Interactive Fuzzy Goal Programming Approach for Optimization of Extended Hub-and-Spoke Regional Port Transportation Networks

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1 Interactive Fuzzy Goal Programming Approach for Optimization of Etended Hub-and-Spoe Regional Port Transportation Networs Chuanu Wang and Liangui Jiang School of Economy and anagement, Shanghai aritime University, Shanghai, 35, China Department of Basic Science, Shanghai aritime University, Shanghai, 35, China Abstract. Based on interactive fuzzy goal programming, a model is introduced for etended hub-and-spoe regional port transportation networ optimization problem where one of the obective functions is considered as a non-linear function. The proposed model considers the imprecise nature of the input data and assumes that each obective function has a fuzzy goal, and aims at ointly minimizing the total costs including the sailing cost and handling cost as well as the total transit times consisting of sailing time and handling time. eanwhile it optimizes the following factors: transportation quantities via a hub port from an original port to a destination port, transportation quantities directly from an original port to a destination port. The solution procedure to the proposed model is then presented. At last, a numerical eample is given to demonstrate the effectiveness of the proposed model and evaluate the performance of the solution procedure. Keywords: Fuzzy goal programming, Hub-and-spoe, Optimization, Port transportation networ. Introduction In regional port transportation networs, the sizes and types of ships calling at every port are different because individual port s natural condition and capacity are different. The route selection is important in regional port transportation networ optimization. The port condition and economy of scale in ship transportation should be considered to select the routes from original ports to destination ports. The ports are connected by direct transports or via a transshipment port. This has led to the formation of hub-and-spoe networs in regional port transportation industry. Cargoes from original ports are usually consolidated at hub port and shipped to different destination ports. Hub-and-spoe transportation can be classified into two types: pure and etended []. The pure hub-and-spoe transportation networ is characterized by transshipment in which direct transport between ports is ecluded and all cargoes have to be transported via a hub port. The etended hub-and spoe transportation networ consists of direct transportation between ports and transshipment via the hub port. Y. Shi et al. (Eds.): ICCS 7, Part IV, LNCS 449, pp , 7. Springer-Verlag Berlin Heidelberg 7

2 Interactive Fuzzy Goal Programming Approach for Optimization 87 In this paper, based on etended hub-and-spoe port transportation system, a decision model is developed to determine the following factors: shipment volume via a hub port from an original port to a destination port, shipment volume directly from an original port to a destination port. The decision problem of hub-and-spoe systems has received many attentions in academic literature. Some researchers solved it as the location allocation problem that determines locations of hubs and assigns shipments to each route[][3][4]. Another researchers eamined it as the pure allocation problem that finds the optimal assignment of shipments under predetermined hub locations[5][6][7]. But the models and methods in these studies are applied in air or truc transportation networs and have been less focused on lining hub-and-spoe system to ports. ost of them only consider transportation (or travel) cost and don t include the handling costs occurring at nodes. Furthermore, these studies only consider one obective function. This paper introduces a model to eamine the pure allocation problem in etended hub-andspoe regional port transportation networ. The model aims at ointly minimizing the total costs consisting of sailing cost and handling cost occurring at the ports as well as total time consisting of sailing time and handling time occurring at the ports, in which the time obective function is a non-linear function. In addition, the eisting decision problem of hub-and-spoe system is a deterministic mathematical programming problem. However, in practice, the input data or decision parameters, such as transportation capacity, cost, time and obective function are often imprecise because some information is uncertain. Therefore, the values of these parameters are rarely constant. To deal with ambiguous parameters in the above decision problem, this paper uses an interactive fuzzy goal programming model to formulate the problem of etended huband-spoe regional port transportation networ optimization, and propose a solution procedure for this model. The Decision Problem of Etended Hub-and-Spoe Port Transportation Networ. Notations, the number of ports ecluding the hub port ; D, transportation demand quantity from port i to port ; C, the transportation cost per unit cargo from port i to port ; S, the transportation capacity in the route from port i to port ; T, the sailing time from port i to port ; C i, the handling cost per unit cargo occurred at port i ; T the handling time per unit cargo occurred at port i ; i, transportation quantity from port i to port ( i ) ; (Decision variables)

3 88 C. Wang and L. Jiang. athematical odel The decision problem of etended hub-and-spoe port transportation system can be formulated as follows: (P) min Z = C Ci i C i min Z = T ( Ti T s.t. i i i S i i = = i i D D i i i i () ) (), i =,,..., i (3) i =,,..., i (4) i =,,..., =,,..., i (5) i =,,..., =,,..., i (6) In real world, the input data or parameters of (P) problem, such as transportation capacity and obective function are often imprecise because some information is unobtainable. Conventional mathematical programming can t capture such vagueness in the critical information. In this case, a fuzzy programming approach is commonly used to treat the information in a fuzzy environment. However, some researchers have presented the shortcomings of using fuzzy programming in solving some multiobective decision problems. Abd El-Wahed () proved that using fuzzy programming in solving such multi-obective transportation problem changes the standard form of the transportation problem [8]. In addition, Li and Lai () proved that using the min-operator does not guarantee an efficient solution[9]. In this paper, we introduce an interactive fuzzy goal programming model for regional port transportation networs optimization, which is the combination of interactive programming, fuzzy programming and goal programming, and leverages the advantages of the three approaches as well as reduces some or all of the shortcomings of each individual approach []. 3 Interactive Fuzzy Goal Programming Approach 3. IFGP odel In our model, we capture the ambiguity about the fuzzy information related to the total transportation cost, total transportation time and transportation capacities by transforming the (P) model into the following (P) model.

4 Interactive Fuzzy Goal Programming Approach for Optimization 89 (P) C T s.t. Ci i ( Ti T i = i C i i < i < Z ) Z (7) (8) < S i =,,..., =,,..., i, (9) as well as constraints given in (3), (4) and (6) where the symbol < indicates essentially smaller than or equal to and allows S denote fuzzy values. one reach some aspiration level, Z, Z, and In this paper, a linear membership function defined in Bellman and Zadeh(97) has been considered for all fuzzy parameters in (P) problem. To formulate model (P) as a goal programming model, we introducing the following positive and negative deviation variables: Z( ) d d = G, =, () where Gi is the aspiration level of the obective function i. By using the given membership functions and introducing an auiliary variable L the fuzzy programming model (P) can be formulated as the following equivalent programming model (P3) (Zimmermann, 978): L ma min ma =, () U L U L( S S ) S i =,,..., =,,..., i () Z( ) d d = G =, (3) L, (4) d, d, =, (5) (P3) a s.t. LZ ( Z ) Z( ) Z as well as constraints given in (3), (4) and (6). 3. The Solution Procedure In order to obtain the solution to odel (P3), the following procedure can be applied. Step: Solve the model (P) as a single obective problem to obtain the initial solutions for each obective function, i.e. = { } and = { }, X X i =,,.., =,,...,,. i. If X = X, select one of them as an optimal solution and go to Step 6. Otherwise, go to Step.

5 9 C. Wang and L. Jiang min ma Step : Determine the best lower ( Z ) and the worst upper ( Z ) for each obective function: Z min = ZX ( ), Z ma = Z( X ), Z min = Z( X ), Z ma = ZX ( ), i =,,.., =,,...,, i. Step 3: Define the membership functions of each obective function. Step 4: Solve problem (P4) as a goal programming and obtain the solution to it. Compare the upper of each obective function with the new value of the obective function. If the new value of each obective function is equal to the upper, go to Step 6. Otherwise, go to Step 5. Step 5: Update the upper of each obective function. If the new value of the obective function is lower than the upper, consider this as a new upper. Otherwise, eep the old one as is. Go to Step 3. Step 6: Stop. 4 Numerical Eample We consider the following numerical eample to demonstrate the application of IFGP. Assume that there are four ports and one hub in regional port transportation networ. There are two types of ships employed in the route between two ports. The transportation capacity of ship for type is 5 Ton, whereas that for type is 8 ton. The handling cost per ton occurring at Port, Port, Port 3, Port 4 and Port are.4,.55,.6,.65 and.4 Yuan, respectively. The handling time per ton occurring at Port, Port, Port 3, Port 4 and Port are.5,.,.,.5 and.5 hour, respectively. The other data of the problem is given in Table, Table and Table3. Table. Transportation demand between ports (ton) Original Destination ports Ports Port Port Port 3 Port 4 Hub Port Port Port Port Hub Table. Transportation costs ( Yuan per Ton) and sailing time(hours) in different routes Destination ports Original Port Port Port 3 Port 4 Hub ports cost time cost time cost time cost time cost time Port Port Port Port Hub

6 Interactive Fuzzy Goal Programming Approach for Optimization 9 Table 3. Transportation capacity provided by carriers for different routes (Ton) Destination ports Original Port Port Port 3 Port 4 Hub ports Lower Lower Lower Lower Lower Port Port Port Port Hub Solution The initial model (P) of the numerical eample can be obtained as in Z = Z = s.t. 78, = 3 4 = = 658, = = 66, 3 4 = = 38, = 956 6,, 5 3, 4 3, 5, 3 3 3, 4 7, 5 3, 3 5, 3 3, , 4 5, 43 4, 6, , i =,,... 4, =,,... 4,i, 9, 6

7 9 C. Wang and L. Jiang (, ) Z Z can By using the above-mentioned solution procedure, the set of solutions be obtained after 7 iterations. The values of obective function are converged at (59., 36638), the corresponding optimal solution is as follows: =, = 79., =., = 57., =., = 8., = 3., =., = 3., = 3., = 6553., = 87., = 6., = 3., = 95., = = 9., = 5., = 4., = 6., d = 34., d =., d = 6445., d = Performance Evaluation To evaluate the performance of the proposed model, the solution to the illustrative eample by using different methods is considered. The fuzzy programming approach gives the following results: Z = 79, Z = The interactive fuzzy goal programming approach provides the following results: Z = 55, Z = To determine the degree of closeness of the IFGP model results to the ideal solution, the family of distance functions presented in [][] is considered. K D ( λ, K) = [ λ ( d ) ] p = p p / p, (6) where d represents the degree of closeness of the solution to the ideal optimal solution with respect to the th obective function, the solution of Z. d = the ideal optimal value of λ is the weight of the th obective and K = Z / λ =. The power p represents a distance parameter p. Thus, we can state the approach is better than others if in Dp ( λ, K) is achieved by its solution with respect to some p. Assuming λ = λ =.5, p =, and, the results of the different approaches are given in Table 4. Table 4. Comparison of results by different approaches Fuzzy programming Interactive fuzzy programming Interactive fuzzy goal programming (Z,Z ) (79, (55, (59., 36888) 36664) 36638) D D D Optimal solution (74.5, ) It is clear from Table that the solution to the proposed approach is better than the solution by the other approaches for all the distance functions.

8 Interactive Fuzzy Goal Programming Approach for Optimization 93 5 Conclusions This paper proposes an interactive fuzzy goal programming model of the etended hub-and-spoe regional port transportation networ optimization problem, where transportation capacity and obective function are considered as o imprecise/fuzzy, and aims at ointly minimizing the total costs including the sailing cost between ports and handling cost occurring at all ports as well as the total transit times consisting of sailing time between ports and handling time in all ports. To obtain the solution of the above-mentioned problem, the solution procedure for the model is illustrated by updating both the membership values and the aspiration levels. At last, a numerical eample is given to demonstrate the effectiveness of the proposed model and evaluate the performance of the solution procedure by comparing its results with those of the fuzzy programming approach and interactive fuzzy programming approach. Acnowledgments. The wor in this paper was supported by National Natural Science Foundation of China ( No ), Shanghai Shuguang Program ( No.6SG48) and Shanghai Puiang Program. References. Zapfel, G., Wasner,., Planning and optimization of hub-and-spoe transportation networs of cooperative third party logistics providers, International Journal of Production Economics, (), 78 :7-.. Ayin,T.,Networing policies for hub-and-spoe systems with applications to the air transportation system, Transportation Science, (995),6:-. 3. O Kelly,.E., A quadratic integer program for location of interacting hub facilities, European Journal of Operational Research, 987, 3: Pirul,H.,Schilling,D.A.,An efficient procedure for designing single allocation hub and spoe systems, anagement Science, (998), 44: S Abdinnourhelm,S.,Venataramanan,.A., Solution approaches to hub location problems, Annals of Operations Research, (998), 78: Bertazzi,L.,Speranza,.G.,Uovich,W.,inimization of logistics costs with given frequencies, Transportation Research B, (997), 3: Liu, J.,Li,C.L.,Chan,C.Y.,ied truc delivery systems with both hub-and-spoe and direct shipment. Transportation Research E, (3), 39: Abd El-Wahed WF, A multi-obective transportation problem under fuzziness. Fuzzy Sets and Systems, (), 7: Li,L.,Lai KK., A fuzzy approach to the multi-obective transportation problem, Computers & Operational Research, (), 7: Abd El-Wahed WF, Lee, S.., Interactive fuzzy goal programming for multi-obective transportation problems, Omega, (6), 34: Steuer, R., ultiple criteria optimization: theory, computation, and application. New Yor: Wiley, (986).