This research aims at solving the special case of multistate assignment problem. The problem includes many special characteristics which are not normally included in the assignment problem. There are many types and conditions of vehicles included in the planning and different road conditions of traveling, which would have different effects on fuel consumption, which is the objective function of the study. The proposed problem is determined as a large and complicated problem making the optimization software unable to find an optimal solution within the proper time. Therefore, the researchers had developed a method for determining the optimal solution by using particle swarm optimization (PSO) in which the methods were developed for solving the proposed problem. This method is called the modified particle swarm optimization (modified PSO). The proposed method was tested with three groups of tested instances, i.e., small, medium, and large groups. The computational result shows that, in small-sized and medium-sized problems, the proposed method performed not significantly different from the optimization software, and in the large-sized problems, the modified PSO method gave 3.61% lower cost than the cost generated from best solution generated from optimization software within 72 hours and it gave 11.03% better solution than that of the best existing heuristics published so far (differential evolution algorithm).
The egg is very important to the Thai economy and its people. Since it is a high nutrition fact food and can be cooked in many dishes, both main courses and desserts, it is therefore widely consumed, resulting in high demand for hen eggs throughout the year and it is likely to increase steadily. During 2012–2018, the average domestic egg consumption tended to increase at a rate of 6.13% per year. In 2018, the consumption of eggs was equal to 14,823.24 million, which is increased from 13,534.98 million eggs at a rate of 9.52% from the year 2017. Egg is cheap compared to other protein resources and easy for cooking. In addition, the government and private sectors have campaigned on the egg consumption suitable for all ages. In 2012–2018, Thai egg production increased at a rate of 5.98% per year, according to the increase in consumption demand. In 2018, the eggs were produced in the amount of 14,915.82 million, which is increased from 13,724.42 million eggs at a rate of 8.68% from the year 2017. Since there was the efficient management of egg farms, resulting in increased productivity, the export value of fresh eggs in 2018 was 92.58 million eggs or worth 319.42 million baht, which tended to increase at the rate of 6.82% per year. In 2018, 4,356.92 tons of chicken eggs were exported which is equivalent to 425.41 million baht. Japan was the main export market.
This research studied the problem of chicken transportation, which was the multistage assignment problem. For the case study, an appropriate vehicle was assigned to transport the chickens directly from its farms to the egg farms, for the purpose of finding the answer with the lowest assignment cost. The mathematical models for the multistage assignment problem were developed which are suitable for the case study. Then, the estimating methods to find the optimal answer were also developed by employing the particle swarm optimization (PSO). When transporting chickens to the purchased farms, some factors must be considered, such as transportation standards, time, and temperature, and also the chickens apart from multiple sources must not blend up. These factors may affect the quality of the chicken. Therefore, if the assigned vehicle is appropriate and meets the needs of the chicken farms, it shall be used to send the chickens directly to the egg farms, without transporting chickens from other farms. The resulting cost of assignments or production costs is at the lowest value, which benefits the chicken farm concerning the decrease in production costs. Since each farm is responsible for all the costs incurred for transportation, they must be managed correctly and efficiently by establishing production schedules and assignments on farms, in both the allocation of chickens and the size of trucks that are used to deliver efficiently and quickly. It shall create the highest possible operating profit and the overall limitation of the chicken production will be raised. As the cost is decreased, chicken and egg farms can employ the time for doing other activities such as feeding, vaccination, or research and development for their farms.
Road transport is an essential means and widely used in the transshipment of agricultural goods in Thailand. However, this type of transportation depends on energy from fuels, which has high energy consumption. Regarding the costs of transportation in 2018, the overall fuel usage of road transportation increased by 0.37% when compared with the previous year. When considering diesel and gasoline usage, it increased by 2.68% and 3.43%, respectively. Besides, the data showed the trends in increasing road transport costs each year (Energy Policy and Planning Office) [
Previously, Srivarapongse and Pijitbanjong [
The research motivation is to increase the profitability of farmers and all stakeholders related to the broiler industry, by reducing operational costs from the current situation. The contributions of this paper are as follows: (1) This research combines environmental care in the assignment problem by considering road categories that impact fuel consumption. This has been a new feature for this kind of problem. (2) The proposed method is a modified metaheuristic method which was developed for solving only this problem. (3) The case study, which is a real-world problem, occurred in Thailand. This paper fulfills the gap in the literature by determining the most appropriate amount of chickens and the most fitted trucks to transport chickens to the factories. Each type of truck had a different capacity to carry chickens.
This research consisted of the following structures. Section
Assignment problem (AP) means the problem of the task allocation to an agent. Each job is different and each employee has different expertise, resulting in unequal time spent in working, and the cost of assigning jobs to each one is different as well. Therefore, the problem is how to assign the task so that the total cost shall be the lowest, with an important condition that the assignment must be one on one basis. In other words, once an assignment has been assigned to an agent, it cannot be assigned to another. On the other hand, if an agent gets assigned a task, he/she does not get assigned another task.
The generalized assignment problem (GAP) is an extended type of the assignment problem (AP), which can assign multiple jobs to an employee, whereby the different assignments might require different resources. Ross and Soland [
GAP has been revealed extensively by plenty of researchers who are trying to solve practical problems. Similarly, Osorio and Laguna [
Dantzig[
The metaheuristic method is necessary to solve GAP problems. Its well-known methods were variable neighborhood search [
Particle swarm optimization (PSO) is a method that used natural imitation behavior by relying on the foraging role of animals, such as birds, fish, or other animals that have the behavior of finding food together. Each animal or particle shall find food by moving from the current point to the new point by using the direction and speed from particle best (
PSO methods have been extensively used to solve various problems, for instance, the assignment problems and multilevel location-allocation problems [
Green logistics have received attention from business organizations in terms of environmental and ecological factors. When making logistics decisions aside from general economic costs, these also included pollution, accidents, resource use, and the risk of climate change [
Nowadays, customers and business organizations place importance on environmental impacts due to the transportation of agricultural products which is a large type of energy consumption and greenhouse gas emissions (GHG). Many organizations are thus aware of the need to assess and reduce the environmental impacts of their activities and services. However, society is still concerned about the impact of human activities and the carelessness of using resources. There are a lot of research studies aiming to reduce the negative effects (i.e., fuel consumption and greenhouse gas emissions) from logistics activities to the environment such as pollution-routing problem (PRP) [
There were previous research studies on the death of chickens, its diets, and growth periods, which could increase production rates or reduce death and weight loss [
The case study is the multistage assignment problem, which is used to assign the appropriate vehicle type suitable for chicken transportation directly from the chicken farms to the egg farms. There are 4 categories of vehicles, which are truck that has ten, six, and four wheels and the modified version of the four-wheel truck aiming to keep the total cost minimum. The cost of the assignment in this case study consists of 3 parts, which are (1) the cost of transshipment depending on the category of vehicle with different fuel consumption rate and the distance in transportation, (2) the cost of transshipment depending on the category of road condition and the distance to travel, and (3) the opportunity cost.
Multistage assignment problem was studied by considering the appropriate vehicle type suitable for chicken transportation straight from the chicken farms to the egg farms, and the limitations are listed as follows: It is direct transportation in which there was no picking the chickens up from different farms and not being transport to other egg farms. The egg farms require quality control and good breed, to protect against communicable diseases. A chicken ranch may transport to many egg ranches. Once the chickens have been produced, all of them can be sold. The egg ranch may obtain chickens from many different farms, but must not over the capabilities of such a farm. Egg ranch shall get not less than 50 percent of the demand of their farm. The time required for transportation should not exceed 8 hours, beginning with loading chicken to the vehicle, transporting, and taking them down. The vehicles employed for transportation are acceptable for needs. Chicken ranch can employ more than one category of vehicles.
This assignment is the multistage assignment problem beginning by assigning the truck type (4 types). Each truck has a different capacity to transport chickens and a different fuel consumption rate, which causes different assignment costs. Therefore, the researcher aims to study the multistage assignment problem by considering the appropriate vehicle type to transport chickens straight from the chicken farms to the egg farms for the cheapest total cost.
The multistage assignment problem of the case study is to assign the layer hen farming to feed chickens starting from hatching and then raise them until they can be sold to the egg farming. Hatching and feeding of layer chickens require different technologies compared to raising chickens to lay eggs. The chicken farm consists of 40 farms, and all are capable of producing chickens differently, as shown in Table
Details of 40 chicken farms.
Types of chicken farms | Production capability (chickens/farm) | Number of farms | Total amount (chickens) |
---|---|---|---|
Large farm | 20,000 | 8 | 160,000 |
Medium farm | 10,000 | 12 | 120,000 |
Small farm | 5,000 | 20 | 100,000 |
Total | 35,000 | 40 | 380,000 |
Types of vehicles used in transportation.
Types of vehicles | Number of chickens | Rate of fuel consumption (liter/kilometer) |
---|---|---|
Ten-wheel truck | 12,000 | 3.2 |
Six-wheel truck | 8,000 | 5.0 |
Four-wheel truck | 4,000 | 8.0 |
Modified four-wheel truck | 2,000 | 10.0 |
Egg production can be obtained by using layer chickens that are obtained from the chicken farms in which there are 60 egg farms. Each farm has a different demand for chickens, but in total there is a demand for 388,000 chickens, as shown in Table
Details of egg farms.
Types of egg farms | Number of laying chickens | Number of farms | Total amount (chickens) |
---|---|---|---|
A | 10,000 | 18 | 180,000 |
B | 8,000 | 14 | 112,000 |
C | 5,000 | 10 | 50,000 |
D | 3,000 | 10 | 30,000 |
E | 2,000 | 8 | 16,000 |
Total | 28,000 | 60 | 388,000 |
The condition of the road used to transport shall affect the speed of the truck and its speed influences the rate of fuel consumption as well. The speed varies according to road conditions. For example, the main roads connecting the province are usually large with 4–6 traffic lanes. The vehicles can speed up than the roads with narrower lanes. The road surface also affects the speed, and the roads with a smooth surface, such as paved roads, can be driven faster than the roads with a rough surface (a concrete road, a damaged road, and bumpy road surface). In this research, the road condition is divided into 5 types [
Road types and fuel consumption rates.
Road types | Average speed (km/hr) | Fuel consumption rate (liter/kilometer) |
---|---|---|
A | <50 | 0.112 |
B | 51–60 | 0.090 |
C | 61–70 | 0.098 |
D | 71–80 | 0.098 |
E | 81–90 | 0.102 |
In general, the cost of transporting goods varies with distance, so the mathematical models shall try to find the shortest route as the answer to the problem, resulting in the lowest total cost. However, this research presents different perspectives with the purpose of finding the lowest grand fuel transportation cost. Therefore, the mathematical model shall try to choose the transportation route with the lowest fuel consumption first regardless of the distance. Calculation examples are shown for a better understanding of the pattern of the problem in this research as follows.
The road types among 6 farms are specified in Table
Example of road type metrics among 6 farms.
Farm | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
1 | — | A | C | B | B | C |
2 | A | — | B | C | A | A |
3 | C | B | — | B | B | C |
4 | B | C | B | — | D | A |
5 | B | A | B | D | — | C |
6 | C | A | C | A | C | — |
Example of distance metrics among 6 farms.
Farm | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
1 | — | 17 | 39 | 37 | 27 | 23 |
2 | 17 | — | 28 | 31 | 19 | 24 |
3 | 39 | 28 | — | 42 | 16 | 26 |
4 | 37 | 31 | 42 | — | 29 | 18 |
5 | 27 | 19 | 16 | 29 | — | 39 |
6 | 23 | 24 | 26 | 18 | 39 | — |
Example of fuel metrics for traveling among 6 farms.
Farm | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
1 | — | 1.904 | 3.822 | 3.330 | 2.430 | 2.254 |
2 | 1.904 | — | 2.520 | 3.038 | 2.128 | 2.688 |
3 | 3.822 | 2.520 | — | 3.780 | 1.440 | 2.548 |
4 | 3.330 | 3.038 | 3.780 | — | 2.842 | 2.016 |
5 | 2.430 | 2.128 | 1.440 | 2.842 | — | 3.822 |
6 | 2.254 | 2.688 | 2.548 | 2.016 | 3.822 | — |
Transportation solutions generally focus on finding the shortest route as it leads to the lowest transportation costs. Nevertheless, shorter routes may consume more fuel. For instance, travel distances from Farm 6 to Farm 2 and Farm 6 to Farm 3 are equal 24 kilometers and 26 kilometers, respectively. When considering the fuel used in both directions, which is 2.688 liters and 2.548 liters, it can be seen that the route from Farms 6–3 is longer than the route from Farms 2–6, but less fuel is used.
Indices Decision variables = 0, other cases = 0, other cases Parameters Objective function: Constraints:
This mathematical model was formulated to execute the multistage assignment problem. The purpose function consists of 3 cost terms: (1) the cost of transshipment depending on the category of vehicle with different fuel consumption rate and the distance in transportation, (2) the cost of transshipment depending on the type of road condition and the distance to travel, and (3) the opportunity cost occurred due to not full capacity of transporting.
Various limitations relating to the decision variables are as follows:
The multistage assignment problem of the case study is to assign the fitted vehicle type for the chickens transporting straight from the chicken ranch to the egg ranch by employing the cheapest total cost. Therefore, the author has applied and developed particle swarm optimization (PSO) and modified particle swarm optimization (modified PSO) in Sections
Particle swarm optimization is one of the most widely used methods, which was first mentioned by Kennedy and Eberhart in 1995. It can find the answer by using the cooperation between the particle and its swarm and each particle searches for the appropriate value from the current location. The direction and velocity for the next position are known by considering the former direction and speed from the particle best (
When the particle recognizes the new velocity (
The encoding method uses the same principle with the differential evolution (DE), which assigns a random number between 0 and 1 for every particle in each particle. Then the random number of each particle is sorted in ascending order. Particles with the lowest random number shall be chosen first. Random numbers of truck types are shown in Table
Initial particle of the truck type.
Particle | Initial particle of the truck type | ||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | |
1 | 0.3662 | 0.0330 | 0.8662 | 0.5309 | 0.4304 |
2 | 0.3738 | 0.2698 | 0.5456 | 0.9380 | 0.4344 |
3 | 0.6267 | 0.6280 | 0.5089 | 0.0926 | 0.3759 |
4 | 0.6083 | 0.7464 | 0.0687 | 0.8766 | 0.5045 |
Initial particle of the chicken farm.
Particle | Initial particle of the chicken farm | ||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | |
1 | 0.2597 | 0.1471 | 0.8420 | 0.4044 | 0.7124 |
2 | 0.3184 | 0.2953 | 0.4526 | 0.8481 | 0.1213 |
3 | 0.1232 | 0.0995 | 0.3469 | 0.3218 | 0.0456 |
4 | 0.9581 | 0.2122 | 0.5713 | 0.8034 | 0.9386 |
Initial particle of the egg farm.
Particle | Initial particle of the egg farm | ||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | |
1 | 0.0472 | 0.6264 | 0.8370 | 0.8489 | 0.2479 |
2 | 0.3940 | 0.7509 | 0.1076 | 0.5824 | 0.5417 |
3 | 0.9313 | 0.2053 | 0.1851 | 0.4747 | 0.0456 |
4 | 0.6833 | 0.5825 | 0.2744 | 0.0746 | 0.5546 |
The decoding method uses the same principle with the differential evolution (DE), where each cycle begins with determining the quantity to be transported by comparing the number of chickens in the first chicken farm with the demand of the first egg farm. If any amount is less, such amount shall be transported. To comply with the constraints of the case study, each egg farm shall receive at least 50 percent according to the chicken demand. However, the egg farms in the last rank may not gain the chickens as they need:
Then the appropriate truck is chosen by considering its capacity that is greater than the quantity to be transported and must be the truck with the capacity closest to the delivered amount.
Once the transportation has been completed, adjustments and validity checks must be recorded to meet the constraints of the case study. If the chicken farm still has chickens left, it will be transported next round until there are no remaining chickens from this farm, and then the next farm will be chosen for further transportation.
The decoding process begins with determining the transport quantity and choosing the appropriate truck type to avoid transportation many times, which results in higher production costs. The chicken farm is able to transport all the chickens without remaining. Egg farms also get the chickens at least 50 percent of the demand. The egg farms in the first particle shall receive all the chickens as they need, while the egg farms in the last order may not receive all the chickens as they need. However, chicken transportation in each round must not be over the capacity of each vehicle and must be used not more than the specified usage hours. When decoding the initial particle, the answer is given in Table
The assignment costs from initial particle decoding.
Particle sequences | Transportation cost | Opportunity cost | Assignment cost |
---|---|---|---|
1 | 13,562 | 3,845 | 17,407 |
2 | 14,610 | 4,846 | 19,456 |
3 | 14,088 | 4,811 | 18,899 |
4 | 14,951 | 4,158 | 19,109 |
5 | 13,433 | 4,152 | 17,585 |
Once the initial particle has been processed (equation (
The particle of the truck type after past the process of particle swarm optimization (PSO).
Particle | The particle of the truck type | ||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | |
1 | 0.4820 | 0.3197 | −0.2520 | 0.1928 | 0.4129 |
2 | 0.5733 | 0.8521 | 0.3194 | 0.5352 | 0.6262 |
3 | 0.3993 | 0.4981 | 0.5763 | 0.4341 | 1.7402 |
4 | 0.2681 | −0.1215 | 0.6809 | 0.3933 | 0.1290 |
The particle of chicken farms after past the process of particle swarm optimization (PSO).
Particle | The particle of the chicken farm | ||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | |
1 | 0.8233 | 0.4177 | 0.6567 | −0.2485 | 0.1032 |
2 | 0.3865 | 0.2223 | 0.6617 | 0.8264 | 0.1277 |
3 | 0.8358 | 1.6121 | 0.1989 | 0.4907 | 0.7508 |
4 | 0.8178 | 0.0477 | 0.6120 | −0.5619 | −0.2719 |
The particle of egg farms after past the process of particle swarm optimization (PSO).
Particle | The particle of the egg farm | ||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | |
1 | 0.6685 | 0.8553 | 0.5730 | 0.8024 | 0.4196 |
2 | 1.2680 | 0.7375 | 0.9217 | 0.6807 | 0.3372 |
3 | −0.1930 | 0.3455 | 0.4981 | 1.0905 | 1.0754 |
4 | 0.1991 | −0.3728 | 0.3575 | 1.3476 | 0.5766 |
The particle sequences of the truck type chosen for the assignment.
Particle | The particle of the truck type | ||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | |
1 | 3 | 2 | 1 | 1 | 2 |
2 | 4 | 4 | 2 | 4 | 3 |
3 | 2 | 3 | 3 | 3 | 4 |
4 | 1 | 1 | 4 | 2 | 1 |
The particle sequences of the chicken farm chosen for the assignment.
Particle | The particle of the chicken farm | ||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | |
1 | 3 | 3 | 3 | 2 | 2 |
2 | 1 | 2 | 4 | 4 | 3 |
3 | 4 | 4 | 1 | 3 | 4 |
4 | 2 | 1 | 2 | 1 | 1 |
The particle sequences of the egg farm chosen for the assignment.
Particle | The particle of the egg farm | ||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | |
1 | 3 | 4 | 3 | 2 | 2 |
2 | 4 | 3 | 4 | 1 | 1 |
3 | 1 | 2 | 2 | 3 | 4 |
4 | 2 | 1 | 1 | 4 | 3 |
The decoding can be conducted by taking the 1st particle of the truck category, the chicken farm, and the egg farm in order to be used to arrange the assignment, which can be put in order as in Table
The particle in the first order to be assigned.
Particle | The particle of the truck type | The particle of the chicken farm | The particle of the egg farm |
---|---|---|---|
1 | 3 | 3 | 3 |
2 | 4 | 1 | 4 |
3 | 2 | 4 | 1 |
4 | 1 | 2 | 2 |
The particle in the first order to be assigned.
No. | Truck | Cycle | Chicken farm | Egg farm | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Type | Space | Farm | Supply | Assign | Remaining | Farm | Demand (50%) | Assign | Remaining | ||
1 | 2 | 3,000 | 1 | 3 | 10,000 | 5,000 | 5,000 | 3 | 5,000 | 5,000 | 0 |
2 | 3 | 1,500 | 1 | 3 | 5,000 | 2,500 | 2,500 | 4 | 2,500 | 2,500 | 0 |
3 | 3 | 1,500 | 1 | 3 | 2,500 | 2,500 | 0 | 1 | 5,000 | 2,500 | 2,500 |
4 | 3 | 1,500 | 1 | 1 | 10,000 | 2,500 | 7,500 | 1 | 2,500 | 2,500 | 0 |
5 | 3 | 0 | 1 | 1 | 7,500 | 4,000 | 3,500 | 2 | 4,000 | 4,000 | 0 |
6 | 3 | 500 | 1 | 1 | 3,500 | 3,500 | 0 | 3 | 5,000 | 3,500 | 1,500 |
7 | 4 | 500 | 1 | 4 | 5,000 | 1,500 | 3,500 | 3 | 1,500 | 1,500 | 0 |
8 | 3 | 1,500 | 1 | 4 | 3,500 | 2,500 | 1,000 | 4 | 2,500 | 2,500 | 0 |
9 | 4 | 1,000 | 1 | 4 | 1,000 | 1,000 | 0 | 1 | 5,000 | 1,000 | 4,000 |
10 | 3 | 0 | 1 | 2 | 5,000 | 4,000 | 1,000 | 1 | 4,000 | 4,000 | 0 |
11 | 4 | 1,000 | 1 | 2 | 1,000 | 1,000 | 0 | 2 | 4,000 | 1,000 | 3,000 |
There are differences in the assignment of chicken farms and egg farms: (1) the chicken farm shall be assigned continuously to run out the chickens and (2) egg farm, where the chicken demand is divided into two equal parts according to the constraint of the case study.
In the decoding process, the conditions must be checked, such as the remaining working hours of each type of vehicle and the time spent on each shipment. Each chicken farm is able to send out most of the chickens and egg farms shall get at least 50% of all chickens as their requirement. The assignment for small-size samples, a particular type of truck usage, does not over the specified number of hours. The details are as in Table
Time for assignment (time unit: hours).
No. | Truck type | Cycle | Chicken farm | Egg farm | Assignment | Time to load up | Time to bring down | Time transport | Total time |
---|---|---|---|---|---|---|---|---|---|
1 | 2 | 1 | 2 | 3 | 5,000 | 0.45 | 0.5 | 2.1 | 4.8 |
2 | 3 | 1 | 4 | 4 | 2,500 | 0.175 | 0.2 | 1.938 | 4.038 |
3 | 3 | 1 | 4 | 1 | 2,500 | 0.175 | 0.225 | 0.788 | 2.988 |
4 | 3 | 1 | 1 | 1 | 2,500 | 0.25 | 0.225 | 2.05 | 4.813 |
5 | 3 | 1 | 1 | 2 | 4,000 | 0.4 | 0.28 | 1.175 | 4.25 |
6 | 3 | 1 | 1 | 3 | 3,500 | 0.35 | 0.28 | 2.175 | 3.9 |
7 | 4 | 1 | 3 | 3 | 1,500 | 0.15 | 0.105 | 0.263 | 2.4 |
8 | 3 | 1 | 3 | 4 | 2,500 | 0.2 | 0.175 | 2.388 | 5.463 |
9 | 2 | 1 | 3 | 1 | 5,000 | 0.45 | 0.45 | 2.388 | 5.563 |
10 | 4 | 1 | 3 | 2 | 1,000 | 0.1 | 0.1 | 2.45 | 5.938 |
Cost of assignment.
No. | Truck | Cycle | Chicken farm | Egg farm | Transportation cost | Opportunity cost | Total cost | |
---|---|---|---|---|---|---|---|---|
Type | Space | |||||||
1 | 2 | 3,000 | 1 | 2 | 3 | 2,304 | 864 | 3,168 |
2 | 3 | 1,500 | 1 | 4 | 4 | 1,211 | 454 | 1,665 |
3 | 3 | 1,500 | 1 | 4 | 1 | 896 | 336 | 1,232 |
4 | 3 | 1,500 | 1 | 1 | 1 | 1,444 | 541 | 1,985 |
5 | 3 | 0 | 1 | 1 | 2 | 1,275 | 0 | 1,275 |
6 | 3 | 500 | 1 | 1 | 3 | 1,170 | 146 | 1,316 |
7 | 4 | 500 | 1 | 3 | 3 | 576 | 144 | 720 |
8 | 3 | 1,500 | 1 | 3 | 4 | 1,639 | 615 | 2,254 |
9 | 2 | 3,000 | 1 | 3 | 1 | 2,670 | 1,032 | 3,702 |
10 | 4 | 1,000 | 1 | 3 | 2 | 1,425 | 713 | 2,138 |
Total | 14,610 | 4,846 | 19,456 |
The update of the particle position has been modified; therefore, we called our modification method as modified PSO. Instead of using formula (
Particle swarm optimization (PSO) is often trapped on the answer which is the local optimal answer. It may lead to not the most appropriate answer. In order to expand, therefore, the answer is improved and there is no change in the answer after processing 200 iterations, by randomly sampling the number of particles that might change in each particle of the truck types, the number of cycles, chicken farms, and egg farms. Then a new random number
Flowchart of modified PSO methodology.
Table
Comparison in terms of features of DE and modified PSO.
Features | DE | Modified PSO |
---|---|---|
Principle | Using the difference among vectors to expand the searching area | Using cooperation among particles to find the best area which contained the best solution |
Number of parameters | High | Low |
Procedure | 4 main steps: | 3 main steps: |
(1) Initial population | (1) Encoding | |
(2) Mutation | (2) Decoding | |
(3) Recombination | (3) Modified particle swarm optimization | |
(4) Selection | ||
Suit for complex problem | Medium | High |
Processing time | Long | Short |
The proposed method is encoded and processed by Visual Studio C# with a mathematical model by Lingo v.11 via Intel Core ™ i5-2450 M CPU 2.50 GHz Ram, 6 GB, compared with solutions provided by Lingo v. 11 software. The particle swarm optimization (PSO) method has been tested with 3 groups of problems (Chicken Farm × Egg Farm): small size (5 × 5), medium size (10 × 10), and large size (20 × 20). The problem of the case study is that the sample has been run 5 times, and the optimal result is recorded. The details are shown in Table
Defining the sample sizes.
Sample | Number of the datasets | Truck type (unit: type) | Round transportation (unit: round) | Chicken farm (unit: farm) | Egg farm (unit: farm) |
---|---|---|---|---|---|
Small size | 12 | 4 | 4 | 5 | 5 |
Medium size | 12 | 4 | 4 | 10 | 10 |
Large size | 12 | 4 | 4 | 20 | 20 |
The case study | 1 | 4 | 4 | 40 | 60 |
The proposed algorithm consists of two methods, i.e., particle swarm optimization (PSO) and modified particle swarm optimization (modified PSO) and the presented problem is compared with the best solution gained from Lingo v.11 (Lingo best solution (LBS)) and the differential evolution algorithm given in [
Explanation of the proposed method.
Algorithms | Definition of the proposed algorithm |
---|---|
PSO | Particle swarm optimization |
Modified PSO | Modified particle swarm optimization |
LBS | Lingo v.11 best solution obtained within predefined time |
DE | Differential evolution algorithm obtained by Kaewman et al. [ |
Datasets consist of 3 groups of problems (Chicken Farm × Egg Farm): small-size groups (5 × 5), medium-size groups (10 × 10), and large-size groups (20 × 20), including the problem of the case study, by using the data from Table
For small- and medium-sized problems, the stopping criterion of the Lingo program is set to run until finding the optimal solution. The stopping criterion of PSO and modified PSO is set to run for 5 minutes to be fairly compared with DE proposed by Kaewman et al. [
From Table
Test results of small-size sample groups (5 × 5).
Dataset | LBS | DE | %gap | PSO | %gap | Modified PSO | %gap |
---|---|---|---|---|---|---|---|
1 | 9,723 | 9,723 | 0.00% | 9,723 | 0.00% | 9,723 | 0.00% |
2 | 8,753 | 8,753 | 0.00% | 8,753 | 0.00% | 8,753 | 0.00% |
3 | 5,330 | 5,330 | 0.00% | 5,330 | 0.00% | 5,330 | 0.00% |
4 | 7,056 | 7,059 | 0.04% | 7,059 | 0.04% | 7,059 | 0.04% |
5 | 7,317 | 7,317 | 0.00% | 7,317 | 0.00% | 7,317 | 0.00% |
6 | 6,098 | 6,107 | 0.15% | 6,107 | 0.15% | 6,102 | 0.07% |
7 | 7,004 | 7,004 | 0.00% | 7,004 | 0.00% | 7,004 | 0.00% |
8 | 7,649 | 7,649 | 0.00% | 7,649 | 0.00% | 7,649 | 0.00% |
9 | 7,761 | 7,761 | 0.00% | 7,761 | 0.00% | 7,761 | 0.00% |
10 | 7,894 | 7,894 | 0.00% | 7,894 | 0.00% | 7,894 | 0.00% |
11 | 7,566 | 7,575 | 0.12% | 7,575 | 0.12% | 7,575 | 0.12% |
12 | 7,683 | 7,683 | 0.00% | 7,683 | 0.00% | 7,683 | 0.00% |
Average | 7,486.17 | 7,487.92 | 0.03% | 7,487.92 | 0.03% | 7,487.50 | 0.02% |
The second experiment has been executed with the medium size of test instances. In this dataset, 15 minutes is used to be the stopping criteria of PS and modified PSO which is equal to that of DE proposed by Kaewman [
Test results of medium-size sample groups (10 × 10).
Dataset | LBS | DE | %gap | PSO | %gap | Modified PSO | %gap |
---|---|---|---|---|---|---|---|
1 | 11,989 | 11,989 | 0.00% | 11,989 | 0.00% | 11,989 | 0.00% |
2 | 11,397 | 11,401 | 0.04% | 11,401 | 0.04% | 11,401 | 0.04% |
3 | 11,572 | 11,572 | 0.00% | 11,574 | 0.02% | 11,572 | 0.00% |
4 | 12,898 | 12,898 | 0.00% | 12,898 | 0.00% | 12,898 | 0.00% |
5 | 12,184 | 12,184 | 0.00% | 12,184 | 0.00% | 12,184 | 0.00% |
6 | 11,315 | 11,315 | 0.00% | 11,315 | 0.00% | 11,311 | −0.04% |
7 | 14,508 | 14,508 | 0.00% | 14,508 | 0.00% | 14,505 | −0.02% |
8 | 12,613 | 12,613 | 0.00% | 12,616 | 0.02% | 12,613 | 0.00% |
9 | 10,902 | 10,921 | 0.17% | 10,921 | 0.17% | 10,921 | 0.17% |
10 | 11,870 | 11,890 | 0.17% | 11,893 | 0.19% | 11,890 | 0.17% |
11 | 15,817 | 15,849 | 0.20% | 15,849 | 0.20% | 15,847 | 0.19% |
12 | 12,114 | 12,114 | 0.00% | 12,117 | 0.02% | 12,114 | 0.00% |
Average | 12,431.58 | 12,437.83 | 0.05% | 12,438.75 | 0.06% | 12,437.08 | 0.04% |
According to Table
The last experiment has been executed with the large size of test instances. In this dataset, the computational time of 30 minutes is used to be the stopping criteria of PS and modified PSO which is equal to that of DE proposed by Kaewman [
Computational result of large-size sample groups (20 × 20).
Dataset | LBS | DE | %gap | PSO | %gap | Modified PSO | %gap |
---|---|---|---|---|---|---|---|
1 | 33,249 | 32,716 | −1.63% | 32,716 | −1.63% | 32,716 | −1.63% |
2 | 29,943 | 29,094 | −2.92% | 29,121 | −2.82% | 29,094 | −2.92% |
3 | 37,672 | 36,128 | −4.27% | 36,146 | −4.22% | 36,106 | −4.34% |
4 | 38,891 | 37,781 | −2.94% | 37,781 | −2.94% | 37,762 | −2.99% |
5 | 39,781 | 37,895 | −4.98% | 37,898 | −4.97% | 37,795 | −5.25% |
6 | 31,480 | 30,084 | −4.64% | 30,084 | −4.64% | 30,084 | −4.64% |
7 | 58,984 | 57,738 | −2.16% | 57,718 | −2.19% | 57,706 | −2.21% |
8 | 89,872 | 87,573 | −2.63% | 87,586 | −2.61% | 87,573 | −2.63% |
9 | 90,164 | 89,079 | −1.22% | 89,104 | −1.19% | 89,007 | −1.30% |
10 | 35,878 | 34,871 | −2.89% | 34,902 | −2.80% | 34,812 | −3.06% |
11 | 29,095 | 28,049 | −3.73% | 28,049 | −3.73% | 28,006 | −3.89% |
12 | 29,892 | 29,152 | −2.54% | 29,198 | −2.38% | 29,110 | −2.69% |
Case study | 125,593 | 114,932 | −9.28% | 114,968 | −9.24% | 114,886 | −9.32% |
Average | 51,576.46 | 49,622.46 | −3.52% | 44,191.92 | −3.49% | 44,147.58 | −3.61% |
Table
According to Table
Statistical test results of answers in large-sized problem groups.
DE | PSO | Modified PSO | |
---|---|---|---|
LBS | 0.021 | 0.022 | 0.020 |
DE | — | 0.021 | 0.002 |
PSO | — | — | 0.001 |
Modified PSO | — | — | — |
Table
Computational results of average of %gap the sample sizes.
No. | Sample | DE (%) | PSO (%) | Modified PSO (%) |
---|---|---|---|---|
1 | Small size | 0.03 | 0.03 | 0.02 |
2 | Medium size | 0.05 | 0.06 | 0.04 |
3 | Large size | −3.05 | −3.01 | −3.13 |
4 | Case study | −9.28 | −9.24 | −9.32 |
Average (%) | −3.06 | −3.04 | −3.10 |
The case study problem was the assignment consisting of the main costs incurred from chicken transportation and depended on the difference of truck types using to deliver throughout the road conditions, which affected the fuel costs as well. Besides, the opportunity cost incurred as there was free space on the transporter truck. In which both costs had to be the lowest and the assignment needed to comply with various conditions as specified, making the problem-solving in this case study more complicated.
The nature of the multistage assignment problem and various conditions caused a complicated problem. In which problem-solving with the exact method either the branch and bound method or the Lingo program could not be conducted in a short time. Therefore, problem-solving using alternative methods or the heuristics method was the appropriate choice for resolving this problem. The heuristics used in this study would use the particle swarm optimization (PSO) method.
The particle swarm optimization (PSO) was the method having a small number of parameters, including a short procedure, which saves time to search for answers, but the appropriate answer was likely to not change to find the answer in other locations. This study added additional steps in order to find answers in other areas which was likely that the answer would be the optimal answer (global optimal). The results showed that the modified particle swarm optimization method provided a better answer than the particle swarm optimization method. This method might be suitable for complicated problems than the exact method of the Lingo program. There were also a small number of procedures and parameters, which make it easier to find the lowest cost as well as the amount of fuel used in operations. From the computational result, we found that modified PSO statistically outperforms the best existing heuristics which is DE proposed by Kaewman [
Future research should study the problem of more complicated assignments and to comply with the conditions in the real-world (Realistic) or study other metaheuristic methods to solve problems more efficiently, by using hybrid methodologies and developing the exact method to find the answer of the assignment problem. This is another interesting way for further study. The others additional factors should be considered, for instance, the study of capability and performance of each type of vehicle and study of driver skills of driving for each type of road.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This paper was supported by KKU-En Grad Camp 2016, Faculty of Engineering, Khon Kaen University, Thailand. This project was funded by the Faculty of Engineering, Khon Kaen University, Thailand.