Acta Mathematicae Applicatae Sinica, English Series Vol. 31, No. 4 (2015) 1091–1102 DOI: 10.1007/s10255-015-0532-y http://www.ApplMath.com.cn & www.SpringerLink.com
Acta Mathemacae Applicatae Sinica, English Series The Editorial Office of AMAS & Springer-Verlag Berlin Heidelberg 2015
Differential Approximation Algorithm of FSMVRP Yu-zhen HU1,2 , Bao-guang XU1,† 1 Institute
of Policy and Management, Chinese Academy of Sciences, Beijing 100190, China
(Email: Email:
[email protected]) 2 School of Economics and Management, Harbin Engineering University, Heilongjiang Harbin 150001, China (Email:
[email protected])
Abstract
We study the fleet size and mix vehicle routing problem with constraints on the capacity of each
vehicle. The objective is to minimize the total cost including fixed utilization cost of vehicles and traveling cost by vehicles. We give differential approximation algorithms for the fleet size and mix vehicle routing problem (FSMVRP) with two kinds of vehicles, the capacities of which are respectively n1 k and n2 k, n2 > n1 ≥ 1, k ≥ 1. Using existing theories for vehicle routing problems and feature of the algorithms represented in the paper, we
also prove that the algorithms give 1 −
and 1 −
6n2 +3n1 (n1 k+n2 k)2 k
Keywords
differential approximation ratio for (k, nk) VRP, n > 1
differential approximation ratio for (n1 k, n2 k) VRP, n2 > n1 > 1.
FSMVRP; differential approximation ratio; approximation algorithm
2000 MR Subject Classification
1
6n+3 (n+1)2 k+n+1
68R05
Introduction
In this paper, we study differential approximation for, a variant of the well-known vehicle routing problem (VRP), the fleet size and mix vehicle routing problem (FSMVRP). In the problem m customers, each of whom has demand qi = 1, i = 1, 2, · · · , m, have to be served by vehicles of limited capacities from a common depot. Two types of vehicles with different capacities can be used, and there are sufficient quantities of each vehicle type. A solution composed of a set of routes is obtained since the following constraints are satisfied: 1. each route starts and then returns to the common depot; 2. each customer is visited exactly once; 3. the total demand of all customers served by a vehicle can not exceed the capacity of the vehicle. The objective is to minimize the total cost, which is composed of vehicle fixed utilization costs and variable traveling costs. Vehicle routing problem and even its simple variants are NP-hard. Approximation algorithms for VRP with only one type of vehicle are considered by some papers and there are two ways to measure their efficiencies, one of which is the standard measure giving the ratio α = val/opt where val and opt are the values of the approximation and the optimal solution respectively, and the other of which is the differential measure giving the ratio Manuscript received July 2010. Revised July 2011. supported by the project of Central University Basic Research Fund (HEUCF150903), the project of the major research task, institute of Policy and Management, Chinese Academy of Sciences (Y201181z01), the National Natural Science Foundation of China(71273072). † Corresponding author.
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α = (wor − val)/(wor − opt) where wor is the value of the worst solution. It is easy to see that 2VRP is polynomial time solvable[3] where 2 means that the capacity of the vehicle is 2, and Haimovich and Rinnooy Kan[11] proved that Metric k VRP[14] for k ≥ 3 is NP-hard where Metric means that the triangle inequality holds in the paper and k is an integer and means that the capacity of the vehicle is k. Many approximation algorithms for kVRP have been studied, for instance, Haimovich et al.[12] gave a 5/2 − 3/(2k) standard approximation for Metric k VRP, Bompadre et al.[4] improved the approximation ratio of the algorithms by Altinkemer and Gavish when the vehicle capacity is fixed, Bazgan et al.[3] considered the first differential approximablity on kVRP and gave the differential ratio of Metric k VRP, Nagoya[17] gave the differential ratio for k VRP without assuming the triangle inequality holds. Many papers focused on the theory and application of differential ratio, such as [1–3,6–10,13,15,16]. However, little research has been done to give the standard or differential approximation on FSMVRP considering not only travel cost but also fixed costs of vehicles having different capacities and consequently different fixed costs. In this paper, an approximation algorithm is given for FSMVRP with two kinds of vehicles of different capacities with coordinating fixed costs, and then is measured by differential approximation ratio α = (wor − val)/(wor − opt), where wor is the value of the worst solution, that is one vehicle with large capacity serves only one customer. This paper is organized as follows: in Section 2, we establish the notations. In Section 3, we will give the differential approximation algorithm for FSMVRP with two kinds of vehicles and measure it.
2
Notations
Let G = (V, E) be a complete undirected graph, in which V = {0, 1, · · · , m} and each edge (i, j) ∈ E has a travel cost dij ≥ 0, and dij is assumed to satisfy the triangle inequality. Node 0 is the depot node and the rest of the nodes i, i = 1, 2, · · · , m are customers. The demand of each customer is denoted by qi , and qi = 1, i = 1, 2, · · · , m. There are two kinds of vehicles with capacity n1 k and n2 k (k ≥ 1) respectively, where n1 and n2 are integers and satisfy n1 < n2 . In this paper, FSMVRP with two types of vehicles is denoted by (n1 k, n2 k) VRP. Fixed costs of two kinds of vehicles are p1 and p2 respectively, where p1 and p2 are real numbers and satisfy 0 < p1 < p2 . The route of each vehicle starts and ends at node 0, and each vehicle cannot deliver more than its capacity n1 k or n2 k. The cost of a solution is the sum of the travel cost and the fixed cost of each vehicle. The following notations are used in the paper as well. - T = (1, 2, · · · , m, 1) is a tour obtained by the heuristic algorithm described in [5], and it satisfies d(T )/opt(T SP ) < 3/2, where d(T ) is the value of the approximation algorithm presented in [5] and opt (TSP) is the value of the optimal solution of TSP. In this paper, m ≡ kq + r, that is r = m mod k, so 0 ≤ r < k. - Let t = q mod (n + 1) in case of (k, nk)VRP and t = q mod (n1 + n2 ) in case of (n1 k, n2 k)VRP. x mod m : x > m - x= , x = 1, 2, . . ., and i and other notations with overline have x : x≤m similar meaning with x. - (i, i + 1) is an edge joining nodes i and i + 1. - T \ (i, i + 1) ∪ (i + r + lk, i + 1 + r + lk) : 0 ≤ l < q, l = j(n + 1), j(n + 1) + n, j = q − 1 is a graph obtained from cycle T by deleting edge (i, i + 1) and edges 0, · · · , n+1
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q (i + r + lk, i + 1 + r + lk) : 0 ≤ l < q, l = j(n + 1), j(n + 1) + n, j = 0, · · · , n+1 − 1, 2q which is a disjoint union of n+1 + 1 paths (components).
- μi,j , i = 1, 2, · · · , m, j = 1, 2, · · · , s denotes a path rather than a cycle,where s is an integer. - (0, μi,j , 0) is a cycle going through node 0 and path μij . - soli =
s j=1
(0, μi,j , 0) is a graph denoted by soli , i = 1, 2, · · · , m including cycles (0, μi,j , 0).
- d(soli ) denotes the value of the solution based on the graph soli , i = 1, 2, · · · , m. - opt(VRP) denotes the value of the optimal solution of (n1 k, n2 k)VRP, n2 > n1 ≥ 1. - wor(VRP) denotes the value of the worst solution of (n1 k, n2 k)VRP, n2 > n1 ≥ 1. - optR (VRP) and worR (VRP) denote the value of the optimal solution and the value of the worst solution of (n1 k, n2 k)VRP respectively, n2 > n1 ≥ 1, if the routing cost is optimized while the fixed cost is not taken into account. - optF (VRP) and worF (VRP) denote the value of the optimal solution and the value of the worst solution of (n1 k, n2 k)VRP respectively, n2 > n1 ≥ 1, if the fixed cost is optimized while the routing cost is not taken into account.
3
The Algorithms for FSMVRP
In this paper, the problem is considered only when p2 /p1 ≥ n2 /n1 , because if p2 /p1 < n2 /n1 , the fixed cost of each customer served by the vehicle of capacity n2 k is lower than that of the vehicle of capacity n1 k, so vehicles of large capacity are preferred in deliveries to all customers if optimizing fixed cost rather than routing cost. vehicles of large capacity are also preferred if considering routing cost rather than fixed cost. It will not make sense due to lack of trade-off between fixed cost and travel cost. Algorithms to be described here are called A(k,nk) and A(n1 k,n2 k) respectively. A(k,nk) is the algorithm for (k, nk)VRP, n > 1, and A(n1 k,n2 k) is the algorithm for (n1 k, n2 k)VRP, n2 > n1 > 1. Before giving formal descriptions, we want to present main ideas about the two algorithms. The core idea is the same essentially. Let S = (q − t)/(n + 1) in case of (k, nk)VRP and S = (q−t)/(n1 k+n2 k) in case of (n1 k, n2 k)VRP. Firstly, kq−kt customers are served alternately by S full-loaded vehicles of large capacity and S full-loaded vehicles of small capacity. Secondly, the remaining kt + r customers are all served by vehicles of large capacity, and, apparently, at most two vehicles of large capacity are needed. The Input of algorithms is the tour T = (1, 2, · · · , m, 1), m ≡ kq + r, 0 ≤ r < k, and then the output is ∪(0, μi,j , 0). In the output of the algorithm A(k,nk) the number of cycles obtained when r = 0 is different from the number of cycles obtained when r = 0, while in the output of algorithm A(n1 k,n2 k) the number stays the same no matter whether r = 0 or r = 0. In algorithm A(k,nk) , r customers are served by a vehicle of large capacity if r = 0, tk customers are visited by a vehicle of large capacity if t = 0,and the remaining customers are served alternately by two types of vehicles nether more nor less. The output of algorithm A(k,nk) when r = 0 and t = 0 is shown in Fig.1, where r, nk represents that r customers are visited by a vehicle of capacity nk, and nk, nk , k, k , tk, nk are interpreted similarly. In the algorithm A(n1 k,n2 k) , (n1 − 1)k + r customers are served by a vehicle of large capacity no matter
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whether r = 0 or r = 0, (t − n1 + 1)k customers are visited by a vehicle of large capacity if n1 ≤ t < n1 + n2 , and the remaining customers are served alternately by two types of vehicles.
node 0 one customer
suspension points
the routing between two customers
Fig.1. The Output of Algorithm A(k,nk)
3.1
Algorithm A(k,nk)
We give the following algorithm for (k, nk)VRP, n > 1, whose running time is O(m). Algorithm A(k,nk) 1. Find a TSP tour of V \{0}, T = (1, 2, · · · , m, 1), m ≡ kq + r, 0 ≤ r < k; 2. If r = 0 (a) If t = 0, for i = 1 to m do - Let (μi,1 , · · · , μ
) 2q i, n+1 +1
= T \[(i, i + 1) ∪ {(i + r + lk, i + 1 + r + lk) : 0 ≤ l
1, n1 +n and kq (6−3n2 k−n1 k)q−(6+2n1 k)t+6(n1 +n2 ) 2(q−t) are decreasing with q and t, − n1 +n2 + 1 is decreasing with 2(n1 +n2 )n1 k q, increasing with t, we deduce:
2(q − t) (6 − 3n2 k − n1 k)q − (6 + 2n1 k)t + 6(n1 + n2 ) p1 − + 1 p2 < 0, 2(n1 + n2 )n1 k n1 + n2 3(q−t) n1 +n2
kq
+3
≤
6n2 + 3n1 , (n1 k + n2 k)2 k
so: val(VRP)
. 2 (n1 k + n2 k) k 2(n1 + n2 )k
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