A Hybrid Genetic Algorithm for Logistics Network Design ... - CiteSeerX

International Journal of Information Systems for Logistics and Management Vol. 3, No. 1, (2007) 1-12

http://www.knu.edu.tw/ijislm

A Hybrid Genetic Algorithm for Logistics Network Design with Flexible Multistage Model Lin Lin*, Mitsuo Gen and Xiaoguang Wang Graduate School of Information, Production and Systems, Waseda University 2-7 Hibikino, Wakamatsu-ku, Kitakyushu 808-0135, Japan Received 5 July 2007; received in revised form 26 October 2007; accepted 10 November 2007

ABSTRACT Logistics network plays a key role in building an efficient and flexible logistics system for companies in the global business environment. A lot of research has been conducted in this field. While the researchers treat logistics networks design problem as a traditional multistage logistics network model, in which arcs should connect the two adjoining echelons in the network and there are no arcs striding over any abutting echelons, thereby the problem can be solved stage by stage. However, in practice this kind of traditional multistage logistics network (tMLN) model sometime causes problems, such as too-long delivery path, slow response etc. In this paper, we address flexible multistage logistics network (fMLN) design problem with nonadjacent structure, i.e. in this problem some non-neighboring echelons are connected with arcs (nonadjacent connecting arcs). In some practical cases, the nonadjacent connecting arcs make the logistics networks cost-effective and adaptable to changes in situation. On the other hand, the existence of them makes the problem much more difficult by traditional optimization methods. We formulate this problem as location-allocation model, and propose an effective hybrid genetic algorithm to solve this problem. Moreover, numerical analysis of case study is carried out to show the effectiveness of the proposed approach. Keywords: logistics network design, multistage network design, distribution channel, location-allocation, genetic algorithm.

1. INTRODUCTION As the development of economic globalization and extension of global electronic marketing, global enterprise services supported by universal supply chain and worldwide logistics become imperative for business world. How to manage logistics system efficiently thus has become a key issue for many companies to control their costs. That is also why an elaborately designed logistics network under the help today’s fully-fledged information technology is catching more and more attentions of business entities, especially that of many multinational companies. *Corresponding author: [email protected]

However, it seems to be quite difficult to do it successfully for these companies due to their huge and extremely complicated logistics network, though they usually have imminent desire to cut down their logistics cost. In this paper, we develop a new flexible multistage logistics network (fMLN) model, which can overcome this kind of difficulty in practice and theory, and its efficiency is proved mathematically. Moreover we propose two genetic algorithms (GA) based approaches to solve it. Finally, the effectiveness and efficiency of the proposed approaches are proved by various measures of performance. Additionally, the effect of different parameter setting on the

2

International Journal of the Information Systems for Logistics and Management (IJISLM), Vol. 3, No. 1 (2007)

performance of the algorithms is also experimentally discussed. The logistics network design problem is defined as the following: given a set of facilities including potential suppliers, potential manufacturing facilities, and distribution centers with multiple possible configurations, and a set of customers with deterministic demands, determine the configuration of the production-distribution system between various subsidiaries of the corporation such that seasonal customer demands and service requirements are met and the profit of the corporation is maximized or the total cost is minimized (Goetschalckx, Vidal and Dogan, 2002). It is an important and strategic operations management problem in supply chain management (SCM), and solving it systemically is to provide an optimal platform for efficient and effective SCM. In this field, numerous researches are conducted (Gen and Syarif, 2005; Gen, Altiparamk and Lin, 2006; Nakatsu, 2005; Altiparamk, Gen and Lin, 2006). However, to date the structures of the logistics network studied in all the literatures are in the framework of the traditional multistage logistics network (tMLN) model. The network structure described by tMLN model is like that shown in Fig. 1. There are usually several facilities organized as 4 echelons (i.e. plants, distribution centers (DCs), retailers and customers (Harland, 1997)) in some certain order, and the product delivery routes should be decided during 3 stages between every two adjoining echelons sequentially. Although the tMLN model and its application had made a big success in both theory and business practices, as time goes on, some faults of the traditional structure of logistics network came to light, making it impossible to fit the fast changing competition environments or meet the diversified customer demands very well. For multinational companies, running their business in a large geographical area, in one or several continents, for instance, these disadvantages are more distinct. For easy understanding, let’s suppose a multinational company (called ABC Co. Ltd., hereafter) selling personal computer

throughout Europe. If ABC establishes its logistics network serving its customers all over Europe following the traditional three-stage logistics network model, its distribution paths will be very long. It results 1) High transportation cost and other related cost, including loading/unloading cost, fixed operating cost of facilities, and labor cost; 2) Long response time, which slows down the response speed to changes in demands and decreases the customer satisfactions; 3) Bullwhip effect (Lee, Padmanabhan, and Whang, 1997), since lots of facilities in the multistage plays a role as mid-process; 4) Difficulties in accurate inventory controlling. Today’s violent business competitions press enterprises to build up a logistics network productively and flexibly. At this point, Dell’s success in SCM may give us some luminous ideas on designing cost-effective logistics network (Logistics & Technology online; Reyes, 2005). Nevertheless to say the company’s entire directto-consumer business model, not just its logistics approach earns remarkable edges over its competitors. Here, however, we pay special attention to its distribution system. The delivery modes employed by Dell and other companies are illustrated in Fig. 2, based on which they realize their strategy in distribution—skipping the mid-process, being close to customers reduces transportation cost and delivery time and increasing customer satisfaction. This is how Dell beats its competitors. In more detail, they are: Normal Delivery, to deliver products from one stage to another adjoining one. Direct Shipment, to transport products from plants to retailers or customers directly. Direct Delivery, to deliver products from DCs to customers not via retailers. In a network, the direct shipment and direct delivery play a role of arcs connecting two nonadjacent echelons. By introducing this kind of direct shipment and direct delivery, we extend the tMLN model to a new logistics network (we name it nonadjacent multistage logistics network or flexible multistage logistics network;

C1 R1

Direct Shipment

DC1 P1

R2

P2

DC2

•

•

Pi •

C2 •

•

DCi

Rk

•

•

Normal Delivery

•

plants

Normal Delivery

Normal Delivery

CI

DCs

retailters

customers

Direct Delivery CL-1

PI DCJ plants DCs

RK retailers

CL

Normal delivery Direct shipment Direct delivery

customers

Fig. 1. The structure of traditional multistage logistics network (tMLN)

Fig. 2. Three kinds of delivering modes in flexible multistage logistics network model

L. Lin et al.: A Hybrid Genetic Algorithm for Logistics Network Design with Flexible Multistage Model

C1 R1 DC1 P1 P2

Pi

C2 R2

DC2

DCj

CL-1

The following notations are used to formulate the mathematical model:

RK

plants

CL DCs

above. A5. Each customer is served by only one facility. A6. Customer demands are known in advance. A7. Customers will get products at the same price, no matter where she/he gets them. It means that the customers have no special preferences.

CI

Rk

PI DCJ

retailers

3

customers

Normal delivery Direct shipment Direct delivery

Fig. 3. The structure of flexible multistage logistics network (fMLN) models

fMLN for abbreviation hereafter) model, shown in Fig. 3. Its application provides a new potential way to shorten the length between the manufactures and final customers, and to serve the customers flexibly. With its help Dell avoided the problems that other computer makers encounter such as the impracticable long supply chains, delivery delays and risks of missing sudden changes in demand or obsolescence of evolving its products. Though the trade-off between direct distribution and indirect distribution is also a critical issue in decision making in distribution channel in sales and marketing (Johnson and Umesh, 2002; Tsay, 2002). However, up to now this kind of flexible logistics network model, with great practical value has been studied at the viewpoint on logistics network design using latest soft computing techniques in few literatures. Basically, multistage transportation problem involving the open/close decision is NP-hard. The existence of the new delivery modes, which distinguish the fMLN from its traditional counterpart, makes the solution space to the problem much larger and more complex. The conventional methods are unable to solve with acceptable computation cost, we thus propose two GA-based approaches to overcome the difficulty. 2. MATHEMATICAL FORMULATION

Notations Indices: i index of plant (i = 1, 2 , …, I) j index of DC (j = 1, 2, …, J) k index of retailer (k = 1, 2, …, K) l index of customer (l =1, 2, …, L) Parameters: I number of plants J number of DCs K number of retailers L number of customers plant i Pi DCj DC j retailer k Rk C customer l b output of plant i demand of customer l dl unit shipping cost of product from Pi to DCj c1ij c2jk unit shipping cost of product from DCj to Rk c3kl unit shipping cost of product from Rk to Cl c4il unit shipping cost of product from Pi to Cl c5jl unit shipping cost of product from DCj to Cl c6ik unit shipping cost of product from Pi to Rk ujD upper bound of the capacity of DCj ukR upper bound of the capacity of Rk fjF fixed part of the open cost of DCj c1V variant part of the open cost (lease cost) of DCj j q1j throughout of DCj q 1j =

fj gkF ck2V qk2

Here, we give some assumptions, based on which we formulate a mathematical model for this problem. A1. In this study, we consider the single product case of a logistics network optimization problem. A2. We consider single time period, such as, one week or one month. A3. In the logistics network, there are maximum four echelons: plants, DCs, retailers and customers. A4. There are three delivery modes: normal delivery, direct shipment and direct delivery, as mentioned

x1ij , ∀ j

open cost of DCj fj = fjF + cj1Vq1j , ∀j fixed part of the open cost of Rk variant part of the open cost (lease cost) of DCj throughout of Rk q k2 =

gk

I

Σ

i=1

L

Σ x3kl , l=1

∀k

open cost of Rk gk = gkF + ck2Vqk2, ∀k

Decision Variables: transportation amount from Pi to DCj x1ij x2jk transportation amount from DCj to Rk x3kl transportation amount from Rk to Cl x4il transportation amount from Pi to Cl x5jl transportation amount from DCj to Cl

4


x6ik

transportation amount from Pi to Rk

capacity of DCs and retailers cannot be surpassed. For tMLN model, the mathematical model (model 2) can be described as follows:

1 , if DC j is opened y 1j = 0 , otherwise 1 , if R k is opened 0 , otherwise

yk1 =

min I

z=

We treat this problem as a location-allocation problem and formulate it as mixed integer programming model (model 1), and the objective function is to minimize the total logistics cost. The open cost of DCs and retailers comprise two parts: fixed cost and variant cost. When one facility is set, the fixed cost is incurred. While the variant cost is dependent on the throughput of the facility, which indicates how large the scale (the capacity) of the facility needs to be established. max z=

J

Σ Σ

i=1 j=1

I

+

K

Σ Σ

c 1ijx1ij +

j=1 k=1

L

J

c 2 jk x2 jk +

L

Σ Σ

k=1 l=1

L

I

J

Σ

j=1

j=1 l=1

f jy 1j +

K

Σ

k=1

T x c 3kl 3kl

J

g k yk2

Σ x1ij ≤ b i , I

I

s. t.

Σ

j=1

L

x1ij +

I

Σ

i=1

Σ

l=1

K

x4il +

K

x1ij +

J

Σ

k=1

Σ

k=1

L

x2 jk +

I

Σ

l=1

L

x6ik ≤ b i , ∀i

(2)

x5 jl , ∀ j

(3)

Σ x2 jk + i Σ= 1 x6ik = l Σ= 1 x3kl , j=1 J

L

Σ

l=1

x4il +

Σ

j=1

I

Σ x1ij ≤ u Dj , i=1 L

Σ x3kl ≤ u kR , l=1

K

x5 jl =

Σ

k=1

∀i

K

Σ x1ij = kΣ= 1 x2 jk , i=1 L

∀k

x3kl ≥ d l, ∀l

(4) (5)

∀j

(6)

∀k

(7)

x1ij, x2jk, x3kl, x4il, x5jl, x6ik ≥ 0, ∀i, j, k, l

(8)

yj1, yk2 ∈ {0, 1}, ∀j, k

(9)

Equation (1) means to minimize the total logistics cost, including shipping cost and open cost of facilities. The forth, fifth and sixth terms are newly introduced to describe the costs of the new delivery modes. Equation (2) means the production limit of plants. Equation (3) and (4) are the flow conservation principle. Equation (5) ensures that the customers’ demand should be satisfied. Equation (6) and (7) make sure the upper bound of the

(10)

(11)

L

Σ

∀j

(12)

∀k

(13)

∀l

(14)

∀j

(15)

x3kl ≤ u kR , ∀k

(16)

Σ x1ij ≥ u Dj , i=1 (1)

L

j=1

K

c 6ikx 6ik

K

s. t.

l=1

J

K

Σ x3kl ≥ d l , k=1

i=1 k=1

K

Σ fjy 1j + kΣ= 1 g k yk2 j=1

Σ x2 jk = l Σ= 1 x3kl , j=1

K

Σ Σ c4ilx4il + Σ Σ c5jlx5jl + Σ Σ i=1 l=1

+

K

J

J

J

+

J

I

J

T x Σ Σ c1ijx1ij + jΣ= 1 kΣ= 1 c2 jk x2 jk + kΣ= 1 l Σ= 1 c3kl 3kl i=1 j=1

x1ij, x2jk, x3kl ≥ 0, ∀i, j, k, l

(17)

yj1, yk2 ∈ {0, 1}, ∀j, k

(18)

This model is quite similar to that of fMLN. Each formula can find its counterpart in the previous model only except the effects of direct shipment and direct delivery. Yesikökcen et al. (1998) proved that to allocate all the customers to available plats can provide a lower bound on the optimal objective function value of model 2. However, his research is conducted in continuous Euclidian space, and the transportation cost is directly proportional to the Euclidian distances between points. In fact data of real-world problem do not satisfy these requirements. (For instance, the direct transportation between two points may be very difficult or expensive due to the traffic situation limits; in international trade when the shipment strides the border among countries, duty can be requested.) All of these make the transportation cost out of direct proportion to the Euclidian distance. How to treat this problem will be explained in detail when describing how to generate the data of the test problems in Section 4. In order to understand the exact effects of direct shipment and direct delivery, let’s conduct a brief survey on the bounds of the objective functions. First of all, let’s consider the transportation costs in both formulas. According to constraint (5), to optimize the transportation flow to each customer usually we have the following


balancing condition: J

L

Σ

l=1

x4il +

Σ

j=1

K

x5 jl +

Σ

(5’)

x3kl = d l , ∀l

k=1

Considering all the available delivery paths, the transportation cost of the product for customer l is

(19)

Here we suppose the delivery paths go through plant I, DC j and retailer k. While for the tMLN model, from equation (12) (13) and (14) we have the following balancing condition for tMLN model: J

L

K

Σ x1ij = jΣ= 1 x2 jk = kΣ= 1 x3kl = d l , l=1

(20)

∀l

The transportation cost of the product for customer l is cTl ′ = (c1ij + c2jk + c3kl)dl, ∀l*

(21)

(*Here we use the same symbols added with ‘’ ‘ to represent the quantity in the tMLN model as those in fMLN model.) Comparing (21) with (13) we can easily get cTl ≤ cTl ′ ,∀l

(22)

Next, let’s study on the open cost of facilities. From (3), we can get q 1j =

I

K

L

L

Σ x1ij = kΣ= 1 x2 jk + l Σ= 1 x5 jl ≤ l Σ= 1 x5 jl , i=1

∀j

Substitute it into (5’) we can learn J

Σ

j=1

q 1j ≤

L

Σ

l=1

(23)

dl

In the same way, we can get K

L

Σ q k2 ≤ l Σ= 1 d l k=1

(24)

While from (20) in the tMLN model and definitions of throughputs of facilities, we thus have J

Σ

j=1

q 1′ j =

K

Σ

k=1

q k2′ =

L

Σ

l=1

dl

Then we can learn J

J

K

K

Σ q 1 ≤ jΣ= 1 q 1′j and kΣ= 1 q k2 ≤ kΣ= 1 q k2′. j=1 j From the definitions of the open costs, by selecting the appropriate delivery paths, it’s reasonable to tell that J

Σ

j=1

f jy 1j +

K

J

K

Σ g k yk2 ≤ jΣ= 1 fj′y 1j ′ + kΣ= 1 g k′yk2′ k=1

It means that the total facility open costs of fMLN model are no more than that of t MLN. In summary, as explained before by taking the two parts of the total logistics cost into consideration we can draw conclusion that the optimal value of objective function of fMLN model is no more than that of the tMLN model. z ≤ z′

clT = min{c4ildl, (c1ij + c5jl)dl, (c6ik + c3ld)dl, (c1ij + c2jk + c3kl)dl}, ∀l

5

(25)

From the mathematical analysis, we conclude that the utilization of nonadjacent model can provides a more cost-effective platform for logistics network design and optimization problem. In another word, selecting more appropriate delivery paths provides chances to cut down the transportation cost and to shorten the length of distribution channel. On the other hand, fMLN model can improve the efficiency of logistics network by guaranteeing more customers supported using limited resources (the capacitated facilities). 3. HYBRID GA DESIGN Although using fMLN model guarantees a flexible and efficient logistics network, the existence of the new delivery modes within makes the problem much more complex. Concretely the complexity of tMLN model is O(IJ*JK*KL), while that is O(IJ*(I+J)K*(I+J+K)L) for nonadjoining MLN model. Here, we employ GA to solve this problem (Wang and Gen, 2006). GA is a search technique used in computer science to find approximate solutions to optimization problems proposed by Holland (1975). GA is a particular class of evolutionary algorithms that use techniques inspired by evolutionary biology such as inheritance, mutation, selection, and crossover (also called recombination). Thanks to its remarkable ability to solve complex problem in science, engineering, economics and management science, GA has emerged as one of the most efficient guided stochastic solution searching procedures for solving various optimization problems. GA is typically implemented as a computer simulation in which a population of abstract representations (called chromosomes) of candidate solutions (called individuals) to an optimization problem evolves toward better solutions (Crosby, 1973). Traditionally, solutions are represented in binary as strings of 0s and 1s, but different encodings are also possible and necessary for different problems (Gen and Cheng, 1997). The evolution starts from an initial which usually consists of completely random individuals and happens in generations. In each generation, the fitness of the whole population is evaluated, multiple individuals are stochastically selected from the current population (based on their fitness), and modified (mutated or recombined) to form a new population. The new population is then used in the next iteration of the algorithm.

6


In this study, we consider a hybrid GA-based approaches that it is called random path-based GA (rpGA) with local optimization, which employs direct encoded chromosome to present a candidate solution. 3.1 Representation of Chromosome We here adopt the random path-based direct encoding method, which has fine express ability to candidate solutions developed by Gen and Cheng (2000). As shown in Fig. 4, in the genes with length 3*L (L is the total number of customers), every 3 loci constitute one unit, each of which represents a delivery route to a customer, from plant via DC and retailer. The allele on first locus of each unit means the ID of a plant, start node of the delivery path. The second is the ID of DCs, and the last one indicates the ID of retailers on the path. 3.2 Initialization We just apply the traditional random initialization method: randomly assign the ID of plant, DC or retailer to the relevant locus as described as following procedure.

method we can obtain the delivery route conveniently. By doing that, computation time can be greatly cut down. Additionally, in order to improve the quantity of the result, many kinds of local optimization methods have been developed including hill climbing method etc (Michalewicz, 1996; Yun and Moon, 2003). In this study, we adopt the local optimization technique based on the concept of neighborhood. A. Neighborhood In each gene unit consisting of three loci, four delivery paths can be formed including normal delivery, direct shipment and direct delivery. All of them form a neighborhood. For instance, we can obtain the neighborhood given in Table 1 from the sample of gene unit shown in Fig. 5, which represents the delivery route to customer 1. In this way, the neighborhood can be decided according to the customer ID and the corresponding gene unit in a chromosome. B. Local Optimization We assign the most economic route in a neighborhood to the customer by calculating and comparing the total transportation through each of these routes.

procedure: Initialization by random path-based encoding input: number of plants (I), DCs (J), retailers (K) and customers (L) output: the chromosome vk[⋅] step 0: for i=1 to L vk[3*j ] ← random (1, I); vk[3*j +1] ← random (0, J); vk[3*j +2] ← random (0, K); step 1: output the chromosome vk[⋅];

Table 1. The delivery paths in corresponding neighborhood

Using this initialization method, a mount of infeasible solutions may be generated, which violate the facility capacity constraints, so a repairing procedure is needed. In the delivery route to a customer, at least one plant is necessary, if the total demand to the plant exceeds its supply capacity, we will assign the customer to another plant with sufficient products supply, the transportation price between which and the customer is the lowest.

C1

#

Delivery Route

1 2 3 4

{P2-DC1-R5-C1} {P2-C1} {P2-R5-C1} {P2-DC1-C1}

2

1

5

A sample of gene unit

3.3 Decoding Using the random path-based direct encoding C1

C2

…

CL

P1 DC3 R8

P2 DC1 R2

…

P3 DC2 R3

… delivery route to C1

delivery route to C2

delivery route to CL

Fig. 4. Representation of Chromosome

DC1

P2

R5

path 1 path 2 path 3 path 4 The paths in corresponding neighborhood Fig. 5. A sample of gene unit and its corresponding neighbor

C1


7

By confining the local optimization process in a relatively narrow area, we can control the trade-off between quantity of the solutions and consumption of computation time well.

C. Evaluation and Selection

procedure: Local Optimization input: problem data and gene unit uk output: a best delivery path to a customer step 0: form all the available paths from uk; step 1: calculate the total transportation cost through each path; step 2: select the most economic one from all the paths as the delivery route for a customer; step 3: output the delivery path;

3.5 Overall Procedure of rpGA with Local Optimization

Using the local optimization technique, we can also deal with the trade-off between local search and global search well. The process is described in following procedure. procedure: Local Optimization based Decoding input: problem data and chromosome vk[⋅] output: a candidate solution step 0: for i=1 to L determine the delivery path to customer i using procedure of Local Optimization; step 1: output a solution; 3.4 Genetic Operators As what have been done in the previous, we use one crossover and mutations to strengthen the search ability of the algorithm. A. Crossover Two-cut point Crossover: We randomly select two cutting points and then exchange the substrings between the two parents. Combinatorial Crossover: Before explaining the Combinatory Crossover, we first introduce the definition of Set of plants, DCs and retailers in a chromosome, which are assembles of the plants, DCs and retailers in each unit in the chromosome separately and are illustrated in Fig. 6 & 7. Then we randomly select some sub-sets from one or more of these sets, and exchange them between the two parents. B. Mutation

Likewise, we use the reciprocal of the total cost as evaluation function and roulette wheel selection.

The overall pseudo-code procedure of random pathbased GA (rpGA) is outlined in the following procedure. procedure: rpGA with Local Optimization input: problem data and GA parameters output: a best solution begin t ←0; // t: generation number initialize P(t) by random path-based direct encoding; // P(t): population of chromosomes fitness eval(P) by local optimization based decoding; while (not terminating condition) do crossover P(t) to yield C(t) by two-cut point crossover & combinatorial crossover; // C(t): offspring mutation P(t) to yield C(t) by insertion mutation & shift mutation; fitness eval(C) by local opti. based decoding; select P(t+1) from P(t) and C(t) by roulette wheel selection; t ← t+1; end output a best solution; end

chromosome

Set of plants: {Pi} = {2, 1, 2, 1, 3, 2, 3} chromosome

2 1 5 1 4 0 2 3 6 1 2 0 3 4 1 2 3 4 3 0 0 Set of DCs: {DCj} = {1, 4, 3, 2, 4, 3, 0}

chromosome

2 1 5 1 4 0 2 3 6 1 2 0 3 4 1 2 3 4 3 0 0 Set of Rs: {Rk} = {5, 0, 6, 0, 1, 4, 0}

Fig. 6. Example of Set of plants, DCs and retailers

parent 1

2 1 5 1 4 0 2 3 6 1 2 0 3 4 1 2 3 4 3 0 0 1

{Pi }={2, 1, 2, 1, 3, 2, 3}, {DCj1}={1, 4, 3, 2, 4, 3, 0}, {Rk1}={5, 0, 6, 0, 1, 4, 0} {Pi2}={3, 2, 2, 1, 1, 3, 2}, {DCj2}={2, 0, 4, 2, 1, 3, 3}, {Rk2}={1, 4, 3, 2, 4, 3, 0} parent 2

Insertion Mutation: The method here used is a little different from that in Algorithm I. As shown in Fig. 8, we randomly select a string which consists of some gene units, and then insert them into another selected locus. Shift Mutation: We select some units at random, and then randomly assign a new value to each locus in it.

2 1 5 1 4 0 2 3 6 1 2 0 3 4 1 2 3 4 3 0 0

3 2 0 2 0 0 2 4 1 1 2 6 1 1 4 3 3 0 2 3 3

2 1 5 1 4 0 2 3 6 1 2 0 1 4 1 3 3 4 3 0 0 3 2 0 2 0 0 2 4 1 1 2 6 3 1 4 2 3 0 2 3 3

Fig. 7. Example of Combinatory Crossover

8


insertion point parent

2 1 5 1 4 0 2 3 6 1 2 0 3 4 1 2 3 4 3 0 0

offspring 2 1 5 1 4 0 2 3 4 1 2 6 1 2 0 3 4 1 3 0 0

Fig. 8. Illustration of Insertion Mutation

generate coordinates of sites of plants, DCs retailers and customers correspondingly (for problem 2, 3 and 4). 2) Calculate the Euclidian distances between relative sites. 3) Because the parameter sensitivity of the problem, to set the cost coefficients reasonably is very critical to the examples. We generate the relative coefficient using the following formula cxy = atdxy + bt

4. NUMERICAL EXPERIMENT In order to investigate the effectiveness of the flexible MLN model and the proposed approaches, we apply them to 4 test problems. Moreover, we also use the tMLN model and pGA-GAL proposed by Gen, Altiparamk and Lin (2006) to the test problems as a comparison. Furthermore, we combined an extended pGA-GAL called epGA for the flexible MLN. The size of the test problems in terms of number of plant, DC, retailer and customer vary from a relatively small one (2, 3, 5, 20) to an excitingly large one (5, 20, 30, 150) for real-world case, respectively. Fig. 9 gives a graphical illustration of problem 1 in Europe. The size of these problems and their computing complexity are given in Table 2. The problems are generated in following steps: 1) Randomly

where cxy is the unit cost coefficient between site x and y. t is the type considering the combination of delivery modes and starting site. at represents a affected unit transportation cost by distance, mainly depended on the delivery mode. bt is the part of unit transportation cost independent of distance, generated from a uniform distribution function between 0 and btmax. The demands of customers are generated randomly between 10 and 100 following the uniform distribution. We use the actual geographical distances for the problems. The data employed in the problem 3 and 4 are almost same, the only difference is that in problem 3 the capacity of each DC is set a relative smaller value (400), while it takes a little larger value (600) in problem 4. The data set for test problems are also available from authors.

UK 2

Ireland Dublin

Edinburgh

2 3

1

1

York

3 4

Netherlands

Germany

London 1

5 1

Amsterdam 15 Berlin 4 3 Bonn 10 12 11 4 16 13 2 Paris Mainz 7 14 17 Lyon 6

6 France 7

Poland 20 Bytom 8 18

2 Warsaw 5 19

Plant 8 DC

9

R Retailer

Monaco 5

Customer

Fig. 9. Graphical illustration of Problem 1

Table 2. The size of test problems Traditional MLN Model Problem #

Number of plants

Number of DCs

1 2 3 4

2 3 5 5

5 10 20 20

Number Number of of retailers customers 8 15 30 30

20 60 150 150

Flexiable MLN Mode

Number of decision variables

Comp. complexity

Number of decision variables

Comp. complexity

223 1105 5200 5200

1.44*1025 2.17*1090 3.79*10274 3.78*10274

379 1930 9150 9150

6.13*1031 2.04*10108 9.38*10316 9.38*10316

9


Table 3. Parameter value in different cases for all the test problems Experiment Case

1

2

3

4

5

6

7

8

9

popSize maxGen pC pM

20 300 0.6 0.6

50 300 0.6 0.6

100 300 0.6 0.6

20 500 0.6 0.6

50 500 0.6 0.6

100 500 0.6 0.6

20 1000 0.6 0.6

50 1000 0.6 0.6

100 1000 0.6 0.6

Experiment Case

10

11

12

13

14

15

16

17

18

popSize maxGen pC pM

100 500 0.1 0.1

100 500 0.1 0.3

100 500 0.1 0.6

100 500 0.3 0.1

100 500 0.3 0.3

100 500 0.3 0.6

100 500 0.3 0.1

100 500 0.3 0.3

100 500 0.3 0.6

300000 GAL-pGA 250000

epGA rpGA

200000 AVG

As known, the parameters tuning and controlling is usually an issue open to debate in theory and application of GA (Eiben, Hinterding and Michalewicz, 1999). To investigate the effect of different parameters combination on GA’s performance, we conducted the experiments in 18 cases for each test problem. In each case we set different permutation of parameter values (see Table 3). The computational results of the test problems are including best value of the objective function (best value), average of best solutions (AVG), and standard deviation of 50 best solution (SD) of the 50 best solutions obtained by 50 runs of each program correspondingly. Fig. 10 & 11 show the minimum AVG and minimum SD considering all the 18 cases for all the problems respectively. (Additionally, problem 3 cannot be solved by using tMLN model and GAL-pGA, in order to make the figures clear, we use the same data obtained in problem 4 to draw the curves.) First of all, we can get the following conclusions:

150000 100000 50000 0

1

2 3 Test Problem

4

Fig. 10. Minimum AVGs of different approaches in all the test problems

3500 3000

(1) Intuitively, with the different parameter permutation, epGA outperforms pGA-GAL in terms of best value (in Table 4), AVG (in Table 5) in all the problems. Comparing with the rpGA, the better results could be some what owing to the utilization of fMLN model, while its SDs (in Table 6) are a little large. (2) rpGA surpasses the other two methods greatly in terms of best value, AVG, SD. Fig. 11 shows the SDs of the three methods in different problems. From the figures we can observe the results more clearly. (3) To examine if the AVGs of pGA-GAL, epGA and rpGA are statistically significantly different under each permutation of the parameters, we carry out t-test for each of them. We used the null hypotheses as the differences between the AVG of the three methods are zero. The test yielded t values (t >> t0.025(98)) in all the 18 cases for each test problem (the two tailed t value with 5% significance is 1.984 < |t0025(98)| < 1.990). Therefore, we can refuse the null hypothesis. In another word, the proposed three approaches are significantly different.

SD

2500 2000 1500 1000

GAL-pGA epGA rpGA

500 0 1

2

3 Test Problem

4

Fig. 11. Minimum SDs of different approaches in all the test problems

(4) To see whether the differences are statistically significant between three methods as a whole in treating all the problems tested, another statistical analysis using sign test, which is a nonparametric version of paired-t test, was also carried out considering all the 18 cases. In the test, we test the hypothesis that the averages between the three methods under any combination of parameters are the same. The test yielded that while rpGA outdid epGA and pGA-GAL with a p value of 2.118*10–22.

10


Table 4. Best values of different approaches in problem 4 with different maxGen and popSize

6000 pGA-GAL MaxGen=300 5000

MaxGen

pGA-GAL

epGA

rpGA

pGA-GAL MaxGen=500 pGA-GAL MaxGen=1000

popSize

300

500

1000

20 50 100 20 50 100 20 50 100

280162.9919 281719.1886 280387.4657 190097.1001 188184.3315 186159.1283 76518.1717 75874.5190 76420.5646

281057.3422 280008.9069 280630.0101 193163.1754 195847.2754 187183.7418 76140.4930 76118.7359 75323.3731

280284.0886 280772.3563 279510.5058 189598.1851 189598.1851 192137.3755 76163.2127 75743.6578 75602.7262

SD

4000

epGA MaxGen=300 epGA MaxGen=500

3000

epGA MaxGen=1000 2000

rpGA MaxGen=300 rpGA MaxGen=500

1000

rpGA MaxGen=1000 0

20

50

100

popSize

(a) The relationship between the SD and the popSize of the three approaches in problem 4 6000

pGA-GAL popSize=20 pGA-GAL popSize=50

5000

Table 5. AVGs of different approaches in problem 4 with different maxGen and popSize

pGA-GAL popSize=100 4000 SD

epGA popSize=20 3000

epGA popSize=50

2000

epGA popSize=100

MaxGen

pGA-GAL

epGA

rpGA

popSize

300

500

1000

20 50 100 20 50 100 20 50 100

280162.9919 281719.1886 280387.4657 190097.1001 188184.3315 186159.1283 76518.1717 75874.5190 76420.5646

281057.3422 280008.9069 280630.0101 193163.1754 195847.2754 187183.7418 76140.4930 76118.7359 75323.3731

280284.0886 280772.3563 279510.5058 189598.1851 189598.1851 192137.3755 76163.2127 75743.6578 75602.7262

Table 6. SDs of different approaches in problem 4 with different maxGen and popSize MaxGen

pGA-GAL

epGA

rpGA

popSize

300

500

1000

20 50 100 20 50 100 20 50 100

1720.4036 1542.7880 2123.7456 4916.5977 4638.8594 4226.4800 1817.6487 1850.5014 1686.6919

2575.3861 2001.0604 1879.2242 4185.3893 3188.8568 3596.7758 1883.6414 1476.9319 1491.0571

2216.0091 1802.5997 2189.2110 4862.8178 4862.8178 3325.9013 1547.2790 1312.4340 1212.4987

Second, feasible solution to problem 3 cannot be obtained by using fMLN model. This shows that fMLN model cannot support logistics network to serve as many customers as the tMLN does with limited resources. Finally, on the effects of the parameters, we can additionally get following conclusions: (1) popSize affects the quality of the results markedly. It means the best value, AGV and SD decreased obviously at almost the same time when popSize increased from 20 to 50 and 100, while other parameters keep

rpGA popSize=20

1000

rpGA popSize=50 0

300

500 maxGen

1000

rpGA popSize=100

(b) The relationship between the SD and the maxGen of the three approaches in problem 4 Fig. 12. The SD of the three approaches in problem 4 with different maxGen and popSize

fixed. To illustrate it we take test problem 4 as an example, and the data are shown in Table 4-6 and Fig. 12(a). (2) maxGen affects the AGV and SD in the same way, but not significantly. What’s more, its action at the best value seems analogously random somewhat. Fig. 12(b) gives the SDs of the three methods in test problem 4. As we can see, in Fig. 12(a), in almost all the cases SD decreases as popSize increases. While in Fig. 12(b), the trend of SD’s varying with MaxGen is not so clear. (3) The effect of pC and pM is a bit complex (see Table 79). In pGA-GAL, as pM or p C increased the SDs were inclined to increase. In epGA, when p C increased, AVGs and SDs trended to decrease, but the relationship between pC and AVGs is quite week. Also, a weak positive relationship between pM and SDs could be observed. In rpGA, when pC got larger, AVGs and SDs are almost improved. Furthermore a positive relationship between pM and best values, AVGs and SDs can also be observed, although the relationship between pM and SDs is not strong. 5. CONCLUSIONS In this paper, we considered logistics network design problem by formulating the problem as location-


Table 7. Best values of different approaches in problem 4 with different pC and pM pc

pGA-GAL

epGA

rpGA

pM

0.1

0.3

0.6

0.1 0.3 0.6 0.1 0.3 0.6 0.1 0.3 0.6

280411.3160 281552.1461 279246.5563 190968.7696 191902.7342 190580.6435 75892.0934 76049.8417 75903.1250

279760.1050 280537.2807 279039.6992 189673.9573 190098.6499 190448.0116 76518.1717 76518.1717 75450.9521

280651.3740 280229.9228 279226.0352 182030.7270 191361.2720 186946.9018 76382.1568 76049.8471 76242.5198

Table 8. AVGs of different approaches in problem 4 with different pC and pM pc

pGA-GAL

epGA

rpGA

pM

0.1

0.3

0.6

0.1 0.3 0.6 0.1 0.3 0.6 0.1 0.3 0.6

283953.5386 284069.1517 284251.9543 199692.9007 197941.5600 199170.7457 80111.4496 78582.9772 78714.9874

284054.7994 284631.6350 284094.6100 198816.2892 199527.1473 198785.1127 80210.3028 78814.6232 78291.7906

283912.0840 283886.0483 283718.5972 199107.5327 198525.7539 198752.4783 80147.7229 78176.0160 78696.4364

Table 9. SDs of different approaches in problem 4 with different pC and pM pc

pGA-GAL

epGA

rpGA

pM

0.1

0.3

0.6

0.1 0.3 0.6 0.1 0.3 0.6 0.1 0.3 0.6

1345.7178 1330.0666 1663.1185 1670.4074 1887.4705 1761.1968 1353.3967 1571.4448 1617.7708

3426.9814 2806.0950 3607.5413 3307.5845 3446.7889 4227.8271 4388.1052 3403.6328 4402.4775

1875.2767 1587.5347 1701.3779 1586.6223 1634.6226 1607.0145 1918.6706 1538.0158 1409.4117

allocation model. A new network model is proposed, flexible multistage logistics network (fMLN) model, as proved that it can make the logistics network more flexible and cost-effective than the traditional multistage logistics network (tMLN) model. Basic location-allocation problem is NP-hard, while the existence of the new delivery modes makes it much more difficult to be solved. We extended pGA approach proposed by Gen, Altiparamk and Lin to address this problem by adding some guiding information into the chromosome. We also proposed an rpGA approach to solve the problem. In this approach, we em-

11

ployed local optimization based decoding method and proposed a new crossover combinatorial crossover to improve the performance of rpGA. Since decoding process takes up most of the computation time in GAs, using this decoding method speeds up the GA procedure. Finally, the computational results using both the fMLN model and tMLN model with corresponding algorithm were computed and compared to prove the effectiveness and efficiency of fMLN model and the proposed GA-based approaches. Additionally, the effects of parameters on the performance of GA are also studied briefly. ACKNOWLEDGMENTS This work is partly supported by the Ministry of Education, Science and Culture, the Japanese Government: Grant-in-Aid for Scientific Research (No.19700071, No. 17510138). REFERENCES Altiparamk, F., Gen, M. and Lin, L. (2006) A genetic algorithm approach for multi-objective optimization of supply chain networks. Computer and Industrial Engineering, 51, 196215. Crosby, J. L. (1973) Computer Simulation in Genetics, John Wiley & Sons, London. Eiben, A. E., Hinterding, R. and Michalewicz, Z. (1999) Parameter control in evolutionary algorithms. IEEE Trans. on EC., 3(2), 124-141. Gen, M. and Syarif, A. (2005) Hybrid genetic algorithm for multi-time period production/distribution planning. Computers & Industrial Engineering, 48(4), 799-809. Gen, M., Altiparamk, F. and Lin, L. (2006) A genetic algorithm for two-stage transportation problem using priority-based encoding. OR Spectrum, 28(3), 337-354. Gen, M. and Cheng, R. (1997) Genetic Algorithms and Engineering Design, John Wiley & Sons, New York. Gen, M. and Cheng, R. (2000) Genetic Algorithms and Engineering Optimization, John Wiley & Sons, New York. Goetschalckx, M., Vidal, C. J. and Dogan, K. (2002) Modeling and design of global logistics systems: A review of integrated strategic and tactical models and design algorithms. European Journal of Operational Research, 143(1), 1-18. Harland, C. (1997) Supply chain operational performance roles. Integrated Manufacturing Systems, 8(2), 70-78. Holland, J. H. (1975) Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor. Johnson, J. L. and Umesh, U. N. (2002) The interplay of task allocation patterns and governance mechanisms in industrial distribution channels. Industrial Marketing Management, 31(8), 665-678. Lee, H. L., Padmanabhan, V. and Whang, S. (1997) The bullwhip effect in supply chains. Sloan Management Review, 38(3), 93-102. Logistics & Technology online: http://www.trafficworld.com/ news/log/112904a.asp Michalewicz, Z. (1996) Genetic Algorithms + Data Structures = Evolution Program, 3rd ed., New York: Spring-Verlag.

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Nakatsu, R. T. (2005) Designing business logistics networks using model-based reasoning and heuristicbased searching. Expert Syst. with Applications, 29(4), 735-745. Reyes, P. M. (2005) Logistics networks: A game theory application for solving the transshipment problem. Applied Mathematics and Computation, 168(2), 1419-1431. Tsay, A. A. (2002) Risk sensitivity in distribution channel partnerships: implications for manufacturer return policies. Journal of Retailing, 78(2), 147-160. Wang, X. and Gen, M. (2006) Flexible multistage logistics net-

works design by priority-based genetic algorithm with an effective decoding method. Proc. of Korea-Japan workshop on Intelligent Logistics Systems, 87-95, Busan, Korea. Yun, Y. S. and Moon, C. U. (2003) Comparison of adaptive genetic algorithms for engineering optimization problems. International Journal of Industrial Engineering, 10(4), 584-590. Yesikokcen, G. N. and Wesolowsky, G. O. (1998) A branchand-bound algorithm for the supply connected locationallocation problem on network. Location Science, 6, 395-415.

A Hybrid Genetic Algorithm for Logistics Network Design ... - CiteSeerX

A Hybrid Genetic Algorithm for Logistics Network Design ... - CiteSeerX

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