Int J Adv Manuf Technol (2006) 29: 1246–1258 DOI 10.1007/s00170-005-0014-0
ORIGINA L ARTI CLE
Lixin Tang . Xianpeng Wang
Iterated local search algorithm based on very large-scale neighborhood for prize-collecting vehicle routing problem
Received: 26 November 2004 / Accepted: 23 February 2005 / Published online: 12 October 2005 # Springer-Verlag London Limited 2005
Abstract This paper presents a new variant of the vehicle routing problem (VRP), the prize-collecting vehicle routing problem (PCVRP), which is derived from the hot rolling production of the iron and steel industry. One major characteristic of the PCVRP is that the customers need not be visited compulsorily but a prize can be collected from each customer when visited. Besides the capacity constraint, a task completion constraint is introduced and requires that the total demand of the visited customers should be no less than a predetermined amount. The main objective is a linear combination of three objectives: minimization of total distance traveled, minimization of vehicles used, and maximization of prizes collected. An iterated local search algorithm (ILS) based on very large-scale neighborhood (VLSN) using cyclic transfer is proposed for the PCVRP. Computational results of problem instances with up to 100 customers show the algorithm is efficient and effective. Keywords Cyclic transfer . ILS . PCVRP . VLSN
1 Introduction The vehicle routing problem, which comprises an important class of combinatorial problems, has drawn considerable attention in operation research because many practical transport logistics problems can be formulated as a vehicle routing problem. The classical vehicle routing problem (VRP) involves a fleet of homogenous vehicles with uniform capacity, a single depot, and a set of customers with known demands. The objective is to design a route that can minimize the total transportation cost while satisfying the demands of all customers. Each vehicle must L. Tang (*) . X. Wang The Logistics Institute, Northeastern University, Shenyang, China e-mail:
[email protected] Tel.: +86-24-83680169 Fax: +86-24-83680169 e-mail:
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begin and terminate at the depot. Each customer can be served exactly once by exactly one vehicle route. Though many variants of the classical VRP have been proposed such as the capacity constrained vehicle routing problem (CVRP) [1], the distance constrained vehicle routing problem (DVRP) [2], the vehicle routing problem with time windows (VRPTW) [3–5], and so on, most of them assume that all customers can and must be visited. The models derived from such an assumption cannot be directly applied in many practical applications. In this paper, we consider a new variant of VRP, which we will call the prize-collecting vehicle routing problem (PCVRP). The PCVRP is closely related to the prize-collecting traveling salesman problem (PCTSP) [6] and classical vehicle routing problems. One major difference with the PCVRP is that the customers need not be visited compulsorily but a prize can be collected from each customer when visited. Besides this, a task completion constraint is introduced and requires that the total demand of the visited customers should be no less than a predetermined amount. In this problem, it is assumed that each vehicle route starts and ends at a central depot within its capacity and that each customer can be visited no more than once by at most one vehicle route. The objective is to determine a subset of all customers to visit so as to minimize the total distance travelled, minimize the vehicles used, and maximize the prizes collected. The prize-collecting vehicle routing problem proposed in this paper is derived from the hot rolling production of the iron and steel industry. In the iron and steel industry, the hot strip mill (HSM) is often considered as the bottleneck of the overall production process while the production sequencing of steel orders through the hot rolling mill is the key to hot strip mill production scheduling. The hot rolling production scheduling problem is, therefore, very important to the iron and steel industry. A process flow of the hot rolling mill production is shown in Fig. 1. Orders for hot rolling are steel plates in rolls that can be sold directly as finished products or be further processed to form highquality cold rolling orders. Rough rolling plate blanks and continuous casting plate blanks are the main raw materials
1247 Hot strip
Slitting shearing
Slab
Reheating furnace
Roughing mil
Slit Coil
Finishing mil Steel plate
Crosscut shearing
Sheet
Fig. 1 Process flow of hot rolling mill production
used in the hot strip rolling process. After some surface treatment, they are continuously reheated via a reheating furnace, rolled into order blanks in rough rolling mill, refined in finishing rolling mill, then cooled, rolled up and polished. Rolling objects between two work rollers, corresponding to a rolling schedule, is called a rolling turn, which can be seen in Fig. 2. The length of a turn should not be more than a certain value, and to avoid edge marks, the slabs of the staple material section should be from wide to narrow. Because of the high temperature, high speed and heavy wear, work rollers and backup rollers on each stand often need to be replaced between two turns to ensure the shape of plates and the flatness of orders, which causes a very high setup cost. So to avoid this setup cost, a work roller should roll as many orders as possible in a turn. In many hot rolling plants, every workday is often divided into three shifts. Schedulers do scheduling for the next shift during the current shift. Each shift contains several turns, and the composition of a hot rolling schedule for a shift is shown in Fig. 3, in which M is the number of turns. The total demand of all orders is usually much bigger than the production line’s capability. Therefore, when doing the scheduling, schedulers select a certain number of orders from a pool, which contains a lot of different orders, and then form as few turns as possible for a shift while taking full use of the production line’s capability. Each order in the pool can be seen as a customer whose demand corresponds to the length of that order and the prize assigned to it corresponds to the desirability to process that order. The prize value assigned to an order is according to the profits
that can be obtained from the order, the importance of the customer who signed the order, the due date of the order and so on. And correspondingly a turn can be seen as a vehicle route in which a set of customers is visited within its capacity. Thus the scheduling problem becomes a prizecollecting vehicle routing problem: the selection of a subset of all customers to visit so that the total cost is minimized while considering the production constraints. The rest of the paper is organized as follows. The literature review on related papers is introduced in Section 2. Section 3 presents the mathematical model of PCVRP derived from hot rolling scheduling. Then the solution methodology is detailed in Section 4. Section 5 reports the computational results of 18 instances, each of which includes 10 problems. Finally, the paper is concluded in Section 6.
2 Literature review There have been many attempts to solve the hot rolling production scheduling problem. The models created and the solution methods proposed are quite varied. Wright and Houck [7] presented a heuristic procedure to generate production schedules for the HSM in the steel industry, but their heuristic can only achieve a local minimum. Balas and Martin [8] reduced the hot rolling production scheduling problems to the knapsack constraint problems and presented a prize collecting traveling salesman problem model, which is a generalization of the traveling salesman
an order
Fig. 2 The forming of a rolling turn
a turn
width of order
warm up material section
staple material section
1248 Fig. 3 The composition of a hot rolling schedule for a shift
The first order of the ith turn Width of order
1
The last order of the ith turn
A turn
2
i
M
M rolling turns in a shift
problem (TSP). In this problem, each city is assigned a prize. When visiting a city, the salesman can obtain a prize and pays a penalty for each city not visited. The objective is to minimize the total travel cost and penalties, while collecting a predetermined prize. This type of scheduling first selects the slabs assigned to orders and then sequences them for processing. Kosiba et al. [9] investigated the hot strip sequencing problem and established a TSP model. Lopez et al. [10] described the hot strip mill production scheduling problem (HSMPSP) and formulated it as a mathematical programming model. To solve the problem, they proposed a tabu search metaheuristic that gave good solutions. However, the models mentioned above are single TSP, which can only arrange for one turn of hot rolling scheduling at a time. Also, the solution method they use is a serial strategy, which is an essentially greedy procedure. Using this strategy, turns in the same shift are arranged one by one. When arranging a turn, their methods select orders from the unscheduled order pool. Other turns are then scheduled by selecting orders from the unscheduled remainder until all the turns are scheduled. Clearly, the turns arranged using the serial strategy would become poorer and poorer because the setup costs for later turns will be larger and larger. Therefore, the single TSP models suffer from the disadvantage of local optimization. Tang et al. [11] proposed a parallel strategy in modeling the hot rolling scheduling problem, and gave a multiple traveling salesman problem (MTSP) model which can simultaneously form m turns for a shift and thus avoids the disadvantage of the local optimization suffered by single TSP models. To solve the problem, they first converted the MTSP into a single TSP model and then constructed a modified genetic algorithm (MGA) to obtain a near-optimal solution to the TSP. However, in their problem they only considered the condition that the orders have been selected, which means they need not worry about orders exceeding the production ability. Besides this, their model did not take into consideration the length constraint on the staple materials of each turn. Cowling [12] proposed a PCVRP model for a steel hot rolling mill and Cowling [13] described a flexible decision support system for steel hot rolling mill scheduling. In his problem, slabs are deposited in the slabyard, which consists of hundreds of steel piles each of which has up to 20 slabs. When required, the slabs will be obtained via an unpiling process and then transferred to one furnace to be reheated to a required tem-
perature, but the unpiling process may be costly and limit the production process. He modeled the slab-yard as a digraph G=(N, A), in which both nodes and arcs are assigned with weights. Each node i∈N denotes a slab and the weight assigned to it represents the desirability to process it. The weight on arc (i, j) measures the desirability to schedule slab j immediately after slab i. Each route of the PCVRP corresponds to a program that is a sequence of slabs to be rolled in a particular shift. The objective is to find a fixed number of routes so as to maximize the sum of nodes visited and arc scores over all paths, plus the non-positive score arising from the unpiling of slabs. So his model concentrates on the unpiling process, which is important in his problem, while ours focuses only on selecting orders from an order pool and simultaneously forming m turns for a shift. A heuristic method based on local search and tabu search is proposed to solve his problem. His heuristic first uses the local search to find an initial solution and then uses the tabu search technique to improve it. The techniques used in the local search procedure are based upon those of a manual planner and thus the neighbourhood searched is relatively small, while in our method it is a very large-scale one obtained by the cyclic transfer method. There are some other works related to prize collecting routing problems. The multiple tour maximum collection problem (MTMCP) is investigated by Butt and Cavalier [14] and a heuristic is proposed for this problem to obtain near optimum solutions. Then Butt and Ryan [15] proposed an optimal solution procedure using column generation. In MTMCP each customer is assigned a prize and the objective is to determine the subset of customers to visit in the time allotted so that the prizes collected is maximized. Chao et al. [16] described the team orienteering problem (TOP) and proposed a fast and effective heuristic. In the TOP, start and end points are specified and others are associated with a score. Each of the m members in the team is given a fixed amount of time, and the goal is to determine m paths from start to end through a subset of points so that the scores collected are maximized. Laporte et al. [17] investigated a class of asymmetrical single depot vehicle routing problems possessing many characteristics. In this problem the nodes are grouped into clusters, each of which contains specified and non-compulsory nodes, and each node is associated with a visiting cost. The objective is to minimize an objective incorporating routing costs, node visiting costs, and vehicle costs. To solve their problem, a
1249
graph extension is used to transform their model into an equivalent TSP and then a branch and bound algorithm is proposed to obtain exact solutions. However, the method can only optimally solve problems involving up to 40 cities. All the papers described above deal with a certain scheduling problem in HSM or a certain routing problem with prize collection, but none of them deal with the same one we consider. For this reason, we will build a PCVRP model for the hot rolling scheduling problem and thus derive a new variant of VRP. And as far as we know, there is no benchmark algorithms presented for this kind of problem. To solve such a problem presented in this paper, we propose an iterated local search algorithm based on a very large-scale neighborhood obtained by the cyclic transfer method. The theory of cyclic transfers, which is a new class of neighbourhood search strategies, is first presented by Thompson and Orlin [18], in which they show that its application to vehicle routing problems also generalizes the edge exchange method of Croes [19] and Lin [20], as well as the famous variable depth method of Lin and Kernighan [21]. Thompson and Psaraftis [22] investigate the application of cyclic transfer to multi-vehicle routing and scheduling problems, and the results they get show that the method is either comparable to or better than the best published heuristic algorithms for several complex and important vehicle routing and scheduling problems. To meet the specificity of the PCVRP, we introduce a virtual vehicle to visit the un-served customers with no cost and prizes. Besides this, in each vehicle route we insert a virtual customer with no demand and prize so that customer transfers can be performed among permutations rather than cyclic permutations of routes. During the procedure of searching for negative cost cyclic transfers, we propose a heuristic using dynamic programming. xijk ¼
3.1 Notations The following symbols are used for defining the problem parameters and variables. Parameters: N P
D
Cij
M Q V a
The set of all nodes, N={0, 1, 2, ..., n}, where 0 denotes the depot and others denote the customers. Each customer represents a slab for scheduling. The set of prizes, P={p1, p2, ..., pn}, where pi denotes the prize assigned to customer i. We define p0=0 for the depot. pi denotes the desirability of processing slab i. The set of demands of customers, D={d1, d2, ..., dn}, where di denotes the demand of customer i. The demand di represents the length of slab i. We define d0=0 for the depot. The travel distance between customers i and j. When i=j, Cij=∞. Cij represents the penalties for the changes of width, hardness and gauge when scheduling slab j directly after slab i. The maximum number of vehicles available, which represents the maximum turns a shift can roll. The maximum capacity each vehicle can load. The large fixed cost of a vehicle if it is used. V represents the large setup cost. The task completion parameter obtained by dividing the predetermined amount of demand by the total demand of all customers.
Decision variables:
1; if vehicle k visits customer j immediately after customer i and i 6¼ j; 0; otherwise:
yik ¼
3 Mathematical model of PCVRP
1; if customer i is visited by vehicle k; 0; otherwise:
i; j 2 N ; k ¼ 1; 2; . . . ; M :
i 2 N nf0g; k ¼ 1; 2; . . . ; M:
3.2 The model Let 1 ; 2 ; 3 2 [0,1] be three preset constants such that 1 þ 2 þ 3 ¼ 1 . Using the above symbols, the PCVRP with a limited number of vehicles is formulated as follows.
Minimize 1
M X n X n X
Cij xijk þ 2
k¼1 i¼0 j¼0
3
M X n X k¼1 i¼1
M X n X k¼1 i¼1
pi yik
Vx0ik (1)
1250
Subject to M X
yik 1; i 2 N nf0g:
(2)
k¼1
M X n X
x0ik M
(3)
k¼1 i¼1
M X n X
x0ik ¼
k¼1 i¼1
n X
M X n X
xi0k
(4)
k¼1 i¼1
yik di Q; k ¼ 1; 2; . . . ; M:
(5)
i¼1
n X
xijk ¼
j¼0
X
n X
xjik ¼ yik ; i2N nf0g; k¼1; 2; . . . ; M :
4 Solution methodology As has been shown in many papers, even the classical VRP is NP-hard, so the PCVRP, which is a complicated variant of the classical VRP, is also NP-hard. Thompson and Psaraftis [22] have shown the efficiency and effectiveness of the cyclic transfer method to multi-vehicle routing and scheduling problems. Thus, we propose an iterated local search algorithm based on cyclic transfer to solve the prize collecting vehicle routing problem. One major difference with our method is that we use dynamic programming to search for negative cost cyclic transfers. Furthermore, we introduce a virtual vehicle that visits the unserved customers into the solution structure and a virtual customer into each vehicle route.
(6)
j¼0
4.1 Main procedure of the proposed algorithm
xijk jS j 1; for all S N nf0g;
i;j2S
2 jS j n; k ¼ 1; 2; . . . ; M : , M X n n X X yik di di a k¼1 i¼1
route. Constraint Eq. 8 is a task completion constraint for real world problems, and it ensures that the demand of visited customers should be no less than a predetermined amount. Constraint Eqs. 9 and 10 specify the integrity conditions on the variables.
(7)
(8)
i¼1
xijk 2 f0; 1g; i; j 2 N ; k ¼ 1; 2; . . . ; M : yik 2 f0; 1g; i 2 N nf0g; k ¼ 1; 2; . . . ; M :
(9) (10)
Each term in Eq. 1 represents one type of cost. The first is the total distance traveled, the second is the cost of total vehicles used, and the third is the total prizes collected from the customers that have been visited. Constraint Eq. 2 ensures that a customer can only be visited one time at most. Constraint Eq. 3 ensures that the number of vehicles used is smaller than that of total vehicles available. Constraint Eq. 4 ensures that the vehicles leaving the depot must return to it in the end. Constraint Eq. 5 is the capacity constraint of vehicles. Constraint Eq. 6 ensures that when a customer is included in a certain vehicle route, there should be exactly one customer (or the depot) before and one customer (or the depot) after it in the route respectively. However, when a customer is not visited, there will be no customers (or the depot) both before and after it. Constraint Eq. 7 is subtour elimination constraint for each vehicle
Although many construction and local improvement methods have been proposed to solve vehicle routing problems, there is no report on how these methods will behave when constrained by a limited vehicle fleet. Furthermore, when using the cyclic transfer method, it is required that the number of vehicles used is fixed. So to deal with limited vehicle availability, we use the strategy proposed by Lau et al. [23]. This strategy is to decrease the vehicle number in stages, where at each stage the number of vehicles used is fixed. So within each stage we first generate an initial solution and then apply the iterated local search based on cyclic transfer to improve it. The main procedure of the proposed algorithm is shown in Fig. 4. SB denotes the best solution found in history and Vn the number of vehicles used at each stage. At the first stage we use the maximum vehicles available, and in the following stages the number of vehicles used decreases. At each stage, the algorithm first checks whether the total capacities of Vn vehicles can satisfy the task completion constraint Eq. 8. If constraint Eq. 8 cannot be satisfied, the algorithm terminates and gives the best solution found SB. Otherwise, an initial solution will be generated and then an ILS algorithm based on cyclic transfer will be used to improve it. 4.2 Generate an initial solution We use the famous savings algorithm proposed by Clark and Wright [24] to get the initial solution. This method is widely known as an efficient heuristic for VRP. It is based on the notion of savings. When there are two routes (0,..., i, 0) and (0, j,..., 0) that can be feasibly merged into a single one
1251
Vn=M
Satisfy constraint (8)?
N
Y
Generate initial routes
Terminate
Improve routes using ILS
Update S*
Vn=Vn-1 Fig. 4 Main procedure of the proposed algorithm
(0,..., i, j,..., 0), a distance savings Sij = ci0 + c0j − cij can be obtained. An initial solution is obtained as follows: Step 0. Create a solution of n routes where each route visits a single customer. Step 1. Calculate the saving Sij=c0i+cj0−cij for all pairs of customers i and j. Note that Sij=Sji for all i, j∈N. Step 2. Order the savings in a non-increasing fashion. Step 3. Starting at the top of the list, if the total deliveries across two routes do not exceed the vehicle’s capacity, Fig. 5 Solution shape with virtual vehicle
then they can be combined into a new feasible route with the highest savings. Step 4. Try the next combination in the list and repeat step 3 until no more combinations are feasible. The number of routes is denoted as r. Step 5. Sequence the customers in each route obtained through step 1–4, using the nearest neighbour method. Step 6. Improve the routes obtained by step 5, using 2-opt. Step 7. Calculate the cost of each route. Step 8. Order the routes in ascending order according to their cost. Step 9. Starting at the top of the list, if M 2 and 2 > 3 : According to the two inequalities, we randomly generate 1 in [0.400, 0.700] and let 2 ¼ 0:52ð1 1 Þ and 3 ¼ 1 1 2 : As described before, there is no benchmark algorithms presented for PCVRP. We compare our algorithm to two other algorithms, whose operating times are set to be equal to that of our algorithm. The main procedure of the first algorithm is the same one as is shown in Fig. 4 except that the routes improvement is accomplished using the exchange operator shown in Prosser and Shaw [25], and we call it the ILS+Exchange. The exchange operator is made as follows: First, randomly select two vehicle routes from all the vehicle routes including the virtual one. Second, exchange two randomly selected customers in the two vehicle routes if this does not exceed the capacity constraint. At last improve the two new routes using 2opt. The second algorithm is the Multi-Start based on VLSN obtained using cyclic transfer (multi-start). The main procedure of this algorithm is shown in Fig. 11. The initial solution is generated using the method described in part 4.2. The results of the experiments, which are tested on a Pentium-IV 2.4 G-Hz PC with 512MB RAM, are shown in Table 1, in which ILS+Cyclic transfer represents our algorithm. The meaning of the instance name such as En51-k4 is that the number of customers is 50 and the number of vehicles available is 4. The results in the f/fmin column are obtained by dividing the function value of one algorithm by the minimum of the three algorithms. The results in the load column are the average of the load densities of used vehicles. Note that the results are the average of 10 problems for each instance and the operating time of the other two algorithms is equal to that of our algorithm. Based upon the results presented in Table 1, we can make the following observations about our methods. (1) With the number of customers increasing in instances tested, the time used increases, but the average performance of our method improves steadily over the other two algorithms for most instances. (2) Our method gives the best solution in all instances when compared to the other two algorithms. This shows the advantage of the cyclic transfer neighborhood and iterated local search.
(3) For most instances, the solutions obtained by our method have a higher load density while the cost is lower.
6 Conclusions In this paper, we proposed a new model for the hot rolling production scheduling problem in which we considered more general conditions, and thus derived a new variant of the vehicle routing problem called PCVRP. This problem has some special characteristics: (1) The number of vehicles available is limited. (2) There are capacity constraints. (3) Not all customers must be visited compulsorily but visiting a customer can result in a prize. (4) The demand of visited customers should be no less than a predetermined amount. (5) The objective is a linear combination of three: minimization of total distance, minimization of vehicles used, and maximization of prizes collected. Though the vehicles used in our problem are homogenous, the formulation of the model and the algorithm we proposed can be easily modified to deal with problems consisting of heterogeneous vehicles. Furthermore, by changing the task completion parameter a as well as the coefficient i , our model can satisfy different kinds of production requirements. To solve this problem we proposed an ILS algorithm based on VLSN obtained by the cyclic transfer method. Computational experiments on 18 standard problem instances, each of which included 10 problems, with extra new generated data show that the method is both effective and efficient. Acknowledgements We would like to thank Professor P. Cowling and Professor S.E. Butt for sending us their papers relative to our research. The research is supported by National Natural Science Foundation of China (Grant No. 60274049) and (Grant No. 70171030), Fok Ying Tung Educaton Foundation, the Excellent Young Faculty Program of the Ministry of Education, China, and National Natural Science Foundation for Distinguished Young Scholars of China (Grant No. 70425003).
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