Genetic Algorithm for Fixed Charge Transportation ...

ArchiveMazandaran of SID University of Science and Technology

Iran

Available online at www.iiec2017.com

13

th

Institute of

International Conference on Industrial Engineering

Industrial Engineering

(IIEC 2017)

Genetic Algorithm for Fixed Charge Transportation Problem with Discount Models Komeil Yousefia, Ahmad J. Afsharib, Mostafa Hajiaghaei-Keshtelic

c

a

Department of Industrial Engineering, Shomal University, Amol, Iran E-mail: [email protected]

b

Department of Industrial Engineering, Shomal University, Amol, Iran E-mail: [email protected]

Department of Industrial Engineering,University of Science and Technology of Mazandaran, Behshahr, Iran E-mail: [email protected]

Abstract Proposing discount offers for products' prices usually directly influences on the process of distributing and selling products. The fixed charge transportation problem (FCTP) is a deployment of the classical transportation problem in which a fixed cost is incurred, independent of the amount transported, along with a variable cost that is proportional to the amount shipped. Since the problem is considered as an NP-hard, in this paper we propose a well-known metaheuristic to solve the FCTP with discount supposition on both fixed and variable charges. In addition, two models with all-units discount and incremental discount is firstly proposed in this study to apply the discount mechanism. Also, as the previous researchers mainly used spanning tree-based and priority-based representations, we used both methods in metaheuristic and compared the results. Furthermore, we apply the Taguchi experimental design method to set the proper values of algorithm in order to improve its performance. Finally, computational results metaheuristic with different encoding approaches, both in terms of the solution quality and computation time, are investigated in different problem sizes. Keywords: Fixed charge transportation problem; Discount model; Genetic algorithm; Spanning tree; Priority-based; Taguchi experimental design.

Introduction Supply chain philosophy has attracted much attention in the past few decades due to increased competition in dynamic environment. Most of manufacturers believe that customer satisfaction is the most important factor to survive in this environment. The main objective of supply chain is manufacturing, distribution, logistics and delivery of goods in the right place, right time and right amount to meet customer satisfaction with minimal time and cost. Thus,

there is a need for technologies and method to reduce the time, cost, and also improving production and operations. Among the various dilemmas in a supply chain, both internal and external transportation problems are considered as important problems rather than the other. As shown in Figure 1, in this particular type of linear programming problems, goods are only allowed to move straight from the source to the destination. Customers Plants DC1 P1

DC2 P2

DC3

Figure 1 - a single-stage FCTP. In the fixed charge transportation problems, there are two types of charges. There is a fixed charge which is independent of the amount of products shipped in each route and also variable charge which is regarded proportional to the amount of shipped goods. The aim is to find a combination of routes that minimizes the total variable and fixed charges. With the growth and development of global markets and more interaction between markets in different countries, companies to attract customers to their goods and services may propose a variety of discounts or promotions. Discounts provided to customers in the case of buying more products are called quantity discounts. Usually, customers may use discounts when they use cash or faster payment, purchase large quantities of products, purchase in off season and etc. Quantity discounts are usually employed in automotive supply chains, food industries, health products and etc.

www.SID.ir

Archive of SID

Literature review Hirsch and Dantzig firstly proposed the fixed charge problem and consequently [1]. Balinski developed the FCTP for the first time in literature [2]. Balinski studied the problem's structures and to solve the problem he developed an approximate algorithm. This problem is later studied about its complexity by Klose [3]. He showed that solution of fixed charge transportation problems is among the nonlinear Programming-hardness category and solution time to the size of the problem increases exponentially. Therefore, approximate algorithms are used to solve such problems. So over the last two decades, several heuristic and metaheuristic methods have been proposed by researchers to solve fixed charge transportation problems (see, e.g., heuristics [4,5]; tabu search [6]; simulated annealing [7,8]; genetic algorithm (GA) [9,10,11]; artificial immune algorithm [12]; hybrid particle swarm algorithm with artificial immune [13]; simplex-based simulated annealing [14]; minimum cost flow-based genetic algorithm [15]). Prüfer number, as one of the effective methods in network problems, has been initially introduced by Gen and Chang [16]. Encoding based on Prüfer number has been used successfully in the spanning tree-based representation (e.g. see [12,17,18]). Gen and Li provided a genetic algorithm by using spanning tree-based representation and Prüfer number to solve two objective transportation problems with fixed cost [19]. Gen and Cheng examined the feasibility of chromosomes produced by using Prüfer number and showed that this method does not lead to produce feasible chromosome in some cases [20]. In the production of random chromosome based on spanning tree, there is the possibility that the production chromosome does not match with transportation network graph. Therefore, Jo et al. investigated the feasibility of chromosomes before decoding and converting it to spanning tree and provided a criterion to evaluate the feasibility and then modifying infeasible chromosomes [21]. But this proposed method may allocate much time for modifications to itself. Finally, Hajiaghaei-Keshteli et al. offered production of feasible chromosomes without the need to modification [9]. In fact, they corrected the procedure developed by Jo et al. [21] and their procedure has been utilized in this research area by researchers until now. Priority-based encoding is also a strong method which is mostly used in recent years in the related research areas. Gen et al. considered the two-stage transportation problem and used the genetic algorithm by using priority-based encoding and provided a new method to design operators [22]. Hwang et al. compared system of direct and U-shaped assembly lines and used the genetic algorithm and prioritybased representation to solve assembly line balancing problem [23]. Lee et al. raised reverse logistics network problem in three stages and used a genetic algorithm with priority-based representation which has a new operator for better search in solution space [24]. They also provided an innovative method for the third stage which is the transportation of materials from distribution center to the

factory. Lotfi and Tavakkoli-Moghaddam presented a genetic algorithm using priority-based encoding for linear and nonlinear transportation problem with fixed cost in which the new operators have been provided for better search in solution space [10]. They compared the problem with two priority representation and spanning tree without setting the parameter of proposed algorithm. Tari and Hashemi used a priority based genetic algorithm to solve the real world size problems in an allocation problem in supply chain [10]. Step fixed charge transportation problems (SFCTP) are developed type of fixed charge transportation problems in which charges are considered as step functions. Kowalski and Lev raised the step fixed charge transportation problem for the first time in which fixed charge is given as a step function dependent on the capacity of route [25]. They presented a simple computational algorithm to solve problems in small sizes. El-Sherbiny considered the fixed charge as a step function and presented an artificial immune algorithm for solving step fixed charge transportation problems [26]. After that, Altassan et al. also considered fixed charge as a step function and provided a heuristic method to solve this problem [27]. Molla-Alizadeh et al. considered the step fixed charge transportation problems in which fixed charge is as a step function and provided two genetic and memetic algorithms with spanning tree-based representation [28]. Also, Molla-Alizadeh et al. examined genetic algorithm operators to solve step fixed charge transportation problem which its fixed charge is as a step function by using Taguchi parameter design and compared performance of genetic algorithm with simulated annealing algorithm by taking seven problems on various aspects [29]. Sanei et al. considered the step fixed charge transportation problem with two types of fixed charge and provided a heuristic method based on Lagrangian relaxation to solve it and showed that their proposed method has better performance than the results of CPLEX software [30]. In this study, we raised fixed charge transportation problems with considering discount models on both fixed and variable charges. To the best of our knowledge, there has not been so far a research by applying a discount on both fixed and variable charges. In addition, two models of All-units discount model and incremental discount model have been used for the first time in this study to apply the discount models. After modeling the desired problem, we proposed with known metaheuristic method such as genetic to solve the proposed models. Also as the researchers mainly used spanning tree-based representation or prioritybased representation to achieve the graph of transportation, we used the both representations in metaheuristic algorithm and compared the results. In addition, Taguchi experimental design has been used in order to set parameters of metaheuristic algorithm and improve their performance. Finally, for evaluated and compared the performance of proposed metaheuristic algorithm and representations method in terms of solution quality and computation time, we use not only the test data from Hajiaghaei-Keshteli et al. [9] but also we added two new different problem sizes to

www.SID.ir

Archive of SID

Mathematical model and descriptions

discounts on variable charges as ( ) percent and on fixed charges as (  ) . Since the stated price includes the total amount of demand, it is called all-units discount model. Allunits discount is as the Table 1.

The problem is considered as a distribution problem with (m) suppliers and (n) customers. A supplier can response to a customer's demand and send its products with the cost of (cij ) for each unit as shipping cost. Besides, there is a fixed

Table 1 – All units discount representation Total Fixed Variable transportation cost cost cost 0 0 0 (cij  xij )  fij cij 0 fij

previous data and reach to nine different problem sizes in three levels; small, medium and large.

cost of ( fij ) considered for opening of a route. The (ai ) and

(b j ) are the value of capacity and demand of each supplier and customer, respectively. The objective minimizes the both variable and fixed costs. Mathematically this problem may be stated as follows: m

Minimize Z 

n

 (cˆ .x ij

ij

 fˆij . yij )

m

ij

 b j ; j  1, 2,..., n

cij1

a1  xij  a2

(cij p  xij )  fij p

fij p

cij p

a p 1  xij  a p

We call the values (a1 , a2 ,..., a p ) as breakpoints of price or

(2)

j 1

x

fij1

price of goods units will be lowers against the order more, we will have relations (6) to (7).

n

 ai ; i  1, 2,...,m

(cij1  xij )  fij1

is assumed that (a1  ...  a p 1  a p ) and since usually the

s.t. ij

a0  xij  a1

discount points and display with (a p ) . In these symbols, it

(1)

i 1 j 1

x

Order quantity

(3)

cij p  ...  cij1  cij 0

(6)

fij p  ...  fij1  fij 0

(7)

i 1

xij  0; i, j

(4)

xij  0  1 yij   ; i, j  0 otherwise

(5)

Thus, fixed charge and variable charge will be considered as formulas (8) and (9).

The objective function (1) asks for minimizing the total variable and fixed cost. Constraints (2) require that all goods available at each origin (i  1, 2,..., m) be delivered. Constraints (3) force, in any feasible solution, delivery (b j ) units of goods to each destination ( j  1, 2,..., n) . Finally, constraints (4) and (5) set the ranges of variables ( xij )

and

( yij ) , respectively. In the problem raised in this study, we have two decision variables. ( xij ) which represents the amount of sent goods from supplier i-th to the customer j-th and the binary variable ( yij ) which shows reopening or lack of reopening the route (ij ) and if ( xij  0) , its value is 1. All-units discount model In this study, a fixed charge transportation problem was considered that the all-units discount model has been applied on both fixed and variable charges of it. In the allunits discount model, cost of transportation per unit of product depends on the demand of destination. This means that when the amount of demand (b) comes to a certain extent of supply ( ) , Phase ( p ) , transmitter applies Table 2 - Incremental discount representation

cˆij p  cij  (1  a p )

(8)

fˆij p  fij  (1   p )

(9)

Incremental discount model In this study, a different kind of discount mechanism called incremental discount model has also been suggested in which discounts will be incrementally awarded to different parts of an order. In this case, prices are not the same in different intervals and items in each interval are purchased at available prices in that interval. Like the all-units discount model, when the amount of demand (b) comes to a certain extent of supply (a), Phase (p), transmitter applies discounts on variable charges as  percent and on fixed charges as (  ) . In this case, the cost of product order is as the Table 2. For example, if the quantity of order is between a0 and a1 , variable charge of each unit product is (cij 0 ) and fixed charge of each unit product is ( fij 0 ) . But if the quantity of order is between (a2 ) and (a3 ) , variable charge of (a1 ) first product (cij 0 ) variable charge of ( a2  a1 ) next product (cij1 ) and variable charge of ( xij  a2 ) next product (cij 2 ) and fixed charge in this intervals is ( fij 2 ) .

www.SID.ir

Archive of SID

Total transportation cost (cij 0  xij )  fij 0



 [cij 0   a1  a0 ]  [cij1   a2  a1 ]  [cij 2   xij  a2 ]  f ij 2 [cij 0   a1  a0 ]  [cij1  xij  a1 ]  f ij1

 p   cij k 1  ak  ak 1   cij p xij  a p   f ij p  k 1 







Fixed cost fij 0

Variable cost cij 0

Order quantity a0  xij  a1

fij1

cij1

a1  xij  a2

fij 2

cij 2

a2  xij  a3

fij p

cij p

a p 1  xij  a p

We call the values (a1 , a2 ,..., a p ) as breakpoints of price or

by having (m) supplier and (n) customers, we act in this

discount points and display with (a p ) . In these symbols, it

way. We produce a string of number with (n  1) digit of

is assumed that (a1  ...  a p 1  a p ) and since usually the

number set of supplier and a string of numbers with ( m  1)

price of goods units will be lowers against the order more, we will have relations (10) to (11).

digit of number set in customers. Finally, the two random strings of numbers are put together and we have a feasible chromosome which is called Prüfer number. As explained earlier, Hajiaghaei-Keshteli et al. [9] proposed a method to generate Prüfer number at random which does not need a repairing procedure. They corrected the procedure developed by Jo et al. [21]. In this paper the modified decoding algorithm of the spanning tree based representation for the FCTP, proposed by HajiaghaeiKeshteli et al. [9], is used. In addition, this type of representation was applied to different transportation problems, such as a bicriteria TP by Gen and Li [31], a bicriteria FCTP by Gen and Li [19], a network design problem by Kim et al. [32] and Gen et al. [33], a two-stage TP by Syarif and Gen [34].

cij p  ...  cij1  cij 0

(10)

fij p  ...  fij1  fij 0

(11)

Thus, fixed charge and variable charge will be considered as formulas (12) and (13). p

cˆij p 

c

ij

k 1

(ak  ak 1 )  cij p ( xij  a p )

(12)

k 1

fˆij p  fij  (1   p )

(13)

Proposed metaheuristics To explain the proposed metaheuristic, at first we describe two used encoding schemes; spanning tree and priority methods. We employ both of these representation methods into the developed genetic algorithm. Encoding scheme The first step in solving the problem model is linking it with metaheuristic algorithm structure, i.e., making a communication bridge between the original problem and solution space in which evolution occurs. In practice, represent a method to feasible chromosomes to be selected. So choosing an appropriate representation method is one of the most important parts of designing an algorithm. While most evolutionary algorithms use a random procedure to generate a set of initial solutions, both spanning tree representation and priority representation have been used to achieve initial feasible solutions in this study. Spanning tree-based representation As mentioned earlier, TP is a network problem which it’s feasible solution has spanning tree topology. The history of using the Prufer number in transportation problems is detailed in the literature. To produce feasible chromosome

Priority-based representation To escape from repair mechanisms required in spanning tree based representation, Gen et al. [22] developed a new method, called priority-based, for a two-stage transportation problem. Later, Lotfi and TavakkoliMoghaddam [10] modified this algorithm to consider both variable and fixed costs in the process of making a transportation tree. In this paper the modified decoding algorithm of the Priority-based representation for the FCTP, developed by Lotfi and Tavakkoli-Moghaddam [10], is used. Priority-based decoding belongs to the category of permutation decoding that does not require to corrective mechanism. So, a chromosome with a permutation of digits from 1 to m  n (total number of producer and consumer) is formed at the early stage. Then, determining the priority for nodes begins from the highest rate ( m  n) and reduces when determining the priority of all nodes to be performed. By using this approach, we reach to a chromosome is consisting the priorities of sources and depots to make a transportation tree. The length of a chromosome is the sum of sources (m) and depots (n) . We begin the priority

www.SID.ir

Archive of SID

assignment from the highest value ( m  n) to the lowest until assigning priority to all nodes. Consequently, the corresponding transportation tree is then generated by sequential arcs appending between sources and depots. There are several related works utilizing Priority-based encoding such as Tari and Hashemi [11] and Lotfi and Tavakkoli-Moghaddam [10] for FCTP, resource constrained project scheduling by Kim et al. [35], multistage logistics network design by Bahrampour et al. [36], reverse logistics network design by Zandieh and Chensebli [37], assembly line management by Hamzadayi and Yildiz [38], and the shortest path routing problem by Chitra and Subbaraj [39]. Genetic algorithm Based on the role of genetics in nature and the natural evolution of living organisms, Holland [40] presented a special type of evolutionary algorithms, i.e. genetic algorithms in the early of 70s. Genetic algorithm is a mathematical model that turns a population of chromosomes to the new ones by using Darwin's operational patterns on replication of survival of superior generation based on the natural process of genetic. The structure of the proposed GA is given in Figure 2. Genetic Algorithm Step 1: Initialize the problem and GA parameters Input: the data instance of the optimization problem and GA parameters; Step 2: Get an initial solution(𝒕), by the spanning treebased or priority-based. Step 3: Evaluation 𝑷(𝒕) Step 4: While (not termination condition) do Crossover 𝑃(𝑡) to yield 𝑂(𝑡) by single point crossover & two point crossover. Mutation 𝑃(𝑡) to yield 𝑂(𝑡) by scramble, insertion and swap mutation. Evaluation 𝑂(𝑡). Step 5: Select 𝑷(𝒕 + 𝟏) from 𝑷(𝒕) and 𝑶(𝒕) by rank selection mechanism. Step 6: Check the stop criterion while (not termination criterion) repeat step 4 and step 5; Output: minimum total cost; Figure 2 - The proposed st-GA and pb-GA procedure for the FCTP. The general structure of a genetic algorithm can be assumed that first of all a mechanism to convert the answer of each problem to a chromosome should be defined. Then, a set of chromosomes, which are in fact a set of answers the problem, are considered as the initial population. After defining the initial answer, new chromosomes so called the child should be created by using the genetic operation. The operation is divided into two main types of crossover and mutation. As well as, the two concepts of crossover operator and mutation operator

are frequently used for the selection of chromosomes which should play the role of parents, that the operator is also defined. After creating the population of children, the best of chromosomes should be selected by using the evaluation. The selection process is based on the fitness value of each string. In fact, evaluation process is the most important debate on the selection process. Accordingly, after the repetition of several generations, the best generation that is the optimal answer of the problem will be created. Four fundamental steps are mostly used in GA: reproduction, selection mechanism, crossover and mutation.

Experimental design Taguchi [41] presented a new approach to the design of experiments. Taguchi, as the first provider of parameter design method, proposed an engineering approach to design a product or process that aims to minimize the changes and sensitivity of disturbance factors. The first goal of an efficient parameter design is to identify and set factors that minimize the changes of answer variable and the next goal is to identify controllable and uncontrollable factors. The ultimate goal of this method is to find the optimal combination of controllable factors. Taguchi has created special set of overall designs for factorial experiments that cover most applications. Orthogonal arrays are part of the set designs. The use of these arrays helps us in determining the minimum number of needed experiments for a series of factors. Instances To cover various types of problems, we considered several levels of influencing inputs. At first, we generated random problem instances for m  5,10,15, 20,30,50 suppliers and n  10,15, 20,30,50,100, 200 customers, respectively. We consider the instances from Hajiaghaei-Keshteli et al. [9] and develop nine instances in three sizes small, medium and large sizes. After specifying the size of problems in a given instance, considering the significant influence of the fixed costs to the solution for each size, four problem types (A–D) are employed. For a given problem size, problem types differ from each other by the range of fixed costs. There are 9  4  36 instances, in which the variable costs have discrete values from 3 to 8 and the fixed costs arise from type A to type D. The problem sizes, types, suppliers/customers, and fixed costs ranges are shown in Table 3. Parameter setting In each algorithm we face several parameters and each of them should be assigned by a discrete or continuous value. In the other hand, we know that the correct choice of the parameters strongly effect on the performance of an algorithm. So, in this section, we study the performance of the algorithm in dealing with different parameters.

www.SID.ir

Archive of SID

Table 3 - Fixed-charge transportation test problems characteristics. Problem size Total supply Problem type Range of variable costs Lower limit Upper limit 5×10 5000 A 3 8 Small 10×10 8000 B 3 8 10×20 10000 C 3 8 15×15 15000 D 3 8 Medium 10×30 15000 20×30 20000 50×50 50000 Large 30×100 30000 50×200 50000

Table 4 - Factors and their levels in GA algorithm. Factors

GA symbols

Type of crossover

A

Type of mutation

B

Population size

C

Crossover percentage

D

Mutation probability

E

GA Levels A (1) – one-point crossover A (2) – two-point crossover B (1) – Swap B (2) – Scramble B (3) – Insertion C (1) – 20 C (2) – 30 C (3) – 40 D (1) – 60% D (2) – 70% D (3) – 80% E (1) – 0.05 E (2) – 0.1 E (3) – 0.15

RPD 

A lg sol  Minsol 100 Minsol

-8 -12 -16 -20 -24 A

D 1 2 3 1 2 3 2 3 1 3 1 2 2 3 1 3 1 2

E 1 2 3 2 3 1 1 2 3 3 1 2 3 1 2 2 3 1

B

C

D

E

Figure 3 - Mean S/N ratio plot for each level of the factors in GA algorithm.

RPD

Table 5 - The modified orthogonal array L18. Trial A B C 1 1 1 1 2 1 1 2 3 1 1 3 4 1 2 1 5 1 2 2 6 1 2 3 7 1 3 1 8 1 3 2 9 1 3 3 10 2 1 1 11 2 1 2 12 2 1 3 13 2 2 1 14 2 2 2 15 2 2 3 16 2 3 1 17 2 3 2 18 2 3 3

(14)

Where Algsol and Minsol are the obtained objective value for each replication of trial in a given instance and the obtained best solution respectively. After converting the objective values to RPDs, the mean RPD is calculated for each trial. We also transform the mean RPDs into the S/N ratios. The S/N ratios of trials are averaged in each level. In accordance with the Table 4, all investigated and effective parameters in algorithm and all its modes are listed. Due to the parameters and levels of the algorithm we reach to L18 orthogonal array as shown in the Table 5. Besides, Figures 3 and 4, shows the best levels for GA parameters as 2, 2, 2, 2 and 2, respectively, according to their alphabetical order. These results are the same in both spanning tree and priority version of the algorithm.

S/N

In order to examine the performance of the presented algorithm thirty six test problems with various sizes are solved. The experiments on GA was based on the L18 orthogonal array. Also in order to achieve the more reliable results five replications were done for each trial. In addition, since we deal various scales of objective functions in each instance, we utilize the relative percentage deviation (RPD) according to formula (14) for each instance.

Range of fixed costs Lower limit Upper limit 50 200 100 400 200 800 400 1600

15 12 9 6 3 A

B

C

D

E

Figure 4 - Mean RPD plot for each level of the factors in GA algorithm. Data generation In order to apply all-units discount and incremental discount, supply quantities of the factories were equally divided into four ranges according to their capacity and when the customer’s order values are in each of these intervals, it will be included the percent discounts on both fixed and variable charges. In this study, the proposed intervals and percentages of discounts have been

www.SID.ir

Archive of SID

0.00  xij  0.25  si

cˆij  cij . 1  0.0  ,

step 2 if

fˆij  fij . 1  0.00 

0.25  xij  0.50  si

cˆij  cij . 1  0.1 ,

step3 if

0.50  xij  0.75  si

cˆij  cij . 1  0.2  ,

step 4 if

fˆij  fij . 1  0.05 

(15)

fˆij  fij . 1  0.10 

0.75  xij  1.00  si

cˆij  cij . 1  0.3 ,

fˆij  fij . 1  0.15 

Then in order to evaluate the performance of proposed Genetic algorithm, the test which was raised to set the parameter in Instances section, was used. In this sample, not only the seven problem sizes raised by HajiaghaeiKeshteli et al. [9] have been used, but also we classified the problems at three small, medium and large levels with the addition of two new problems. Given the significant impact of fixed charges in these problems, the experiment is formed of 9 different sizes in four types of (A-D). Variable charges in all aspects among [3-8] and fixed charges vary according to the type of problem from A-D. Data considered for the test has been shown in Table 3. Numerical experiments To evaluate the performance of suggested solutions in allunits discount and incremental discount models, nine problems at three small, medium and large levels with different values of supplier (m) and the customers (n) in four types A-D have been produced. To establish the same conditions in the proposed metaheuristic, search time has been determined according to equation 0.02  m  n (seconds). Hence with this condition stop, search time increases by increasing the size of the problem. Also due to the random nature of metaheuristic algorithm to achieve reliable results, 30 repetitions are considered for each of the samples. Then using these repetitions, the best cost, average cost and worst cost are specified and the average cost is used to compare the results in terms of response quality. Figures 5 and 6 shows all-units discount and incremental discount computational results respectively. These results obtained from the quality of the answer given to the suggested solution methods for test sample. As seen in the figures, Genetic algorithm which its representation is encrypted based on the priorities has better performance than the Genetic algorithm which was based on the spanning tree. Objective function in Genetic algorithm due

400 350 300 250 200 150 100 50 0

340.10 214.82 72.13 88.76 47.73 57.50 23.62 30.22 145.72

182.75 210.78 122.52

49.27 59.33 67.45 22.51 27.04 40.39

pb-GA

st-GA

Figure 5 - Comparison of the mean of optimal solution for different problem sizes for the FCTP with All-units discount mechanism.

Thousands

step1 if

to random nature of these methods represents the average value of objective function from 30 times of the algorithm implementation. In order to compare the results of suggested solution methods, the average objective function for each test sample sizes obtained has been shown in Figures. In order to the overall performance of each solution methods is also determined by calculating the average values. From this Figures, we can conclude that genetic algorithm based on priority representation (pb-GA) has better performance than other solution method.

Thousands

considered in accordance with the equations (15) as samples. For example, if the customer order is between 25 to 50 percent of the production of supplier in the second stage, discount is applied in the variable charge as (  10%) and in the fixed charge as (   5%) . So, other steps can be defined as well.

400 350 300 250 200 150 100 50 0

349.29 242.05 188.65 76.30 96.24 50.48 65.70 165.69 25.52 34.51

218.30 127.77

55.46 62.78 74.04 24.20 30.68 43.46

pb-GA

st-GA

Figure 6 - Comparison of the mean of optimal solution for different problem sizes for the FCTP with incremental discount mechanism. Finally, to compare the results of suggested solution methods, the average gap for each test sample sizes obtained has been shown in Figures 7 and 8. As seen in the figure, Genetic algorithm which their representation are encrypted based on the priorities has better performance than the Genetic algorithm which was based on the spanning tree. As it is obvious, pb-GA exhibits robust performance, meanwhile the problems size increases. It also shows remarkable performance improvements of pbGA in large size problems versus GA.

www.SID.ir

Archive of SID

62.41

95% CI for the Mean

49.04 50.34

50

32.88 12.57 5.90 0.91

0.71

40

19.24 18.05 22.66 0.90

1.16

0.89

0.99

1.10

0.79

29.5288

30

0.65 RPD

70.00 60.00 50.00 40.00 30.00 20.00 10.00 0.00

20

10

pb-GA

st-GA

0.698544

0

Figure 7 – Comparison of the mean of optimal solution gap for different problem sizes for the FCTP with All-units discount mechanism. 70.00 60.00 50.00 40.00 30.00 20.00 10.00 0.00

60.52 47.11 48.31 31.14 13.09 5.88 0.40

0.53

22.78 17.27 19.67 0.95

1.02

1.02

pb-GA

0.89

0.70

0.45

0.32

st-GA

Figure 8 - Comparison of the mean of optimal solution gap for different problem sizes for the FCTP with incremental discount mechanism. In order to more study about the results obtained, analysis of variance (ANOVA) was used. The statistical results show that there is a significant difference in the performance of different solution methods. Figures 9 and 10 are showing the average charts and least significant difference (LSD) for the proposed solution methods at 95% confidence level. As can be seen from these Figures, it can be concluded the genetic algorithm based on priority representation (pb-GA) has better performance than the other solution method. 95% CI for the Mean 50

40

30.3416

RPD

30

20

10 0.900969

0 pb-GA

st-GA

Figure 9 - Means plot and LSD intervals for the proposed solution methods in All-units discount.

pb-GA

st-GA

Figure 10 - Means plot and LSD intervals for the proposed solution methods in incremental discount mechanism.

Conclusion and future works Fixed charge transportation problem is one of the nonpolynomial hard problems that its solution by old methods is difficult. In this paper, this problem has been initially considered with two mechanisms of all-units discount and incremental discount. To solve the fixed charge transportation problem model by applying mechanisms of all-units discount and incremental discount, genetic algorithms based on spanning tree-based representation and priority-based representation has been used. Therefore, an overview of the history of fixed charge transportation problems and most widely used types of it was initially conducted in this paper and classifications for fixed charge transportation problems and its methods of solving are provided. Then, the fixed charge transportation problem model by applying two mechanisms of all-units discount and incremental discount was developed. In the following, spanning tree-based representation and prioritybased representation and solution mechanism of genetic algorithm was examined. In the next section, Taguchi experimental design was used to adjust the parameters of the proposed genetic algorithm and better performance of it. Finally by providing several experimental prototypes, performance of solution methods in different models was reviewed and evaluated. To compare the overall performance of response spanning tree-based representation and response priority-based representation and efficiency of the proposed genetic algorithm, an experiment with thirty-six problems in different levels and dimensions has been proposed. For each of all-units discount and incremental discount models, results show the very sensible superiority of response priority-based representation to the response spanning tree-based representation. The future work is to extend our approach to the case of multistage, other optimization objectives or fuzzy data in which direct costs, fixed charges, supplies, demands, conveyance capacities and transported quantities (decision variables) are fuzzy. Also, we recommend utilizing recent and strong metaheuristics and the other representation methods in this research area.

www.SID.ir

Archive of SID

References [1] Hirsch, Warren M., and George B. Dantzig. "The fixed charge problem." Naval Research Logistics Quarterly 15, no. 3 (1968): 413-424. [2] Balinski, Michel L. "Fixed‐cost transportation problems." Naval Research Logistics Quarterly 8, no. 1 (1961): 41-54. [3] Klose, Andreas. "Algorithms for solving the singlesink fixed-charge transportation problem." Computers & Operations Research 35, no. 6 (2008): 2079-2092. [4] Aguado, Jesús Sáez. "Fixed Charge Transportation Problems: a new heuristic approach based on Lagrangean relaxation and the solving of core problems." Annals of Operations Research 172, no. 1 (2009): 45-69. [5] Loch, Gustavo Valentim, and A. C. L. Silva. "A computational experiment in a heuristic for the Fixed Charge Transportation Problem." International Refereed Journal of Engineering and Science 3, no. 4 (2014). [6] Sun, Minghe, Jay E. Aronson, Patrick G. McKeown, and Dennis Drinka. "A tabu search heuristic procedure for the fixed charge transportation problem." European Journal of Operational Research 106, no. 2 (1998): 441-456. [7] Jawahar, N., Angappa Gunasekaran, and N. Balaji. "A simulated annealing algorithm to the multi-period fixed charge distribution problem associated with backorder and inventory." International Journal of Production Research50, no. 9 (2012): 2533-2554. [8] Yadegari, Ehsan, Mostafa Zandieh, and Hesamaddin Najmi. "A hybrid spanning tree-based genetic/simulated annealing algorithm for a closedloop logistics network design problem." International Journal of Applied Decision Sciences 8, no. 4 (2015): 400-426. [9] Hajiaghaei-Keshteli, M., S. Molla-Alizadeh-Zavardehi, and Reza Tavakkoli-Moghaddam. "Addressing a nonlinear fixed-charge transportation problem using a spanning tree-based genetic algorithm." Computers & Industrial Engineering 59, no. 2 (2010): 259-271. [10] Lotfi, M. M., and Reza Tavakkoli-Moghaddam. "A genetic algorithm using priority-based encoding with new operators for fixed charge transportation problems." Applied Soft Computing 13, no. 5 (2013): 2711-2726. [11] Tari, Farhad Ghassemi, and Zahra Hashemi. "A priority based genetic algorithm for nonlinear transportation costs problems." Computers & Industrial Engineering 96 (2016): 86-95. [12] Molla-Alizadeh-Zavardehi, S., M. HajiaghaeiKeshteli, and Reza Tavakkoli-Moghaddam. "Solving a capacitated fixed-charge transportation problem by artificial immune and genetic algorithms with a Prüfer number representation." Expert Systems with Applications 38, no. 8 (2011): 10462-10474. [13] El-Sherbiny, Mahmoud M., and Rashid M. Alhamali. "A hybrid particle swarm algorithm with artificial

immune learning for solving the fixed charge transportation problem." Computers & Industrial Engineering 64, no. 2 (2013): 610-620. [14] Yaghini, Masoud, Mohsen Momeni, and Mohammadreza Sarmadi. "A simplex-based simulated annealing algorithm for node-arc capacitated multicommodity network design." Applied Soft Computing 12, no. 9 (2012): 2997-3003. [15] Xie, Fanrong, and Renan Jia. "Nonlinear fixed charge transportation problem by minimum cost flow-based genetic algorithm." Computers & Industrial Engineering 63, no. 4 (2012): 763-778. [16] Gen, Mitsuo, and Runwei Cheng. "Foundations of genetic algorithms." Genetic Algorithms and Engineering Design (1997): 1-41. [17] Hajiaghaei-Keshteli, M. "The allocation of customers to potential distribution centers in supply chain networks: GA and AIA approaches." Applied Soft Computing 11, no. 2 (2011): 2069-2078. [18] Mahmoodi-Rad, A., S. Molla-Alizadeh-Zavardehi, R. Dehghan, M. Sanei, and S. Niroomand. "Genetic and differential evolution algorithms for the allocation of customers to potential distribution centers in a fuzzy environment." The International Journal of Advanced Manufacturing Technology 70, no. 9-12 (2014): 19391954. [19] Gen, Mitsuo, and Yinzhen Li. "Spanning tree-based genetic algorithm for the bicriteria fixed charge transportation problem." In Evolutionary Computation, 1999. CEC 99. Proceedings of the 1999 Congress on, vol. 3. IEEE, 1999. [20] Gen, Mitsuo, and Runwei Cheng. Genetic algorithms and engineering optimization. Vol. 7. John Wiley & Sons, 2000. [21] Jo, Jung-Bok, Yinzhen Li, and Mitsuo Gen. "Nonlinear fixed charge transportation problem by spanning tree-based genetic algorithm." Computers & Industrial Engineering 53, no. 2 (2007): 290-298. [22] Gen, Mitsuo, Fulya Altiparmak, and Lin Lin. "A genetic algorithm for two-stage transportation problem using priority-based encoding." OR spectrum28, no. 3 (2006): 337-354. [23] Hwang, Rea Kook, Hiroshi Katayama, and Mitsuo Gen. "U-shaped assembly line balancing problem with genetic algorithm." International Journal of Production Research 46, no. 16 (2008): 4637-4649. [24] Lee, Jeong-Eun, Mitsuo Gen, and Kyong-Gu Rhee. "Hybrid priority-based genetic algorithm for multistage reverse logistics network." Industrial Engineering and Management Systems 8, no. 1 (2009): 14-21. [25] Kowalski, Krzysztof, and Benjamin Lev. "On step fixed-charge transportation problem." Omega 36, no. 5 (2008): 913-917. [26] El-Sherbiny, Mahmoud Moustafa. "Alternate mutation based artificial immune algorithm for step fixed charge transportation problem." Egyptian Informatics Journal 13, no. 2 (2012): 123-134. [27] Altassan, Khalid M., Mahmoud M. El-Sherbiny, and

www.SID.ir

Archive of SID

Bokkasam Sasidhar. "Near Optimal Solution for the Step Fixed Charge Transportation Problem." Applied Mathematics & Information Sciences 7, no. 2 (2013): 661-669. [28] Molla-Alizadeh-Zavardehi, S., S. Sadi Nezhad, Reza Tavakkoli-Moghaddam, and M. Yazdani. "Solving a fuzzy fixed charge solid transportation problem by metaheuristics." Mathematical and Computer Modelling 57, no. 5 (2013): 1543-1558. [29] Molla-Alizadeh-Zavardehi, S., A. Mahmoodirad, and M. Rahimian. "Step fixed charge transportation problems via genetic algorithm." Indian Journal of Science and Technology 7, no. 7 (2014): 949-954. [30] Sanei, Masoud, Ali Mahmoodirad, Sadegh Niroomand, Ali Jamalian, and Shahin Gelareh. "Step fixed-charge solid transportation problem: a Lagrangian relaxation heuristic approach." Computational and Applied Mathematics (2015): 1-21. [31] Gen, Mitsuo, and Yin-Zhen Li. "Spanning tree-based genetic algorithm for bicriteria transportation problem." Computers & industrial engineering 35, no. 3 (1998): 531-534. [32] Kim, J. R., Mitsuo Gen, and K. Ida. "Bicriteria network design using a spanning tree-based genetic algorithm." Artificial Life and Robotics 3, no. 2 (1999): 65-72. [33] Gen, Mitsuo, Anup Kumar, and Jong Ryul Kim. "Recent network design techniques using evolutionary algorithms." International Journal of Production Economics 98, no. 2 (2005): 251-261. [34] Syarif, Admi, and Mitsuo Gen. "Double spanning

tree-based genetic algorithm for two stage transportation problem." International Journal of Knowledge Based Intelligent Engineering Systems 7, no. 4 (2003): 214-221. [35] Kim, Kwan Woo, Mitsuo Gen, and Genji Yamazaki. "Hybrid genetic algorithm with fuzzy logic for resource-constrained project scheduling." Applied Soft Computing 2, no. 3 (2003): 174-188. [36] Bahrampour, Peyman, Mansoureh Safari, and Mahmood Baghban Taraghdari. "Modeling MultiProduct Multi-Stage Supply Chain Network Design." Procedia Economics and Finance 36 (2016): 70-80. [37] Zandieh, M., and Abdolhay Chensebli. "Reverse logistics network design: a water flow-like algorithm approach." OPSEARCH (2016): 1-26. [38] Hamzadayi, Alper, and Gokalp Yildiz. "A genetic algorithm based approach for simultaneously balancing and sequencing of mixed-model U-lines with parallel workstations and zoning constraints." Computers & Industrial Engineering 62, no. 1 (2012): 206-215. [39] Chitra, Chinnasamy, and P. Subbaraj. "A nondominated sorting genetic algorithm solution for shortest path routing problem in computer networks." Expert Systems with Applications 39, no. 1 (2012): 1518-1525. [40] Holland, John H. Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence. U Michigan Press, 1975. [41] Taguchi, Genichi. Introduction to quality engineering: designing quality into products and processes. 1986.

www.SID.ir