation by a minimal component, contours of the aggregated function are quadrate in a two-dimensional problem and all of the objective functions are impartially maximized because values of the aggregated function correspond to the Tchebycheff metric. Furthermore, the aggregation by a minimal component is also regarded as the fuzzy decision rule [3] in decision making under fuzzy environments. Concerning the aggregation by weighting coefficients, each of the weights for the objectives is interpreted as a relative degree of importance which represents the trade-off ratio between the objective functions. Because contours of the aggregated function by weighting coefficients are straight lines in a two-dimensional problem, values of the objective functions may be biased for some of the objectives. In the aggregation by a distance from aspiration levels, it follows that a solution closest to the aspiration levels specified by a decision maker is found. Because we employ the Tchebycheff metric
as a measure of distance, this aggregation has characteristics similar to the aggregation by a minimal component in which solutions farthest from the origin are searched. The aggregation by a minimal component. The membership function aggregated by a minimal component is represented as lm1 ðz11 ; z12 Þ ¼ minfl11 ðz11 ðy; zÞÞ; l12 ðz12 ðu; vÞÞg: ð17Þ Employing this aggregation implies that Firm U maximizes a membership function with a smaller value. The aggregation by weighting coefficients. When Firm U specifies weighting coefficients for the two fuzzy goals and tries to maximize the weighted sum of the satisfactory degrees, the following aggregated membership function is employed: lw1 ðz11 ; z12 Þ ¼ al11 ðz11 ðy; zÞÞ þ ð1 aÞl12 ðz12 ðu; vÞÞ;
ð18Þ
where a is a weighting coefficient of the first fuzzy goal l11 and then 1 a is that of the second one. The aggregation by a distance from aspiration levels. When Firm U has aspiration levels to the two kinds of the satisfactory degrees for the fuzzy goals, the two membership functions are
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aggregated by using a distance from the aspiration levels as follows: z1 ðl11 ; l12 Þ ¼ maxf l11 l11 ðz11 ðy; zÞÞ; l12 l12 ðz12 ðu; vÞÞg;
ð19Þ
where l11 and l12 are membership values representing the aspiration levels to the fuzzy goals prescribed by the membership functions l11 and l12 , respectively. Moreover, we interpret z1 ðl11 ; l12 Þ as a newly defined objective function and identify the linear membership function ld1 ðz11 ; z12 Þ in a similar way. 3.3. The first phase of the interactive fuzzy programming In this section, examining the three models of aggregation presented in Section 3.2, we apply the interactive fuzzy programming method to the transportation problem in the housing material manufacturer. In the first phase, the upper level decision maker subjectively specifies a minimal satisfactory level. Taking into consideration overall satisfactory balance between the two levels, the upper level decision maker updates the minimal satisfactory level if necessary, and a tentative solution is derived.
3.3.1. The model with the aggregation by a minimal component We apply the first phase of the interactive fuzzy programming method to the problem with the fuzzy goal of Firm U aggregated by a minimal component. The first phase of the interactive fuzzy programming method starts to solve the following problem for obtaining a solution which maximizes the smaller degree of satisfaction between both levels: maximize min lm1 ðz11 ; z12 Þ; minfl21 ðz21 ðy; uÞÞ; l22 ðz22 ðz; vÞÞg subject to ðy; z; u; vÞ 2 S:
ð20Þ
Problem (20) becomes consequently a problem maximizing the smallest of the four degrees of satisfaction, i.e., the two membership functions of Firm U and the two membership functions of Firms L1 and L2. The obtained optimal solution to Problem (20) is shown in Table 9. If Firm U is not satisfied with the obtained solution, consulting the related information of the solution, Firm U specifies the minimal satisfactory level d^ and formulates the following problem, in which the smaller satisfactory degree of Firms L1 and L2 is maximized under the condition that
Table 9 An optimal solution to Problem (20) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
210 90
90 60
139 111
31 69
50 50
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
1 1 0
0 0 1
0 0 1
1 0 0
0 0 1
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 1
1 0 0
0 2 0
0 0 1
1 0 0
Objective function value Satisfactory degree lðzÞ Aggregated satisfactory degree
z11
z12
z21
z22
1,840,200 0.52
5 0.60 0.52
311,420 0.52 –
443,720 0.52 –
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177
the aggregated satisfactory degree of Firm U is larger than or equal to the minimal satisfactory level: maximize minfl21 ðz21 ðy; uÞÞ; l22 ðz22 ðz; vÞÞg ð21Þ
subject to ðy; z; u; vÞ 2 S; lm1 ðz11 ; z12 Þ P d^:
An optimal solution to Problem (21) with d^ ¼ 0:65 is shown in Table 10. Comparing the solution to the maximin problem shown in Table 9, to satisfy the condition of the minimal satisfactory level, the satisfactory degrees of both fuzzy goals of Firm U are increased and those of Firms L1 and L2 are decreased. Moreover, drivers with higher ability level are assigned largely. The ratio of satisfactory degrees between the two levels is D ¼ 0:68 and it follows that a tentative solution is derived if the ratio is in the interval ½Dmin ; Dmax specified by Firm U and Firm U is satisfied with the solution. To examine the characteristics of the model with the aggregation by a minimal component, we solve Problem (21), varying the parameter d^, and suppose that Firm U specifies the bounds of the ratio of satisfaction at ½Dmin ; Dmax ¼ ½0:6; 0:8 . The
Fig. 2. The relation between the satisfactory degrees and d^ in the model with the aggregation by a minimal component.
result is shown in Fig. 2 depicting the relation between the satisfactory degrees and the minimal satisfactory level d^. The range ½0:61; 0:68 of the abscissa shown in Fig. 2 is the interval in which the ratio D is in the bounds ½0:6; 0:8 , i.e., if Firm U specifies the minimal satisfactory level d^ in the interval ½0:61; 0:68 , the ratio D is in the bounds ½Dmin ; Dmax ¼ ½0:6; 0:8 . As seen in Fig. 2, the satisfactory degree l12 of the fuzzy goal for the opportunity loss with respect to the transportation time takes a fixed value 0:8 for any d^ 2 ½0:61; 0:68 . Because the aggregated satisfactory degree is represented by (17) and l11 < l12 ¼ 0:8 for any d^ 2 ½0:61; 0:68 , the aggregated
Table 10 An optimal solution to Problem (21) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
262 38
130 20
88 162
32 68
40 60
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 0 2
0 0 1
0 1 0
1 0 0
0 1 0
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 1
1 0 0
0 1 1
0 0 1
1 0 0
Objective function value Satisfactory degree lðzÞ Aggregated satisfactory degree
z11
z12
z21
z22
1,762,400 0.65
4 0.8 0.65
280,840 0.46 –
377,840 0.44 –
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M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
satisfactory degree is always equal to the satisfactory degree l11 and therefore it follows that the satisfactory degree for the second objective function cannot affect the aggregated satisfactory degree at all. 3.3.2. The model with the aggregation by weighting coefficients We apply the first phase of the interactive fuzzy programming method to the problem with the fuzzy goal of Firm U aggregated by weighting coefficients. In a way similar to the model with the aggregation by a minimal component, the first phase of the interactive fuzzy programming method starts to solve the following problem for obtaining a solution which maximizes the smaller degree of satisfaction between both levels: maximize min lw1 ðz11 ; z12 Þ; minfl21 ðz21 ðy; uÞÞ; l22 ðz22 ðz; vÞÞg subject to ðy; z; u; vÞ 2 S:
ð22Þ
An optimal solution to Problem (22) with the weighting coefficients a ¼ 1 a ¼ 0:5 of the fuzzy goals l11 and l12 in (18) is shown in Table 11. If Firm U is not satisfied with the obtained solution, consulting the related information of the
solution, Firm U specifies the minimal satisfactory level d^ and formulates the following problem, in which the smaller satisfactory degree of Firms L1 and L2 is maximized under the condition that the aggregated satisfactory degree of Firm U is larger than or equal to the minimal satisfactory level: maximize minfl21 ðz21 ðy; uÞÞ; l22 ðz22 ðz; vÞÞg ð23Þ
subject to ðy; z; u; vÞ 2 S; lw1 ðz11 ; z12 Þ P d^:
An optimal solution to Problem (23) with d^ ¼ 0:72 is shown in Table 12. Comparing the solution in Table 12 with the solution in Table 11, the small lot transportation of Firm L2 becomes larger and Firm L1 decreases the number of assigned drivers to cope with the reduction of transportation ordered by Firm U. To examine the characteristics of the model with the aggregation by weighting coefficients, we solve Problem (23), varying the parameter d^, and assume that Firm U specifies the bounds of the ratio of satisfaction at ½Dmin ; Dmax ¼ ½0:6; 0:8 . The result is shown in Fig. 3 depicting the relation between the satisfactory degrees and the minimal satisfactory level d^.
Table 11 An optimal solution to Problem (22) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
120 180
30 120
230 20
32 68
62 38
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 1 0
0 0 1
0 2 0
1 0 0
0 0 1
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 2
2 0 0
0 1 0
0 0 1
1 0 0
Objective function value Satisfactory degree lðzÞ Aggregated satisfactory degree
z11
z12
z21
z22
1,960,000 0.32
4 0.8 0.56
325,920 0.55 –
466,080 0.55 –
M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
179
Table 12 An optimal solution to Problem (23) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
120 180
91 59
122 128
62 38
75 25
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 1 0
0 0 1
0 0 1
1 0 0
0 0 1
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 2
1 0 0
0 2 0
0 0 1
1 0 0
Objective function value Satisfactory degree lðzÞ Aggregated satisfactory degree
z11
z12
z21
z22
1,888,200 0.44
3 1.0 0.72
316,000 0.53 –
450,740 0.53 –
The range ½0:65; 0:78 of the abscissa shown in Fig. 3 is the interval in which the ratio D is in the bounds ½0:6; 0:8 in a way similar to Fig. 2. As seen in Fig. 3, the satisfactory degree l12 of the fuzzy goal for the opportunity loss with respect to the transportation time takes a fixed value 1:0 for any d^ 2 ½0:65; 0:78 and that of the cost of transportation takes some value such that 0:4 6 l11 6 0:6. The value of l11 is smaller than that of the model with the aggregation by a minimal component and there is a larger difference between the two satisfactory degrees l11 and l12 . Moreover, the interval
of d^ in which the ratio D is in the specified bounds ½0:6; 0:8 is 0:65 6 d^ 6 0:78 and it is wider than that of the previous model. 3.3.3. The model with the aggregation by a distance from aspiration levels We apply the first phase of the interactive fuzzy programming method to the problem with the fuzzy goal of Firm U aggregated by a distance from aspiration levels. In a way similar to the previous two models, the first phase of the interactive fuzzy programming method starts to solve the following problem for obtaining a solution which maximizes the smaller degree of satisfaction between both levels: maximize min ld1 ðz11 ; z12 Þ; minfl21 ðz21 ðy; uÞÞ; l22 ðz22 ðz; vÞÞg subject to ðy; z; u; vÞ 2 S:
Fig. 3. The relation between the satisfactory degrees and d^ in the model with the aggregation by weighting coefficients.
ð24Þ
Assume that Firm U specifies the aspiration levels in (19) at l11 ¼ 1:0 and l12 ¼ 0:8, and the Zimmermann method [19] is employed and the parameters of the linear membership function are determined as z1 ¼ 0:12 and z0 ¼ 1:0. Then, an optimal solution to Problem (24) with these
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Table 13 An optimal solution to Problem (24) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
120 180
122 28
107 143
73 27
48 52
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 1 0
0 0 1
0 1 0
1 0 0
0 1 0
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 2
1 0 0
0 2 0
0 1 0
1 0 0
Objective function value Satisfactory degree lðzÞ Aggregated satisfactory degree
z11
z12
z21
z22
1,869,400 0.47
6 0.4 0.54
320,660 0.54 –
457,920 0.54 –
parameters is shown in Table 13. Especially, it should be noted that the difference between the two satisfactory degrees of Firm U, l11 and l12 , is smaller than those of the previous two models. If Firm U is not satisfied with the obtained solution, consulting the related information of the solution, Firm U specifies the minimal satisfactory level d^ and formulates the following problem, in which the smaller satisfactory degree of Firms L1 and L2 is maximized under the condition that the aggregated satisfactory degree of Firm U is
larger than or equal to the minimal satisfactory level: maximize minfl21 ðz21 ðy; uÞÞ; l22 ðz22 ðz; vÞÞg ð25Þ
subject to ðy; z; u; vÞ 2 S; ld1 ðz11 ; z12 Þ P d^:
An optimal solution to Problem (25) with d^ ¼ 0:72 is shown in Table 14.
Table 14 An optimal solution to Problem (25) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
249 51
128 22
88 162
40 60
41 59
Assignaed drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
1 0 1
0 0 1
0 0 1
0 1 0
0 1 0
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 1 0
1 0 0
0 1 1
0 0 1
1 0 0
Objective function value Satisfactory degree lðzÞ Aggregated satisfactory degree
z11
z12
z21
z22
1,773,200 0.63
5 0.6 0.72
284,820 0.47 –
401,420 0.47 –
M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
Comparing the solution in Table 14 with the solution in Table 13, to satisfy the condition of the minimal satisfactory level, some parts of the regular transportation to customers 1 and 2 are switched to the small lot transportation, some parts of the small lot transportation to customers 3, 4 and 5 are switched to the regular transportation and drivers with higher ability level are assigned for increased orders in both forwarding agents. Moreover, the difference between l11 and l12 is still small. To examine the characteristics of the model with the aggregation by a distance from aspiration levels, we solve Problem (25), varying the parameter d^, and assume that Firm U specifies the bounds of the ratio of satisfactory degrees at ½Dmin ; Dmax ¼ ½0:6; 0:8 . The result is shown in Fig. 4 depicting the relation between the satisfactory degrees and the minimal satisfactory level d^. The range ½0:66; 0:75 of the abscissa shown in Fig. 4 means the interval in which the ratio D is in the bounds ½0:6; 0:8 in a way similar to Figs. 2 and 3. As seen in Fig. 4, the satisfactory degree l11 takes some value in 0:58 6 l11 6 0:7 and the satisfactory degree l12 for the opportunity loss is smaller than l11 for the cost of transportation. The interval of d^ in which the ratio D is in the specified bounds ½0:6; 0:8 is 0:66 6 d^ 6 0:75 and it is wider than that of the model with the aggregation by a minimal component.
181
3.4. The second phase of the interactive fuzzy programming In the first phase of the interactive fuzzy programming, Firms L1 and L2 are treated impartially and they can be regarded as a group. However, if Firm U wants to give either of Firms L1 and L2 special consideration, Firm U might specify different intervals ½Djmin ; Djmax , j ¼ 1; 2 for them separately. In such a case, it is necessary to start the second phase of the interactive fuzzy programming. In the second phase, by coordinating the satisfactory degrees of Firm U and each of Firms L1 and L2, the final satisfactory solution is derived. First, we calculate the following individual ratios of satisfactory degrees of Firm U and each of Firms L1 and L2 with respect to the solution obtained in the first phase: Di ¼
l2i ðz2i Þ ; l1 ðz11 ; z12 Þ
i ¼ 1; 2;
ð26Þ
where l1 ðz11 ; z12 Þ denotes the aggregated satisfactory degree of Firm U. For a forwarding agent whose ratio of satisfactory degrees with Firm U is not in the interval ½Djmin ; Djmax determined by Firm U, Firm U specifies a minimal satisfactory level d~ and/or a maximal satisfactory level d and formulates the problem: maximize l2i ðz2i Þ subject to ðy; z; u; vÞ 2 S; l1 ðz11 ; z12 Þ P d^; d~ 6 l2j ðz2j Þ 6 d;
Fig. 4. The relation between the satisfactory degrees and d^ in the model with the aggregation by a distance from aspiration levels.
ð27Þ
j ¼ 1; 2;
j 6¼ i:
Assume that Firm U adopts a policy that Firm U gives favorable treatment to Firm L1 dealing with the regular transportation, and specifies the bounds of the ratios of satisfactory degrees for Firms L1 and L2 at ½D1min ; D1max ¼ ½0:7; 0:8 and ½D2min ; D2max ¼ ½0:6; 0:7 , respectively. From the examination in Section 3.3, because it is found that, in the model with the aggregation by a minimal component, the satisfactory degree for the second objective function does not affect the
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aggregated satisfactory degree at all, we will deal with only the model with the aggregation by weighting coefficients and the model with the aggregation by a distance from aspiration levels in the second phase of the interactive fuzzy programming method. 3.4.1. The model with the aggregation by weighting coefficients Assume that, in the first phase, Firm U is satisfied with the solution which is obtained by solving Problem (23) with d^ ¼ 0:72. The corresponding individual ratios of satisfactory degrees are D1 ¼ D2 ¼ 0:74, and we have D1 2 [0.7, 0.8] and D2 62 ½0:6; 0:7 . Because the ratio D2 is over the upper bound 0.7, assume that Firm U specifies the minimal satisfactory level d~ for l22 . Then, the following problem is formulated: maximize l21 ðz21 ðy; uÞÞ subject to ðy; z; u; vÞ 2 S; lw1 ðz11 ; z12 Þ P d^ ¼ 0:72; 0:504 ¼ d~ 6 l22 ðz22 ðz; vÞÞ;
ð28Þ
where the minimal satisfactory level d~ for l22 is determined at d~ ¼ 0:72 0:7 ¼ 0:504. An optimal solution to Problem (28) is shown in Table 15.
As seen in Table 15, comparing the transportation planning corresponding to the tentative solution obtained in the first phase which is shown in Table 12, some part of the small lot transportation to customer 3 is switched to the regular transportation and conversely some part of the regular transportation to customer 4 is switched to the small lot transportation. From these facts, it is thought that the satisfactory degree of Firm L1 increases and the ratio D2 is in the specified interval. To examine the characteristics of this model in the second phase of the interactive fuzzy programming, we solve Problem (28), varying the parameter d~. The result is shown in Fig. 5, depicting the relation between the ratios of satisfactory degrees and the minimal satisfactory level d~. As seen in Fig. 5, because D1 decreases and D2 increases as d~ increases, we can understand the conflict between l21 and l22 . Moreover, the interval of d~ leading to possible satisfactory solutions in the second phase is 0:489 6 d~ 6 0:504 and we have verified that the range of the parameter d~ which we can control is very narrow. 3.4.2. The model with the aggregation by a distance from aspiration levels Assume that, in the first phase, Firm U is satisfied with the solution which is obtained by solving Problem (25) with d^ ¼ 0:72. The corresponding
Table 15 An optimal solution to Problem (28) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
120 180
90 60
150 100
32 68
74 26
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 1 0
0 0 1
0 0 1
1 0 0
0 0 1
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 2
1 0 0
0 2 0
0 0 1
1 0 0
Objective function value Satisfactory degree lðzÞ Ratio of satisfactory degrees
z11
z12
z21
z22
1,887,200 0.44
3 1.0 lw1 ¼ 0:72
339,520 0.57 D1 ¼ 0:79
426,520 0.50 D2 ¼ 0:69
M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
Fig. 5. The relation between the ratios of satisfactory degrees and d~ in the model with the aggregation by weighting coefficients.
individual ratios of satisfactory degrees are D1 ¼ D2 ¼ 0:65, and we have D1 62 ½0:7; 0:8 and D2 2 ½0:6; 0:7 . Because the ratio D1 is smaller than the lower bound 0.7, assume that Firm U specifies the minimal satisfactory level d~ for l21 . Then, the following problem is formulated: maximize l22 ðz22 ðz; vÞÞ subject to ðy; z; u; vÞ 2 S; ld1 ðz11 ; z12 Þ P d^ ¼ 0:72; 0:504 ¼ d~ 6 l21 ðz21 ðy; uÞÞ;
ð29Þ
183
where the minimal satisfactory level d~ for l21 is determined at d~ ¼ 0:72 0:7 ¼ 0:504. An optimal solution to Problem (29) is shown in Table 16. As seen in Table 16, comparing the transportation planning corresponding to the tentative solution obtained in the first phase which is shown in Table 14, by increasing some amount of the order to the regular transportation, we can increase the satisfactory degree of Firm L1 with the small reduction of the satisfactory degree of Firm L2 so as to satisfy the conditions of the ratios of satisfactory degrees. It is also seen that the orders to the regular transportation becomes larger than in the transportation planning corresponding to the solution of the model with the aggregation by weighting coefficients shown in Table 15. To examine the characteristics of this model in the second phase of the interactive fuzzy programming, we solve Problem (29), varying the parameter d~. The result is shown in Fig. 6 depicting the relation between the ratios of satisfactory degrees and the minimal satisfactory level d~. As seen in Fig. 6, because D1 increases and D2 decreases as d~ increases, we can also understand the conflict between l21 and l22 . Moreover, the interval of d~ leading to possible satisfactory solutions in the second phase is 0:504 6 d~ 6 0:573 and it is found that the range of the parameter d~ which
Table 16 An optimal solution to Problem (29) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
249 51
120 30
106 144
45 55
40 60
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
1 0 1
0 0 1
0 0 1
0 1 0
0 1 0
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 1 0
1 0 0
0 2 0
0 0 1
1 0 0
Objective function value Satisfactory degree lðzÞ Ratio of satisfactory degrees
z11
z12
z21
z22
1,773,200 0.63
5 0.60 ld1 ¼ 0:72
302,740 0.50 D1 ¼ 0:70
394,500 0.46 D2 ¼ 0:64
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by weighting coefficients, and the aggregation by a distance from aspiration levels. Finally, we have concluded that the aggregation by a distance from aspiration levels is effective for our problem.
References
Fig. 6. The relation between the ratios of satisfactory degrees and d~ in the model with the aggregation by a distance from aspiration levels.
we can control is wider than that of the model with the aggregation by weighting coefficients. From the above mentioned analysis, in the model with the aggregation by a distance from aspiration levels, it is easier to control the two satisfactory degrees of Firm U and the range of the parameter d~ which we can control is wider than the others. It seems reasonable to conclude, from these facts, that employing the model with the aggregation by a distance from aspiration levels is appropriate for the decentralized two-level transportation problem in the housing material manufacturer.
4. Conclusions In this paper, we have dealt with the transportation problem in the housing material manufacturer and, applying the interactive fuzzy programming method, we have derived a satisfactory solution to the problem by taking into account not only the degree of satisfaction with respect to the objectives of the housing material manufacturer but also those of the two forwarding agents to which the housing material manufacturer entrusts transportation of products. As the housing material manufacturer has the two objective function, we have aggregated the two satisfactory degrees for the two objectives have examined three aggregation methods: the aggregation by a minimal component, the aggregation
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Production, Manufacturing and Logistics
A decentralized two-level transportation problem in a housing material manufacturer: Interactive fuzzy programming approach Masatoshi Sakawa
a,*
, Ichiro Nishizaki a, Yoshio Uemura
b
a
b
Department of Artificial Complex Systems Engineering, Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima 739-8527, Japan System Analysis EDP Department, Juken Sangyou Co., Ltd., 1-1 Mokuzaikou-minami, Hatsukaichi, Hiroshima 738-8502, Japan Received 21 November 2000; accepted 23 July 2001
Abstract In this paper, we deal with a transportation problem in a housing material manufacturer and derive a satisfactory solution to the problem by taking into account not only the degree of satisfaction with respect to objectives of the housing material manufacturer but also those of two forwarding agents to which the housing material manufacturer entrusts transportation of products. The two forwarding agents handle the regular transportation and the small lot transportation, and each of them assigns work force, the transformation tasks properly. There are two objectives in the housing material manufacturer: one is to minimize the transportation cost and the other is to minimize the opportunity loss with respect to transportation time. The objectives of the two forwarding agents are to maximize their profits. We formulate the transportation planning and work force assignment problem as a decentralized two-level integer programming problem, and derive a satisfactory solution by applying an interactive fuzzy programming method. Ó 2002 Published by Elsevier Science B.V. Keywords: Optimization; Transportation planning; Work force assignment; Two-level programming; Interactive fuzzy programming
1. Introduction In this paper, we treat a transportation problem in a housing material manufacturer. The housing material manufacturer does not transport its
* Corresponding author. Tel.: +81-82-424-7695; fax: +81-82424-7695. E-mail address: [email protected] (M. Sakawa).
products from its factory or warehouse to customers on its own account but entrusts the transportation to forwarding agents. There are two kinds of forwarding agents: one handles the regular transportation and the other handles the small lot transportation. Minimizing the transportation cost and the opportunity loss with respect to transportation time, the housing material manufacturer gives the two forwarding agents orders for the transportation of the products to the customers. The two forwarding agents assign work force
0377-2217/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 3 7 7 - 2 2 1 7 ( 0 1 ) 0 0 2 7 3 - 9
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so as to maximize their profits, taking ability of drivers into account. Such a transportation planning and work force assignment problem can be formulated as a decentralized two-level integer programming problem. To make rational and efficient plan for transportation of the products and to take into account cooperative relation between the housing material manufacturer and the two forwarding agents, by applying interactive fuzzy programming, we try to derive a satisfactory solution. In a simple two-level mathematical programming problem, there are two decision makers at the upper and the lower levels, and the upper level decision maker determines a decision and thereafter the lower level decision maker chooses a decision. Moreover, assuming that each decision maker completely knows the objective functions of the upper and the lower level decision makers and the constraints and, for a decision of the upper level decision maker, the lower level decision maker responds rationally, the upper level decision maker specifies a decision so as to optimize the objective. Then a solution defined as the above mentioned procedure is called the Stackelberg (equilibrium) solution. The Stackelberg solution has been employed as a solution concept when decision making problems are formulated as twolevel mathematical programming problems [1,2,5, 15,18], whether there is a cooperative relationship between the decision makers or not. Furthermore, it should be noted that the Stackelberg solution does not always satisfy Pareto optimality because of its noncooperative nature. In situations where the Stackelberg solution is employed, it is supposed that the two decision makers do not have motivation to cooperate mutually. However, if there are commercial transactions between two firms in the long term, or a parent firm orders a subsidiary firm some tasks, the two firms may have motivation to cooperate each other and be willing to derive a solution satisfactory to both firms. In the case study dealt with in this paper, the housing material manufacturer has been connected in business with each of the forwarding agents over a long period of time and there exists cooperative relationship between them. For this cooperative relationship, it is nat-
ural that the housing material manufacturer does not optimize only its own objectives but makes decisions cooperatively by balancing its own satisfaction with those of the two forwarding agents. Thus, requiring a cooperative decision making method different from the Stackelberg solution models, we employ interactive fuzzy programming approach to the transportation planning and work force assignment problem. Methods for obtaining solutions to two-level programming problems where there is cooperative relationship between decision makers have been presented within interactive fuzzy programming by Lai [7], Shih et al. [14], and Sakawa et al. [9–13]. Especially, the interactive fuzzy programming method for a decentralized two-level linear programming problem developed by Sakawa et al. [13] is applicable to the transportation problem in the housing material manufacturer by revising it partly because the transportation problem in the housing material manufacturer can be formulated as a decentralized two-level integer programming problem with the two objectives of the upper level decision maker. In Section 2, we formulate the transportation problem in the housing material manufacturer and solve four individual programming problems with a single objective function which is each of the two objectives of the housing material manufacturer and the two objectives of the two forwarding agents in order to understand characteristics of the transportation problem and to specify parameters of membership functions of fuzzy goals. In Section 3, the fuzzy goals for the objective functions are identified and the two fuzzy goals of the housing material manufacturer are aggregated by three methods. Then, we derive a satisfactory solution to the transportation problem by applying the interactive fuzzy programming method composed of two interactive phases.
2. Formulation of the transportation problem The transportation problem in the housing material manufacturer is formulated as a decentralized two-level integer programming problem. The upper level decision maker is the housing
M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
169
Table 1 Data on transportation of Firm U Customer
Xi
ai (U/product)
bi (U/product)
ci (days)
di (days)
1 2 3 4 5
300 150 250 100 100
1000 2000 2200 800 2000
2000 4000 3000 1400 3400
10 2 2 7 2
5 1 1 3 1
material manufacturer and there are two decision makers at the lower level who are the two forwarding agents handling the regular transportation and the small lot transportation. The housing material manufacturer determines allocation of the transportation tasks to five customers between the two forwarding agents, and they assign work force, taking three levels of ability of drivers into account. The housing material manufacturer and the two forwarding agents have the following objectives. 2.1. Objective functions To describe concisely, let Firm U, Firm L1 and Firm L2 denote the housing material manufacturer, the forwarding agent handling the regular transportation and the forwarding agent handling the small lot transportation, respectively. The first objective function of Firm U. Coordinating orders to Firm L1 and Firm L2, Firm U minimizes the transportation cost, z11 ðy; zÞ ¼
5 X
ðai yi þ bi zi Þ;
ð1Þ
i¼1
where ai and bi are unit costs of the regular and the small lot transportation to customer i, respectively, and are shown in Table 1; yi and zi denote decision variables which are the numbers of the products to be sent to customer i by Firm L1 and Firm L2, respectively. Because the decision variables y ¼ ðy1 ; . . . ; y5 Þ and z ¼ ðz1 ; . . . ; z5 Þ of Firm U are the numbers of the products to be sent to the five customers, they are integer variables. The second objective function of Firm U. Firm U also minimizes the opportunity loss with respect to the transportation time which is a delay from the earliest delivery time,
z12 ðu; vÞ ¼
5 X 3 X i¼1
ðeij uij þ fij vij Þ;
ð2Þ
j¼1
where eij and fij are, respectively, the values of the opportunity loss when drivers with ability level j in Firms L1 and L2 transport the products to customer i, and they are defined by eij ¼ oij oi3 and fij ¼ pij pi3 ; oij and pij are transportation times when the drivers with ability level j of Firms L1 and L2 transport the products to customer i. Because we use a larger index j for drivers with higher ability, oi3 and pi3 are the smallest values. oij and pij are shown in Tables 2 and 3, respectively. uij and vij denote decision variables which are the numbers of the assigned drivers with ability level j in Firms L1 and L2 who transform the products to customer i. Because the decision variables u ¼ ðu11 ; . . . ; u53 Þ and v ¼ ðv11 ; . . . ; v53 Þ of Firms L1 and L2
Table 2 Drivers data of Firm L1a qj o1j , e1j (days) o2j , e2j (days) o3j , e3j (days) o4j , e4j (days) o5j , e5j (days) s1j s2j s3j s4j s5j a
(U) (U) (U) (U) (U)
Level 1
Level 2
Level 3
100 10, 2
120 9, 1
150 8, 0
3* , –
2, 1
1, 0
3*, –
2, 0
2, 0
7, 2
6, 1
5, 0
3*, –
2, 1
1, 0
40,000 – – 34,000 –
57,000 36,000 36,000 48,000 36,000
72,000 44,000 44,000 60,000 44,000
The symbol * denotes drivers who cannot be assigned.
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Table 3 Drivers data of Firm L2a rj p1j , e1j (days) p2j , e2j (days) p3j , e3j (days) p4j , e4j (days) p5j , e5j (days) t1j t2j t3j t4j t5j a
(U) (U) (U) (U) (U)
2.2. Constraints
Level 1
Level 2
Level 3
60 5, 2
72 4, 1
90 3, 0
1, 0
1, 0
1, 0
3*, –
2, 0
2, 0
Constraints on the amount of transportation. Because the amount of transportation to customer i is fixed at Xi , the sum yi þ zi of the numbers of the products to be sent to customer i by Firms L1 and L2 equals Xi and the constraints on the amount of transportation are represented as
2*, –
1, 0
1, 0
Xi ¼ yi þ zi ;
1, 0
1, 0
1, 0
40,000 32,000 – – 32,000
52,000 43,000 43,000 49,000 43,000
62,000 54,000 54,000 58,000 54,000
The values of Xi are shown in Table 1. Constraints on the minimal orders. Firm U must order Firms L1 and L2 to transport more than 30 and 20 products to each customer, respectively. The constraints on the minimal orders are represented as
The symbol * denotes drivers who cannot be assigned.
are the numbers of drivers, they are integer variables. The objective function of Firm L1. By assigning drivers efficiently, Firm L1 maximizes the following profit defined by taking necessary expenses and costs on drivers from the income from Firm U: z21 ðy; uÞ ¼
5 X i¼1
( ai yi
0:3ai yi þ
3 X
!) sij uij
:
j¼1
i¼1
j¼1
where tij denotes the cost on a driver with ability level j who transports the products to customer i, and values of tij are shown in Table 3.
ð5Þ
yi P 30;
i ¼ 1; . . . ; 5;
ð6Þ
zi P 20;
i ¼ 1; . . . ; 5:
ð7Þ
Constraints on assignment of work force . Firms L1 and L2 must transport the products to the customers before the specified date of delivery and they cannot assign transportation of the products to drivers who take longer time than the specified date of delivery. The constraints on assignment of work force are represented as X yi 6 qj uij ; i ¼ 1; . . . ; 5; ð8Þ
ð3Þ The necessary expense is defined as 30% of the income from Firm U. In the objective function (3), sij denotes the cost on a driver with ability level j who transports the products to customer i, and values of sij are shown in Table 2. The objective function of Firm L2. Similarly to Firm L1, Firm L2 also maximizes the following profit defined by taking necessary expenses and costs on drivers from the income from Firm U: ( !) 5 3 X X z22 ðz; vÞ ¼ bi zi 0:3bi zi þ tij vij ; ð4Þ
i ¼ 1; . . . ; 5:
j2Ki
zi 6
X
rj vij ;
i ¼ 1; . . . ; 5;
ð9Þ
j2Li
where qj and rj are capacities of transportation of a driver with ability level j in Firms L1 and L2, respectively. Let Ki ¼ fj j oij 6 ci g and Li ¼ fj j pij 6 di g, where ci and di denote the specified dates of delivery to customer i for Firms L1 and L2, respectively, and they are shown in Table 1. Constraints on the numbers of drivers. For each ability level, Firms L1 and L2 must assign from one to three drivers and the constraints on the numbers of drivers are represented as 16
5 X i¼1
uij 6 3;
j ¼ 1; 2; 3;
ð10Þ
M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
16
5 X
vij 6 3;
j ¼ 1; 2; 3:
ð11Þ
i¼1
Constraints on redundant assignment of work force. To prevent redundant assignment of work force, the following constraints, meaning that actual transportation to each customer must be smaller than the sum of the volume of the order and the capacity of transportation of a driver with the lowest ability level, are imposed: X qj uij 6 yi þ q1 1; i ¼ 1; . . . ; 5; ð12Þ j2Ki
X
rj vij 6 zi þ r1 1;
i ¼ 1; . . . ; 5:
ð13Þ
j2Li
3. Interactive fuzzy programming We apply the interactive fuzzy programming method to the transportation planning and work force assignment problem, which is formulated as the decentralized two-level integer programming problem, because the housing material manufacturer denoted by Firm U makes decisions by taking into account not only its own satisfaction but also the satisfaction of the two forwarding agents denoted by Firm L1 and L2. To effectively support cooperative decision making in decentralized organizations, the interactive fuzzy programming methods have been developed. In the methods, it is assumed that multiple decision makers can coordinate satisfaction of them and arrive at an agreement, and interactive process is performed by a delegate of them or an analyst who makes contact with the decision makers. Consequently, the delegate or the analyst controls all of the decision variables, paying more attention or respect to the upper level decision maker and considering satisfactory balance between the two levels. Before examining the application, we show an outline of the interactive fuzzy programming method. The interactive fuzzy programming method by Sakawa et al. [13] was developed for obtaining a satisfactory solution to a decentralized
171
two-level linear programming problem in which there are a single decision maker at the upper level and two or more decision makers at the lower level and each decision maker has a single objective function. The algorithm is composed of two phases. In the first phase, the decision makers at both levels identify membership functions of their fuzzy goals for the objective functions. Let l0 denote the membership function of the upper level decision maker and li , i ¼ 1; . . . ; k, the membership functions of the k lower level decision makers. Taking into consideration overall satisfactory balance between the two levels by consulting the ratio of satisfactory degrees between the two levels, D¼
mini¼1;...;k li ðzi Þ ; l0 ðz0 Þ
ð14Þ
the upper level decision maker specifies the minimal satisfactory level d^ and updates it if necessary, and then a tentative solution is obtained. In this phase, the lower level decision makers are treated impartially, and therefore they can be regarded as a group. In the second phase, consulting the ratios of satisfaction between the upper level decision maker and each of the lower level decision makers, the upper level decision maker specifies minimal and/or maximal satisfactory levels to some of the lower level decision makers and updates them if necessary. By coordinating the satisfactory degrees of the decision makers individually, the final satisfactory solution can be derived. The interactive procedure is depicted by a flowchart in Fig. 1. Through this interactive process, although more attention or respect to the upper level decision maker is paid, we can obtain the final satisfactory solution well-balanced between the both levels, and the satisfactory degree of the upper level decision maker and that of each lower level decision maker are coordinated appropriately. First, to incorporate fuzziness of human judgment of the decision makers, we identify membership functions of fuzzy goals for their objective functions. To do so, we solve four singleobjective programming problems in which four objective functions shown in the previous section
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Fig. 1. Interactive procedures in the fuzzy programming.
are optimized separately, and determine parameters of the membership functions by consulting the optimal values and solutions of the four problems.
minimize z11 ðy; zÞ subject to ðy; z; u; vÞ 2 S;
3.1. Individual optimization problems
where S denotes the feasible region satisfying the constraints (5)–(13). Problem (15) is solved by using ‘lp_solve’ developed by Berkelaar [4], which is a software for solving mixed integer program-
We first solve the problem minimizing the transportation cost z11 of Firm U:
ð15Þ
M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
ming problems and is based on the branch-andbound techniques, and subsequent problems formulated in this paper are also solved by using lp_solve. The obtained optimal solution to Problem (15) is shown in Table 4. For the obtained solution, the transportation planning is one-sided to the regular transportation of Firm L1, and Firm L2 receives only the minimum orders. Therefore, the profit of Firm L2, z22 , is extremely small. Similarly, the results of solving the rest of the problems, which minimize or maximize the opportunity loss z12 of Firm U and the profits of Firms L1 and L2, are shown in Tables 5–7. For the result of minimization of the opportunity loss z12 of Firm U shown in Table 5, the opportunity loss of the small lot transportation in Firm L2 is zero while that of the regular transportation in Firm L1 becomes three days because of the constraints (12). Moreover, the objective function (2) does not include the decision variables y and z and it is found that the transportation planning is not one-sided between the regular transportation and the small lot transportation in the planning shown in Table 5. Therefore, this planning produces impartial profits to Firms L1 and L2. For the result of maximization of the profit z21 of Firm L1 shown in Table 6, because the P transportation is planned such that the income 5i¼1 ai yi from Firm U becomes larger, the solution shown
173
in Table 6 is one-sided to the regular transportation. On the other hand, to keep expenses to the drivers as low as possible, Firm L1 assigns the transportation to drivers with relatively lower ability, and consequently the transportation time and the opportunity loss of Firm U become large. Similarly, for maximization of the profit z22 of Firm L2 shown in Table 7, the solution is onesided to the small lot transportation. Because the capacity of transportation of Firm L2 is not so large, the transportation to customer 1 by Firm L1, y1 ¼ 120, is more than its minimal order. We identify the membership functions of the fuzzy goals for the objective functions, consulting the obtained solutions and related information of the four individual programming problems with a single objective function. In this paper, we use linear membership functions such as (16) which is for objective functions to be minimized, and employ the Zimmermann method for determining parameters of the membership functions [19]. 8 0; z > z0 ; > > < 0 zz 1 0 lðzÞ ¼ ð16Þ > z1 z0 ; z < z 6 z ; > : 1; z 6 z1 : In the Zimmermann method, parameter z1 is specified as the optimal value of the corresponding individual programming problem and parameter z0 is specified as the worst objective function value among the optimal solutions to the other
Table 4 An optimal solution to the problem minimizing z11 Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
280 20
130 20
229 21
70 30
80 20
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 0 2
0 0 1
0 2 0
1 0 0
0 1 0
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
1 0 0
1 0 0
0 1 0
0 0 1
1 0 0
Objective function value
z11
z12
z21
z22
(U) 1,552,800
7 (days)
(U) 551,860
(U) 100
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Table 5 An optimal solution to the problem minimizing z12 Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
150 150
130 20
150 100
70 30
80 20
Assigned drivers in Firm L1. (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 0 1
0 0 1
0 0 1
1 0 0
0 1 0
Assigned drivers in Firm L2. (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 2
1 0 0
0 2 0
0 0 1
1 0 0
Objective function value
z11
z12
z21
z22
(U) 1,746,000
3 (days)
(U) 439,260
(U) 221,000
Table 6 An optimal solution to the problem minimizing z21 Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
250 50
130 20
230 20
80 20
80 20
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
1 0 1
0 0 1
0 2 0
1 0 0
0 1 0
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 1
1 0 0
0 1 0
0 1 0
1 0 0
Objective function value
z11
z12
z21
z22
(U) 1,576,000
8 (days)
(U) 570,000
(U) 17,200
individual programming problems. The obtained parameters are shown in Table 8. Next, by using the interactive fuzzy programming method [13], we try to derive a satisfactory solution such that the satisfactory degree of the housing material manufacturer, Firm U, is regarded as important and the satisfaction of Firm U and those of the forwarding agents, Firms L1 and L2, are well-balanced. 3.2. Aggregation of the fuzzy goals of Firm U In our problem, the upper level decision maker, Firm U, has the two objective function while there
are a single decision maker at the upper level and two decision makers at the lower level. Therefore, to apply the interactive fuzzy programming method, we aggregate the fuzzy goals for the two objective functions of Firm U. In this paper, we examine three aggregation methods: the aggregation by a minimal component, the aggregation by weighting coefficients, and the aggregation by a distance from aspiration levels. The three aggregation methods are known as methods for scalarizing multiobjective mathematical programming problems [6,8,16,17]. The characteristics of the three methods are summarized as follows.
M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
175
Table 7 An optimal solution to the problem minimizing z22 Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
120 180
30 120
30 220
30 70
40 60
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 0 1
0 1 0
0 1 0
1 0 0
0 1 0
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 2
2 0 0
0 2 1
0 1 0
1 0 0
Objective function value
z11
z12
z21
z22
(U) 2,152,000
6 (days)
(U) 31,000
(U) 852,400
Table 8 Parameters z1 and z0 of the linear membership functions 1
z z0
z11
z12
z21
z22
1,552,800 2,152,000
3 8
570,000 31,000
852,400 100
In the aggregation by a minimal component, contours of the aggregated function are quadrate in a two-dimensional problem and all of the objective functions are impartially maximized because values of the aggregated function correspond to the Tchebycheff metric. Furthermore, the aggregation by a minimal component is also regarded as the fuzzy decision rule [3] in decision making under fuzzy environments. Concerning the aggregation by weighting coefficients, each of the weights for the objectives is interpreted as a relative degree of importance which represents the trade-off ratio between the objective functions. Because contours of the aggregated function by weighting coefficients are straight lines in a two-dimensional problem, values of the objective functions may be biased for some of the objectives. In the aggregation by a distance from aspiration levels, it follows that a solution closest to the aspiration levels specified by a decision maker is found. Because we employ the Tchebycheff metric
as a measure of distance, this aggregation has characteristics similar to the aggregation by a minimal component in which solutions farthest from the origin are searched. The aggregation by a minimal component. The membership function aggregated by a minimal component is represented as lm1 ðz11 ; z12 Þ ¼ minfl11 ðz11 ðy; zÞÞ; l12 ðz12 ðu; vÞÞg: ð17Þ Employing this aggregation implies that Firm U maximizes a membership function with a smaller value. The aggregation by weighting coefficients. When Firm U specifies weighting coefficients for the two fuzzy goals and tries to maximize the weighted sum of the satisfactory degrees, the following aggregated membership function is employed: lw1 ðz11 ; z12 Þ ¼ al11 ðz11 ðy; zÞÞ þ ð1 aÞl12 ðz12 ðu; vÞÞ;
ð18Þ
where a is a weighting coefficient of the first fuzzy goal l11 and then 1 a is that of the second one. The aggregation by a distance from aspiration levels. When Firm U has aspiration levels to the two kinds of the satisfactory degrees for the fuzzy goals, the two membership functions are
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aggregated by using a distance from the aspiration levels as follows: z1 ðl11 ; l12 Þ ¼ maxf l11 l11 ðz11 ðy; zÞÞ; l12 l12 ðz12 ðu; vÞÞg;
ð19Þ
where l11 and l12 are membership values representing the aspiration levels to the fuzzy goals prescribed by the membership functions l11 and l12 , respectively. Moreover, we interpret z1 ðl11 ; l12 Þ as a newly defined objective function and identify the linear membership function ld1 ðz11 ; z12 Þ in a similar way. 3.3. The first phase of the interactive fuzzy programming In this section, examining the three models of aggregation presented in Section 3.2, we apply the interactive fuzzy programming method to the transportation problem in the housing material manufacturer. In the first phase, the upper level decision maker subjectively specifies a minimal satisfactory level. Taking into consideration overall satisfactory balance between the two levels, the upper level decision maker updates the minimal satisfactory level if necessary, and a tentative solution is derived.
3.3.1. The model with the aggregation by a minimal component We apply the first phase of the interactive fuzzy programming method to the problem with the fuzzy goal of Firm U aggregated by a minimal component. The first phase of the interactive fuzzy programming method starts to solve the following problem for obtaining a solution which maximizes the smaller degree of satisfaction between both levels: maximize min lm1 ðz11 ; z12 Þ; minfl21 ðz21 ðy; uÞÞ; l22 ðz22 ðz; vÞÞg subject to ðy; z; u; vÞ 2 S:
ð20Þ
Problem (20) becomes consequently a problem maximizing the smallest of the four degrees of satisfaction, i.e., the two membership functions of Firm U and the two membership functions of Firms L1 and L2. The obtained optimal solution to Problem (20) is shown in Table 9. If Firm U is not satisfied with the obtained solution, consulting the related information of the solution, Firm U specifies the minimal satisfactory level d^ and formulates the following problem, in which the smaller satisfactory degree of Firms L1 and L2 is maximized under the condition that
Table 9 An optimal solution to Problem (20) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
210 90
90 60
139 111
31 69
50 50
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
1 1 0
0 0 1
0 0 1
1 0 0
0 0 1
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 1
1 0 0
0 2 0
0 0 1
1 0 0
Objective function value Satisfactory degree lðzÞ Aggregated satisfactory degree
z11
z12
z21
z22
1,840,200 0.52
5 0.60 0.52
311,420 0.52 –
443,720 0.52 –
M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
177
the aggregated satisfactory degree of Firm U is larger than or equal to the minimal satisfactory level: maximize minfl21 ðz21 ðy; uÞÞ; l22 ðz22 ðz; vÞÞg ð21Þ
subject to ðy; z; u; vÞ 2 S; lm1 ðz11 ; z12 Þ P d^:
An optimal solution to Problem (21) with d^ ¼ 0:65 is shown in Table 10. Comparing the solution to the maximin problem shown in Table 9, to satisfy the condition of the minimal satisfactory level, the satisfactory degrees of both fuzzy goals of Firm U are increased and those of Firms L1 and L2 are decreased. Moreover, drivers with higher ability level are assigned largely. The ratio of satisfactory degrees between the two levels is D ¼ 0:68 and it follows that a tentative solution is derived if the ratio is in the interval ½Dmin ; Dmax specified by Firm U and Firm U is satisfied with the solution. To examine the characteristics of the model with the aggregation by a minimal component, we solve Problem (21), varying the parameter d^, and suppose that Firm U specifies the bounds of the ratio of satisfaction at ½Dmin ; Dmax ¼ ½0:6; 0:8 . The
Fig. 2. The relation between the satisfactory degrees and d^ in the model with the aggregation by a minimal component.
result is shown in Fig. 2 depicting the relation between the satisfactory degrees and the minimal satisfactory level d^. The range ½0:61; 0:68 of the abscissa shown in Fig. 2 is the interval in which the ratio D is in the bounds ½0:6; 0:8 , i.e., if Firm U specifies the minimal satisfactory level d^ in the interval ½0:61; 0:68 , the ratio D is in the bounds ½Dmin ; Dmax ¼ ½0:6; 0:8 . As seen in Fig. 2, the satisfactory degree l12 of the fuzzy goal for the opportunity loss with respect to the transportation time takes a fixed value 0:8 for any d^ 2 ½0:61; 0:68 . Because the aggregated satisfactory degree is represented by (17) and l11 < l12 ¼ 0:8 for any d^ 2 ½0:61; 0:68 , the aggregated
Table 10 An optimal solution to Problem (21) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
262 38
130 20
88 162
32 68
40 60
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 0 2
0 0 1
0 1 0
1 0 0
0 1 0
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 1
1 0 0
0 1 1
0 0 1
1 0 0
Objective function value Satisfactory degree lðzÞ Aggregated satisfactory degree
z11
z12
z21
z22
1,762,400 0.65
4 0.8 0.65
280,840 0.46 –
377,840 0.44 –
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M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
satisfactory degree is always equal to the satisfactory degree l11 and therefore it follows that the satisfactory degree for the second objective function cannot affect the aggregated satisfactory degree at all. 3.3.2. The model with the aggregation by weighting coefficients We apply the first phase of the interactive fuzzy programming method to the problem with the fuzzy goal of Firm U aggregated by weighting coefficients. In a way similar to the model with the aggregation by a minimal component, the first phase of the interactive fuzzy programming method starts to solve the following problem for obtaining a solution which maximizes the smaller degree of satisfaction between both levels: maximize min lw1 ðz11 ; z12 Þ; minfl21 ðz21 ðy; uÞÞ; l22 ðz22 ðz; vÞÞg subject to ðy; z; u; vÞ 2 S:
ð22Þ
An optimal solution to Problem (22) with the weighting coefficients a ¼ 1 a ¼ 0:5 of the fuzzy goals l11 and l12 in (18) is shown in Table 11. If Firm U is not satisfied with the obtained solution, consulting the related information of the
solution, Firm U specifies the minimal satisfactory level d^ and formulates the following problem, in which the smaller satisfactory degree of Firms L1 and L2 is maximized under the condition that the aggregated satisfactory degree of Firm U is larger than or equal to the minimal satisfactory level: maximize minfl21 ðz21 ðy; uÞÞ; l22 ðz22 ðz; vÞÞg ð23Þ
subject to ðy; z; u; vÞ 2 S; lw1 ðz11 ; z12 Þ P d^:
An optimal solution to Problem (23) with d^ ¼ 0:72 is shown in Table 12. Comparing the solution in Table 12 with the solution in Table 11, the small lot transportation of Firm L2 becomes larger and Firm L1 decreases the number of assigned drivers to cope with the reduction of transportation ordered by Firm U. To examine the characteristics of the model with the aggregation by weighting coefficients, we solve Problem (23), varying the parameter d^, and assume that Firm U specifies the bounds of the ratio of satisfaction at ½Dmin ; Dmax ¼ ½0:6; 0:8 . The result is shown in Fig. 3 depicting the relation between the satisfactory degrees and the minimal satisfactory level d^.
Table 11 An optimal solution to Problem (22) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
120 180
30 120
230 20
32 68
62 38
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 1 0
0 0 1
0 2 0
1 0 0
0 0 1
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 2
2 0 0
0 1 0
0 0 1
1 0 0
Objective function value Satisfactory degree lðzÞ Aggregated satisfactory degree
z11
z12
z21
z22
1,960,000 0.32
4 0.8 0.56
325,920 0.55 –
466,080 0.55 –
M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
179
Table 12 An optimal solution to Problem (23) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
120 180
91 59
122 128
62 38
75 25
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 1 0
0 0 1
0 0 1
1 0 0
0 0 1
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 2
1 0 0
0 2 0
0 0 1
1 0 0
Objective function value Satisfactory degree lðzÞ Aggregated satisfactory degree
z11
z12
z21
z22
1,888,200 0.44
3 1.0 0.72
316,000 0.53 –
450,740 0.53 –
The range ½0:65; 0:78 of the abscissa shown in Fig. 3 is the interval in which the ratio D is in the bounds ½0:6; 0:8 in a way similar to Fig. 2. As seen in Fig. 3, the satisfactory degree l12 of the fuzzy goal for the opportunity loss with respect to the transportation time takes a fixed value 1:0 for any d^ 2 ½0:65; 0:78 and that of the cost of transportation takes some value such that 0:4 6 l11 6 0:6. The value of l11 is smaller than that of the model with the aggregation by a minimal component and there is a larger difference between the two satisfactory degrees l11 and l12 . Moreover, the interval
of d^ in which the ratio D is in the specified bounds ½0:6; 0:8 is 0:65 6 d^ 6 0:78 and it is wider than that of the previous model. 3.3.3. The model with the aggregation by a distance from aspiration levels We apply the first phase of the interactive fuzzy programming method to the problem with the fuzzy goal of Firm U aggregated by a distance from aspiration levels. In a way similar to the previous two models, the first phase of the interactive fuzzy programming method starts to solve the following problem for obtaining a solution which maximizes the smaller degree of satisfaction between both levels: maximize min ld1 ðz11 ; z12 Þ; minfl21 ðz21 ðy; uÞÞ; l22 ðz22 ðz; vÞÞg subject to ðy; z; u; vÞ 2 S:
Fig. 3. The relation between the satisfactory degrees and d^ in the model with the aggregation by weighting coefficients.
ð24Þ
Assume that Firm U specifies the aspiration levels in (19) at l11 ¼ 1:0 and l12 ¼ 0:8, and the Zimmermann method [19] is employed and the parameters of the linear membership function are determined as z1 ¼ 0:12 and z0 ¼ 1:0. Then, an optimal solution to Problem (24) with these
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Table 13 An optimal solution to Problem (24) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
120 180
122 28
107 143
73 27
48 52
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 1 0
0 0 1
0 1 0
1 0 0
0 1 0
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 2
1 0 0
0 2 0
0 1 0
1 0 0
Objective function value Satisfactory degree lðzÞ Aggregated satisfactory degree
z11
z12
z21
z22
1,869,400 0.47
6 0.4 0.54
320,660 0.54 –
457,920 0.54 –
parameters is shown in Table 13. Especially, it should be noted that the difference between the two satisfactory degrees of Firm U, l11 and l12 , is smaller than those of the previous two models. If Firm U is not satisfied with the obtained solution, consulting the related information of the solution, Firm U specifies the minimal satisfactory level d^ and formulates the following problem, in which the smaller satisfactory degree of Firms L1 and L2 is maximized under the condition that the aggregated satisfactory degree of Firm U is
larger than or equal to the minimal satisfactory level: maximize minfl21 ðz21 ðy; uÞÞ; l22 ðz22 ðz; vÞÞg ð25Þ
subject to ðy; z; u; vÞ 2 S; ld1 ðz11 ; z12 Þ P d^:
An optimal solution to Problem (25) with d^ ¼ 0:72 is shown in Table 14.
Table 14 An optimal solution to Problem (25) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
249 51
128 22
88 162
40 60
41 59
Assignaed drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
1 0 1
0 0 1
0 0 1
0 1 0
0 1 0
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 1 0
1 0 0
0 1 1
0 0 1
1 0 0
Objective function value Satisfactory degree lðzÞ Aggregated satisfactory degree
z11
z12
z21
z22
1,773,200 0.63
5 0.6 0.72
284,820 0.47 –
401,420 0.47 –
M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
Comparing the solution in Table 14 with the solution in Table 13, to satisfy the condition of the minimal satisfactory level, some parts of the regular transportation to customers 1 and 2 are switched to the small lot transportation, some parts of the small lot transportation to customers 3, 4 and 5 are switched to the regular transportation and drivers with higher ability level are assigned for increased orders in both forwarding agents. Moreover, the difference between l11 and l12 is still small. To examine the characteristics of the model with the aggregation by a distance from aspiration levels, we solve Problem (25), varying the parameter d^, and assume that Firm U specifies the bounds of the ratio of satisfactory degrees at ½Dmin ; Dmax ¼ ½0:6; 0:8 . The result is shown in Fig. 4 depicting the relation between the satisfactory degrees and the minimal satisfactory level d^. The range ½0:66; 0:75 of the abscissa shown in Fig. 4 means the interval in which the ratio D is in the bounds ½0:6; 0:8 in a way similar to Figs. 2 and 3. As seen in Fig. 4, the satisfactory degree l11 takes some value in 0:58 6 l11 6 0:7 and the satisfactory degree l12 for the opportunity loss is smaller than l11 for the cost of transportation. The interval of d^ in which the ratio D is in the specified bounds ½0:6; 0:8 is 0:66 6 d^ 6 0:75 and it is wider than that of the model with the aggregation by a minimal component.
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3.4. The second phase of the interactive fuzzy programming In the first phase of the interactive fuzzy programming, Firms L1 and L2 are treated impartially and they can be regarded as a group. However, if Firm U wants to give either of Firms L1 and L2 special consideration, Firm U might specify different intervals ½Djmin ; Djmax , j ¼ 1; 2 for them separately. In such a case, it is necessary to start the second phase of the interactive fuzzy programming. In the second phase, by coordinating the satisfactory degrees of Firm U and each of Firms L1 and L2, the final satisfactory solution is derived. First, we calculate the following individual ratios of satisfactory degrees of Firm U and each of Firms L1 and L2 with respect to the solution obtained in the first phase: Di ¼
l2i ðz2i Þ ; l1 ðz11 ; z12 Þ
i ¼ 1; 2;
ð26Þ
where l1 ðz11 ; z12 Þ denotes the aggregated satisfactory degree of Firm U. For a forwarding agent whose ratio of satisfactory degrees with Firm U is not in the interval ½Djmin ; Djmax determined by Firm U, Firm U specifies a minimal satisfactory level d~ and/or a maximal satisfactory level d and formulates the problem: maximize l2i ðz2i Þ subject to ðy; z; u; vÞ 2 S; l1 ðz11 ; z12 Þ P d^; d~ 6 l2j ðz2j Þ 6 d;
Fig. 4. The relation between the satisfactory degrees and d^ in the model with the aggregation by a distance from aspiration levels.
ð27Þ
j ¼ 1; 2;
j 6¼ i:
Assume that Firm U adopts a policy that Firm U gives favorable treatment to Firm L1 dealing with the regular transportation, and specifies the bounds of the ratios of satisfactory degrees for Firms L1 and L2 at ½D1min ; D1max ¼ ½0:7; 0:8 and ½D2min ; D2max ¼ ½0:6; 0:7 , respectively. From the examination in Section 3.3, because it is found that, in the model with the aggregation by a minimal component, the satisfactory degree for the second objective function does not affect the
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aggregated satisfactory degree at all, we will deal with only the model with the aggregation by weighting coefficients and the model with the aggregation by a distance from aspiration levels in the second phase of the interactive fuzzy programming method. 3.4.1. The model with the aggregation by weighting coefficients Assume that, in the first phase, Firm U is satisfied with the solution which is obtained by solving Problem (23) with d^ ¼ 0:72. The corresponding individual ratios of satisfactory degrees are D1 ¼ D2 ¼ 0:74, and we have D1 2 [0.7, 0.8] and D2 62 ½0:6; 0:7 . Because the ratio D2 is over the upper bound 0.7, assume that Firm U specifies the minimal satisfactory level d~ for l22 . Then, the following problem is formulated: maximize l21 ðz21 ðy; uÞÞ subject to ðy; z; u; vÞ 2 S; lw1 ðz11 ; z12 Þ P d^ ¼ 0:72; 0:504 ¼ d~ 6 l22 ðz22 ðz; vÞÞ;
ð28Þ
where the minimal satisfactory level d~ for l22 is determined at d~ ¼ 0:72 0:7 ¼ 0:504. An optimal solution to Problem (28) is shown in Table 15.
As seen in Table 15, comparing the transportation planning corresponding to the tentative solution obtained in the first phase which is shown in Table 12, some part of the small lot transportation to customer 3 is switched to the regular transportation and conversely some part of the regular transportation to customer 4 is switched to the small lot transportation. From these facts, it is thought that the satisfactory degree of Firm L1 increases and the ratio D2 is in the specified interval. To examine the characteristics of this model in the second phase of the interactive fuzzy programming, we solve Problem (28), varying the parameter d~. The result is shown in Fig. 5, depicting the relation between the ratios of satisfactory degrees and the minimal satisfactory level d~. As seen in Fig. 5, because D1 decreases and D2 increases as d~ increases, we can understand the conflict between l21 and l22 . Moreover, the interval of d~ leading to possible satisfactory solutions in the second phase is 0:489 6 d~ 6 0:504 and we have verified that the range of the parameter d~ which we can control is very narrow. 3.4.2. The model with the aggregation by a distance from aspiration levels Assume that, in the first phase, Firm U is satisfied with the solution which is obtained by solving Problem (25) with d^ ¼ 0:72. The corresponding
Table 15 An optimal solution to Problem (28) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
120 180
90 60
150 100
32 68
74 26
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
0 1 0
0 0 1
0 0 1
1 0 0
0 0 1
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 0 2
1 0 0
0 2 0
0 0 1
1 0 0
Objective function value Satisfactory degree lðzÞ Ratio of satisfactory degrees
z11
z12
z21
z22
1,887,200 0.44
3 1.0 lw1 ¼ 0:72
339,520 0.57 D1 ¼ 0:79
426,520 0.50 D2 ¼ 0:69
M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
Fig. 5. The relation between the ratios of satisfactory degrees and d~ in the model with the aggregation by weighting coefficients.
individual ratios of satisfactory degrees are D1 ¼ D2 ¼ 0:65, and we have D1 62 ½0:7; 0:8 and D2 2 ½0:6; 0:7 . Because the ratio D1 is smaller than the lower bound 0.7, assume that Firm U specifies the minimal satisfactory level d~ for l21 . Then, the following problem is formulated: maximize l22 ðz22 ðz; vÞÞ subject to ðy; z; u; vÞ 2 S; ld1 ðz11 ; z12 Þ P d^ ¼ 0:72; 0:504 ¼ d~ 6 l21 ðz21 ðy; uÞÞ;
ð29Þ
183
where the minimal satisfactory level d~ for l21 is determined at d~ ¼ 0:72 0:7 ¼ 0:504. An optimal solution to Problem (29) is shown in Table 16. As seen in Table 16, comparing the transportation planning corresponding to the tentative solution obtained in the first phase which is shown in Table 14, by increasing some amount of the order to the regular transportation, we can increase the satisfactory degree of Firm L1 with the small reduction of the satisfactory degree of Firm L2 so as to satisfy the conditions of the ratios of satisfactory degrees. It is also seen that the orders to the regular transportation becomes larger than in the transportation planning corresponding to the solution of the model with the aggregation by weighting coefficients shown in Table 15. To examine the characteristics of this model in the second phase of the interactive fuzzy programming, we solve Problem (29), varying the parameter d~. The result is shown in Fig. 6 depicting the relation between the ratios of satisfactory degrees and the minimal satisfactory level d~. As seen in Fig. 6, because D1 increases and D2 decreases as d~ increases, we can also understand the conflict between l21 and l22 . Moreover, the interval of d~ leading to possible satisfactory solutions in the second phase is 0:504 6 d~ 6 0:573 and it is found that the range of the parameter d~ which
Table 16 An optimal solution to Problem (29) Customer i
1
2
3
4
5
Orders to Firm L1: yi Orders to Firm L2: zi
249 51
120 30
106 144
45 55
40 60
Assigned drivers in Firm L1 (Level 1): ui1 (Level 2): ui2 (Level 3): ui3
1 0 1
0 0 1
0 0 1
0 1 0
0 1 0
Assigned drivers in Firm L2 (Level 1): vi1 (Level 2): vi2 (Level 3): vi3
0 1 0
1 0 0
0 2 0
0 0 1
1 0 0
Objective function value Satisfactory degree lðzÞ Ratio of satisfactory degrees
z11
z12
z21
z22
1,773,200 0.63
5 0.60 ld1 ¼ 0:72
302,740 0.50 D1 ¼ 0:70
394,500 0.46 D2 ¼ 0:64
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M. Sakawa et al. / European Journal of Operational Research 141 (2002) 167–185
by weighting coefficients, and the aggregation by a distance from aspiration levels. Finally, we have concluded that the aggregation by a distance from aspiration levels is effective for our problem.
References
Fig. 6. The relation between the ratios of satisfactory degrees and d~ in the model with the aggregation by a distance from aspiration levels.
we can control is wider than that of the model with the aggregation by weighting coefficients. From the above mentioned analysis, in the model with the aggregation by a distance from aspiration levels, it is easier to control the two satisfactory degrees of Firm U and the range of the parameter d~ which we can control is wider than the others. It seems reasonable to conclude, from these facts, that employing the model with the aggregation by a distance from aspiration levels is appropriate for the decentralized two-level transportation problem in the housing material manufacturer.
4. Conclusions In this paper, we have dealt with the transportation problem in the housing material manufacturer and, applying the interactive fuzzy programming method, we have derived a satisfactory solution to the problem by taking into account not only the degree of satisfaction with respect to the objectives of the housing material manufacturer but also those of the two forwarding agents to which the housing material manufacturer entrusts transportation of products. As the housing material manufacturer has the two objective function, we have aggregated the two satisfactory degrees for the two objectives have examined three aggregation methods: the aggregation by a minimal component, the aggregation
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