Available online at ScienceDirect. Procedia Manufacturing 11 (2017 ) Wan Chen Chiang, Chen Yang Cheng*

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1 Available online at ScienceDirect Procedia Manufacturing 11 (2017 ) th International Conference on Flexible Automation and Intelligent Manufacturing, FAIM2017, June 2017, Modena, Italy Considering the performance bonus balance in the Vehicle Routing Problem with Soft Time Windows Wan Chen Chiang, Chen Yang Cheng* National Taipei University of Technology, 1, Sec. 3, Zhongxiao E. Rd., Taipei Taiwan, R.O.C. Abstract In the field of operations research, the Vehicle Routing Problem with Time Windows (VRPTW) has been widely studied because it is extensively used in practical applications. Some situations in the practice are discussed in the most of past relevant research, i.e., time window and vehicle capability. However, the performance bonus is not considered. In most logistics companies, performance bonus is calculated by the piece. But it is a problem that the method of calculation is not fair for all staff. In this paper, the model not only considers the performance bonus into the VRPTW, but also changes the calculation method to make a load balance between every staff. In the same time, it also makes the calculation of performance bonus more fair for all staff Published The Authors. by Elsevier Published B.V. by This Elsevier is an open B.V. access article under the CC BY-NC-ND license ( Peer-review under responsibility of the scientific committee of the 27th International Conference on Flexible Automation and Peer-review under responsibility of the scientific committee of the 27th International Conference on Flexible Automation and Intelligent Manufacturing Manufacturing. Keywords: VRPTW ; performance bonus ; soft time windows 1. Introduction In the field of operations research, the Vehicle Routing Problem with Time Windows (VRPTW) has been widely studied because it is extensively used in practical applications. In the most of past researches, the objective function minimizes the number of vehicles or minimize the transit time. Some restrictions and situations in the practice are discussed, i.e., time window and vehicle load limits. Considering these restrictions could make the mathematical * Corresponding author. Tel.: (+886) ext address: cycheng@ntut.edu.tw Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of the scientific committee of the 27th International Conference on Flexible Automation and Intelligent Manufacturing doi: /j.promfg

2 Wan Chen Chiang and Chen Yang Cheng / Procedia Manufacturing 11 ( 2017 ) model more realistic. However, some factors from human are less discussed, such as the performance bonus. In the most logistics companies, manager establishes the performance bonuses to maintain the efficiency and to balance the work load of each staff. But the performance bonus has never been considered. That is why the model consider the performance bonus into the VRPTW in this paper. The performance bonus is usually calculated by the piece or the commission based on the freight charges. For example, when each staff delivers a goods, the staff could get a fixed amount of performance bonuses. Also, when each staff receives a goods, the staff could get the commission based on the freight charges. However, this method of calculation is not fair for all staff. The fair means that the income of the staff is consistent with the labor and time paid by the staff. However, actually the service areas of logistics companies are divided into many areas, and each staff is responsible for each area. But the characteristics of each area may be very different, like the Fig. 1. In practice, the staffs of some area have to transport a lot of goods to customers whose distributions are very concentrated, so the staff could finish a lot of orders. In contrast to that, there are fewer orders in the other areas such as suburbs and remote districts, and the customers may be very far away from each other. So the staff could only complete very few orders. However, it does not mean that the staff in this area is not hard-working. In this paper, the model not only considers the performance bonus into the VRPTW, but also changes the calculation method to make a load balance between every staff. In the same time, it also makes the calculation of performance bonus more fair for all staff. Fig. 1. The work load and path length of different area. The paper is organized as follows. In the next section we discuss the relevant literature. In Section3 presents a model that considers the performance bonus into the VRPTW. In Section4 show the computational results of our mathematical model. Finally, we make the conclusions of this research. 2. Literature review 2.1. Vehicle Routing Problem (VRP) In this paper, the model considers some restrictions to solve some practical problems faced by logistics companies. This type of transportation problems can be traced back to travelling salesman problem (TSP). The TSP is a question: "there are some cities and the distances between each city, and to solve if each city need to be visited exactly once and finally return to the beginning city, how far is the shortest route?"[1]. TSP is also a special instance

3 2158 Wan Chen Chiang and Chen Yang Cheng / Procedia Manufacturing 11 ( 2017 ) of the vehicle routing problem (VRP). It is classified as an NP-hard problem and has been widely studied in the field of operations research[2]. In order to make the VRP close to practical applications, past studies have used different target types, such as minimizing the number of vehicles, minimizing transit time, etc.[3, 4], or adding more restrictions, such as soft window considerations, different vehicle load limits[5, 6]. The classification of the characteristics for the VRP is as Table 1[7]. However, some factors from human are less discussed in the past researches, such as the performance bonus. So this paper adds the constraint of the performance bonus into the mathematical model to understand its impact on this type of problem. Table 1. The characteristics for the VRP. Characteristics Factors Number of vehicles Single vehicle, multiple vehicles Vehicle type Only one type, multiple types, special type Plant stations Only one plant station, multiple plant stations Demand pattern Demand is known, random, and allow partially to meet demand Demand settings Nodes, section line, mixed the nodes and section line Network type No direction, direction, mixed type, Euclid Vehicle capacity All vehicle capacity limits are the same, different vehicles have different capacity limits, unlimited capacity Maximum travel time All routes are the same, different routes have different times, no limitation Operating procedure Receipt, delivery, mixed type, batch delivery Cost Variable costs, fixed costs, opportunity costs 2.2. Time Windows The time windows usually are considered as a kind of demand restriction in the VRP. It is also known as the vehicle routing problem with Time Windows (VRPTW). The time windows can be divided into two types: hard time windows and soft time windows[8, 9]. Hard time windows: for this type of time windows, the service of the demand has time limit, and the staff has to serve the customers in the limited time interval. Soft time windows: for this type of time windows, the service of the demand has flexible time limit, so the staff can serve the customers outside of the limited time interval. Therefore, if the staff is early or late to serve the customers, it will lead to the penalties. For the VRPTW, using soft time window is closer to practice than hard time windows[10], and the penalties can be adjusted to ensure a feasible solution. So, in this paper, we choose to use the soft time window to formulate a mathematical model for the VRPTW Soft Time Windows In past researches, the soft time windows can be divided into many types by the penalties calculation method. In general, the penalties are calculated for the outside both early and late of the limited time interval. But in some model, the limited time intervals are considered only one side of the exceeding time, that is referred to as the semi soft time windows. For this kind of the soft time window usually considered the upper side[11]. In addition, there is a kind of the soft time window that is more like a hard time windows. It has a certain degree of allowance, and also has to calculate the penalties. Once more than a certain upper and lower boundary, the penalties will be set to infinity. It regarded as an unacceptable part[12]. The penalties are calculated by the following

4 Wan Chen Chiang and Chen Yang Cheng / Procedia Manufacturing 11 ( 2017 ) Fig. 2. In the function, the is the penalty of node, the and are the unit penalty cost for early and late of the limited time interval, the is the arrival time of node. (1) Fig. 2. The soft time window has an unacceptable part. 3. Mathematical model In this paper, we considered the performance bonus to formulate a mathematical model. All symbol for the model as shown in the following. Notation for the model

5 2160 Wan Chen Chiang and Chen Yang Cheng / Procedia Manufacturing 11 ( 2017 ) In this mathematical model, the objective function minimizes the total cost, including traveling cost, performance bonus cost, waiting cost, and overtime cost. For the basic limitation of the path, the constraint (3) and (4) are conservation of every node. The constraint (5) means that every staff at least have one customer. The constraint (6) let every node be served by only one staff. The constraint (7) and (8) are that every staff starts from and returns to the warehouse. In addition, the constraint (9) limit the capacity of vehicle. For the time window restrictions, we could get the waiting time and overtime of every node in the constraint (10) and (11), and the arrival time of the next node is calculated by the constraint (12). For the finally part of the performance bonus restrictions, the constraint (13) is that every staff s performance bonus must meet the minimum of performance bonus. The constraint (14) is to calculate the performance bonus of every node. The constraint (15) is the performance bonus for returning to the warehouse of every staff. The constraint (16) is the total performance bonus for every staff. At last, the constraint (17) is to make a performance bonus balance between every staff. s.t. (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

6 Wan Chen Chiang and Chen Yang Cheng / Procedia Manufacturing 11 ( 2017 ) (12) (13) (14) (15) (16) (17) 4. Computational Results In the last section, we considered the performance bonus to formulate a mathematical model. In this section, we use the C++ to call the Gurobi library to solve our model. At the first, the model need to be transformed into a linear model because our model is nonlinear. Then we select twenty nodes from the Solomon's VRPTW benchmark problems to be simple examples. The twenty nodes will be assigned to five staff to transport. We used two sets of data: C101 and C102. The data of the coordinate, demand, and service time are same in the C101 and C102. The time window setting is different between C101 and C102. In the C101, every customer's time window has a clear starting time and end time. However, there are no starting times of time window for some customers in the C102. The starting times of these customers are set to 0 or. It also indicates that there are no additional waiting costs in our model when the goods of these customers are delivered in advance. We use these data to compare the differences in the model that whether it has a constraint for the balance of the performance bonuses or not. The computational results for the model as shown in the following Table 2 and Table 3. Table 2. The computational results for the cost. Data type C101 C102 Unbalanced Balanced Unbalanced Balanced Transport cost (dollars) Performance bonus cost (dollars) Total cost (dollars) In the Table 2, it is the results results for the cost from two sets of data, although the cost of unbalanced model is lower than balanced model, but there is no big difference between these two models. The Table 3 shows the results for the performance bonus, we could find that adding the restriction could indeed achieve a certain effect on the balance of performance bonus. The results from the C102 is particularly evident. In the C102, the proportion of the unbalanced model reaches

7 2162 Wan Chen Chiang and Chen Yang Cheng / Procedia Manufacturing 11 ( 2017 ) Table 3. The computational results for the performance bonus. Data type C101 C102 Unbalanced Balanced Unbalanced Balanced Max performance bonus (dollars) Min performance bonus (dollars) Proportion = (max/min) In the Fig. 3, it shows the comparison of costs and proportions. Whether the C101 or the C102, the costs of unbalanced model are slightly higher than the balanced model. However, the proportions of C102 have a large difference between the two models. The proportion of unbalanced model in the C102 is 1.721, it reduced to by the balanced model. Despite the increasing of some additional costs, it makes the performance bonus of staffs reach a stable balance. 5. Conclusion Fig. 3. The comparison of costs and proportions. In this paper, different from the past, we consider the performance bonus that has never been discussed into the VRPTW problem. In the most logistics companies nowadays, the performance bonus is usually established to maintain the efficiency and to balance the work load. However, affected by the actual situation, the performance calculation method is not fair for all staff. So this paper changes the calculation method to make the calculation of performance bonus more fair for all staff and the income from the performance bonus between staffs more balanced. Finally, from the results, we could find that adding the performance bonus restriction in our model could indeed achieve a certain effect on the balance of performance bonus.

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