New Heuristic Approach to Solve the Fixed Charge Transportation Problem with Incremental Discount Komeil Yousefi1,*, Ahmad J. Afshari2, Mostafa Hajiaghaei-Keshteli3 1
Department of Industrial Engineering, Shomal University, Amol, Iran Lecturer.Email:
[email protected]
2
Department of Industrial Engineering, Shomal University, Amol, Iran Email:
[email protected]
3
Department of Industrial Engineering,University of Science and Technology of Mazandaran, Behshahr, Iran Email:
[email protected]
Abstract 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 new heuristic along with well-known metaheuristic to solve the FCTP with incremental discount on both fixed and variable charges. In addition, since the researchers recently used the priority-based representation to encode the transportation graphs and achieved very good results, we consider this representation in metaheuristic and compare the results with the proposed heuristic. Furthermore, we apply the Taguchi experimental design method to set the proper values of algorithm in order to improve its performance. Finally, computational results of heuristic and metaheuristic, both in terms of the solution quality and computation time, are studied in different problem sizes. Keywords: Fixed charge transportation problem, Incremental discount, Heuristic, Priority-based.
1. 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. In this particular type of linear programming problems, goods are only allowed to move straight from the source to the destination. In the fixed charge transportation problems (FCTP), 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.
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2. 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]). 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 [16]. Hwang et al. compared system of direct and U-shaped assembly lines and used the genetic algorithm and priority-based representation to solve assembly line balancing problem [17]. 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 [18]. 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]. In this study, we raised fixed charge transportation problems with considering incremental discount 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. After modeling the desired problem, we proposed a new heuristic method along with known metaheuristic such as genetic to solve the proposed model. Since the researchers recently used the priority-based representation to encode the transportation graphs and achieved very good results, we consider this representation in metaheuristic and compare the results with the proposed heuristic. Proposed heuristic is developed based on the nature of the problem which is detailed in related section. Furthermore, Taguchi experimental design method is utilized to set and estimate the proper values of the algorithms' parameters to improve their performance. Finally, for evaluated and compared the performance of proposed solutions 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 previous data and reach to nine different problem sizes in three levels; small, medium and large.
3. Mathematical model and descriptions 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 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.
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3.1. Notations Index list i source indice j Parameters list m number of sources ai capacity of source cij variable transportation cost
depot indice
cˆij
variable transportation cost by applying discount
fˆij
fixed transportation cost by applying discount
p
percentage discount of variable charges on level p
p
discount level
n bj
number of depots demand of depot
f ij
fixed transportation cost
p percentage discount of fixed charges on level p Decision variables list xij unknown quantity to be transported from source i to depot j a binary variable which is 1 if xij 0 yij 3.2. Mathematical model Mathematically this problem may be stated as follows: Minimize Z
m
n
(cˆ .x ij
ij
fˆij . yij )
(1)
i 1 j 1
s.t. n
x
ij
ai ; i 1, 2,...,m
(2)
b j ; j 1, 2,..., n
(3)
j 1 m
x
ij
i 1
xij 0; i, j
(4)
xij 0 1 yij ; i, j 0 otherwise
(5)
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. 3.3. Incremental discount model In this study, a kind of discount mechanism called incremental discount model has 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
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all-units discount model, when the amount of demand (b) comes to a certain extent of supply (a), level (p), transmitter applies discounts on variable charges as ( p ) percent and on fixed charges as ( p ) . In this case, the cost of product order is as the Table 1. 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 ) . [19] Table 1 - Incremental discount representation Discount level Total transportation cost (cij 0 xij ) fij 0 0 1 2
p
[cij 0 a1 a0 ] [cij1 a2 a1 ] [cij 2 xij a2 ] fij 2 [cij 0 a1 a0 ] [cij1 xij a1 ] fij1
p cij k 1 ak ak 1 cij p xij a p k 1
fij p
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 discount points and display with (a p ) . In these symbols, it is assumed that (a1 ... a p 1 a p ) and since usually the price of goods units will be lowers against the order more, we will have relations (6) to (7). cij p ... cij1 cij 0 fij p ... fij1 fij 0 (6) Thus, fixed charge and variable charge will be considered as formulas (8) and (9). cˆij p cij (1 a p ) fˆij p fij (1 p ) (8)
(7) (9)
4. Proposed heuristic Integration of the fixed costs with the variable costs, that we are called it the consolidated cost (cc), is the basis of the proposed heuristic. This means that we integrate these two types of cost by two methods to achieve the cc. Then, we solved the problem using the cc like a classical transportation problem and calculate FCTPs objective function after obtaining xij and replace them into the main objective function. In a nutshell, we propose the heuristic based on the two types of costs exist in the FCTP. We consider them from different points of view that they influence on the objective functions. To explain exactly about the developed heuristic and four methods to achieve the cc, we depict the procedure 1, with the following Indexes and parameters. i source indice j depot indice p discount level m capacity of source n demand of depot bj ai variable transportation cost fixed transportation cost cij adjusted variable transportation cost associated with route (i,j) f ij
adjusted fixed transportation cost associated with route (i,j)
cˆij
variable transportation cost associated with route (i,j) by applying discount in level p
4
fˆij
fixed transportation cost associated with route (i,j) by applying discount in level p
p
percentage discount of variable charges on level p
p
percentage discount of fixed charges on level p
rij
rating of transported product cost from source i to depot j
trij
total rating of transported product cost for source i and depot j
frij
final rating of transported product cost from source i to depot j
ccij
consolidated cost The decision variables are: xij unknown quantity to be transported from source i to depot j a binary variable which is 1 if xij 0 yij Procedure 1: Heuristic Input: indices, decision variables and parameters Output: The amounts of transported product from source i to depot j (xij ) and the lowest total cost; Step 1: obtaining adjusted costs: cij 0 cij1 cij p cij p 1 Step 2: Method I: Step 1: ccijI
fij
Min a ,b c f i
j
ij
Min ai ,b j
ij
fij 0 fij1 fij p p 1
, for each i,j;
Method II: Step 1: rmini, j cij fij , for each i,j;
rmaxi, j cij max ai ,b j
rava i, j cij min ai ,b j
f
f
ij
, for each i,j;
, for each i,j;
ij
Step 2: trmin i , j
n
r
j 1 i 1,..,m
min i , j
m
r
i 1 j 1,..,n
trmax i , j
min i , j
trava i , j
n
rava i , j
j 1 i 1,..,m
Step 3:
5
m
r
i 1 j 1,..,n
ava i , j
n
r
j 1 i 1,..,m
max i , j
m
r
i 1 j 1,..,n
max i , j
frmin i , j
trmin i , j rmin i , j
frmax i , j
, for each i,j;
frava i , j
trava i , j rava i , j
trmax i , j rmax i , j
, for each i,j;
, for each i,j;
Step 4:
ccijII
frmin i , j
frmax i , j frava i , j frmini , j
, for each i,j;
Step 3: Solving like the classical transportation problem (TP) using each consolidated costs (ccijI and ccijII ) ; Step 4: obtaining amount of xij from each consolidated costs in step 3; Step 5: set discount level (p), percentage discount of variable charges ( p ) and percentage discount of fixed charges ( p ) according to amount of xij from step 4;
Step 6: calculation of the total cost using cˆ ijp xij ˆfijp yij ; Step 7: choosing the lowest total cost from step 6;
5. Proposed metaheuristic To explain the proposed metaheuristic, at first we describe priority-based encoding scheme. We employ this representation method into the developed genetic algorithm. While most evolutionary algorithms use a random procedure to generate a set of initial solutions, priority representation has been used to achieve initial feasible solutions in this study. 5.1. Priority-based representation Priority-based encoding is a relatively new and is considered as strong method used in recent works. Initially Gen et al. [16] used this this type of representation in transportation problems and then Lotfi and Tavakkoli-Moghaddam [10] employed it for FCTP. 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. Consequently, the corresponding transportation tree is then generated by sequential arcs appending between sources and depots. 5.2. Genetic algorithm Based on the role of genetics in nature and the natural evolution of living organisms, Holland [20] 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 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
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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. The structure of the proposed GA is given in Algorithm 1. Algorithm 1: The proposed GA procedure for the FCTP with Incremental Discount Step 1: Initialize the problem and GA parameters Input: the data instance of the optimization problem and GA parameters; Step 2: Initialize P t , by the 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 or two point crossover. Mutation P t to yield O t by scramble or insertion or 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;
6. Experimental design Taguchi [21] 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. 6.1. 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.
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6.2. 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. In order to examine the performance of the presented algorithms thirty six test problems with various sizes are solved. We implement the algorithms in Matlab 14a, and run on a PC with 2.5 GHz Intel Core i5-3210M CPU and 4GB of RAM memory. The experiment on the 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 (10) for each instance. Using the average of RPD measures of trials, the parameters and operators that have minimum RPD average are selected as the best ones. Alg sol - Minsol RPD = ×100 (10) Minsol 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 the Taguchi method, the S/N ratio of the minimization objectives is as such formula (11). The aim is to maximize the signal-to-noise ratio. (11) S/ N ratio = - 10 log10 (objective function)2 In accordance with the Table 2, all investigated and effective parameters in GA and all its modes are listed. Table 2 - 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
Using an L18 orthogonal array there are totally 18 experiments to be conducted. Therefore, the parameters of GA were set as follows: Type of crossover two point crossover, Type of mutation Scramble, Population size 30, Crossover percentage 70%, Mutation probability 0.1.
7. Data generation In order to apply 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 considered in accordance with the equations (12) as samples. For example, 8
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. level 1 if 0.00 si xij 0.25 si cˆ ij cij .( 1 0.0 ) , ˆf ij fij .( 1 0.00 ) level 2 if 0.25 si xij 0.50 si cˆ ij cij .( 1 0.1 ) , ˆfij fij .( 1 0.05 ) (12) level 3 if 0.50 si xij 0.75 si cˆ ij cij .( 1 0.2 ) , ˆfij fij .( 1 0.10 ) level 4 if 0.75 si xij 1.00 si cˆ ij cij .( 1 0.3 ) , ˆfij fij .( 1 0.15 ) Then in order to evaluate the performance of proposed heuristic and metaheuristic algorithm, the test which was raised to set the parameter in section (6.1), was used. In this sample, not only the seven problem sizes raised by Hajiaghaei-Keshteli 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. We implement the proposed heuristic and metaheuristic in Matlab 14a, and run on a PC with 2.5 GHz Intel Core i5-3210M CPU and 4GB of RAM memory. Data considered for the test has been shown in Table 3. Table 3 - Fixed-charge transportation test problems characteristics. Problem size Total supply Problem type Range of variable costs Lower limit Upper limit Small 5×10 5000 A 3 8 10×10 8000 B 3 8 10×20 10000 C 3 8 Medium 15×15 15000 D 3 8 10×30 15000 20×30 20000 Large 50×50 50000 30×100 30000 50×200 50000
Range of fixed costs Lower limit Upper limit 50 200 100 400 200 800 400 1600
7.1. Numerical experiments To evaluate the performance of suggested solutions in incremental discount model, nine problems at three small, medium and large levels with different values of supplier (m) and the applicant (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 Eq. 0.001 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 charge, average charge and worst charge are specified and the average charge is used to compare the results in terms of response quality. In order to compare the results of suggested solution methods, the average gap for each test sample sizes obtained has been shown in Figure 1. Gap represents the percentage of deviations from the best solution. The best solution for each size of the test problem is the best objective function that it has been achieved by the all proposed solution methods. As seen in the figure, our proposed heuristic gives the best results.
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3.00 2.50 2.00 1.50 1.00 0.50 0.00
1.89 1.36
2.31 2.48
2.03 1.60 1.67
1.59
1.08
0.95 0.57
1.25 0.19 0.11
pb-GA
0.91 0.08 0.00 0.00
Heuristic
Figure 1. 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. Figure 2 shows the average chart and least significant difference (LSD) for the proposed solution methods at 95% confidence level. As can be seen from this Figure, it can be concluded the proposed heuristic based on consolidated cost has better performance than the other solution method. 95% CI for the Mean 2.0
GAP
1.5
1.58686
1.0
0.643321
0.5
0.0 pb-GA
Heuristic
Figure 2. Means plot and LSD intervals for the proposed solution methods in incremental discount mechanism.
8. Conclusion Fixed charge transportation problem is one of the non-polynomial difficult problems that its solution by old methods is difficult. A classification for fixed charge transportation problems and its solution methods in the literature is provided. So, in this paper, this problem has been initially considered with incremental discount. To solve the fixed charge transportation problem by applying incremental discount, genetic algorithm with priority-based representation has been used. Also new heuristic method has been proposed due to the nature of the problem, as well. Besides, Taguchi experimental design was used to adjust the parameters of the proposed GA and better performance of them. Finally, by providing several experimental tries, performance of solution methods in different models was reviewed and evaluated. To compare the overall performance of the proposed heuristic and metaheuristic algorithm, an experiment with thirty-six problems in different levels and dimensions has been proposed. results show that our heuristic which were proposed for these models enjoy relatively better performance compared with Genetic algorithm that are written base priority. This superiority in problems with medium and large size is more tangible. In addition not only heuristic algorithm provides better responses, but also they spend much less computation time compared to GA. 10
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