SOLVING THE CAPACITATED VEHICLE ROUTING PROBLEM WITH LEXI-SEARCH APPROACH

: In this paper we present a variant vehicle routing problem called “Solving the Capacitated Vehicle Routing Problem with Lexi-Search Approach” (CVRP). The purpose of this article is to propose an efficient Lexi-Search Algorithm using pattern recognition technique for solving CVRP on a scalable multicomputer platform and to obtain an optimal solution. Our results show that the proposed algorithm is highly competitive on a set of benchmark problems. In this paper we focus our investigation on solving the capacitated VPR (CVPR) and considered a variant vehicle routing problem called as “ Solving the Capacitated Vehicle Routing Problem with Lexi-Search Approach”. First the model is formulated into a zero-one programming problem. A Lexi-Search Algorithm using Pattern Recognition Technique is developed for getting an optimal solution. The problem is discussed with suitable numerical illustration. We have programmed the proposed algorithm using C-language. The computational details are reported. As an observation the CPU run time is fairly less for higher values to the parameters of the problem to obtain optimal solutions


INTRODUCTION
The general trend in the transportation sector is that transportation companies are merging to larger units which can be provided a large number of delivery services. In order to get the most possible benefit from the vehicle fleet, it can be attractive to serve conceptually different transportation tasks by the same fleet, thus models are needed that can handle all additional constraints associated with a transportation task, for example, provide a unified approach for several Vehicle Routing Problems with Time Windows.
There are objectives other than minimizing the transportation cost that may arise in vehicle routing problems such as minimizing the number of vehicles required to serve all customers, balancing the routes, or minimizing the waiting time of a customer.
The Vehicle Routing Problem (VRP) is a generic name given to a whole class of problems in which a set of routes for a fleet of vehicles based at one or several depots must be minimized for a number of geographically dispersed cities or customers. In the VRP, the road network is represented by a graph with arcs and vertices. Arcs represent roads and vertices represent road intersections, junctions, customer locations, and the depot. Each arc has an associated cost. Each customer location vertex has an associated number of goods to be delivered.
Each vehicle has its own capacity and cost associated with its utilization.
Another well-known generalization of the VRP is the Multi-Depot Vehicle Routing Problem (MDVRP). In this extension every customer is visited by a vehicle based at one of several depots. In the standard MDVRP every vehicle route must start and end at the same depot.
There exist only a few exact algorithms for this problem.  (1977) and Raft (1982). More recently, Chao et al. (1993) have proposed a search procedure combining Dueck's (1993) record-to-record local method for the reassignment of customers to different vehicle routes, followed by Lin's 2opt procedure (1965) for the improvement of individual routes. Renaud et al. (1996)

described a
Tabu search heuristic in which an initial solution is built by first assigning every customer to its nearest depot. A Petal algorithm has developed by the latter authors (1996) is then used for the solution of the VRP associated with each depot. It finally applies an improvement phase using either a subset of the 4-opt exchanges to improve individual routes, swapping customers between routes from the same or different depots, or exchanging customers between three routes. The Tabu search approach of Cordeau et al. (1997) is probably the best known algorithm for the MDVRP. An initial solution is obtained by assigning each customer to its nearest depot and a VRP solution is generated for each depot by means of a sweep algorithm. Improvements are performed by transferring a customer between two routes incident to the same depot, or by relocating a customer in a route incident to another depot. Reinsertions are performed by means of the GENI heuristic (1992). One of the main characteristics of this algorithm is that infeasible solutions are allowed throughout the search. Continuous diversification is achieved through the penalization of frequent moves. In this paper we focus our investigation to solving the capacitated VPR (CVPR).
As in most NP-hard problems, three approaches are typically employed to solve these types of problems: heuristics, approximation methods and exact methods. While heuristics do not provide guarantees about the solution quality, they are useful in practical contexts because of their speed and ability to handle large instances. A special class of heuristics is Meta-Heuristics, which are general frameworks for heuristics. Approximation algorithms are a special class of heuristic that provides a solution and an error guarantee.
We are aware of a number of VRP algorithms based on this approach. One of the first attempts to apply Tabu search to the VRP is due to Willard (1989). Here, the problem is first transformed into a TSP by replication of the depot, and the search is restricted to neighbour solutions that can be reached by means of 2-opt or 3-opt interchanges while satisfying the VRP constraints. In Pureza and Franca (1991), the search proceeds from one solution to the next by swap-ping vertices between two routes. Osman (1991,1993) uses a combination of 2-opt moves, 4769 CAPACITATED VEHICLE ROUTING PROBLEM vertex reassignments to different routes, and vertex interchanges between routes. Another algorithm was developed by Semet and Taillard (1993) for the solution of a real-life VRP containing several features, and different from the version considered in this paper. Here the basic Tabu move consists of moving a city from its current route into an alternative route. Finally, Taillard (1992) partitions the vertex set into clusters separately through vertex moves from one route to another. Clusters are updated throughout the algorithm. Note that in all these algorithms, a feasible solution is never allowed to be-come infeasible with respect to side constraints. Exact methods guarantee that the optimal solution is found if the method is given sufficiently time and space. The VRP is a hard combinatorial problem, and to this day only relatively small VRP instances can be solved to optimality. Interesting exceptions are the problems solved to optimality by Fisher (1989), using minimum k-trees.
In this paper we present a variant vehicle routing problem called "Solving the Capacitated Vehicle Routing Problem with Lexi-Search Approach" (CVRP). There are 'n' cities in that city '1' is head quarter city, each city have positive demand. Multiple vehicles with uniform capacity starts from head quarter and supply according to their demand with total minimum distance and return to head quarter. The purpose of this paper is to propose an efficient Lexi-Search Algorithm using pattern recognition technique for solving CVRP on a scalable multicomputer platform and to obtain an optimal solution. Our results show that the proposed algorithm is highly competitive on a set of benchmark problems. The remainder of this paper is organized as follows.

PROBLEM DESCRIPTION
In this discussion we considered a variant vehicle routing problem called as "Solving the Capacitated Vehicle Routing Problem with Lexi-Search Approach". There are some vertices available. Vertex '1' considered a depot and the remaining vertices are considered as cities, each city has known demand. Few vehicles with uniform capacity are available at a depot. Each vehicle starts at depot and supply according to their demand in different routes with minimum cost/distance and return to depot. Each vehicle visits each city only once. Here, there is a restriction that the vehicle capacity is always greater than or equal to total demand of cities in that vehicle route. Let G = (N, A) be a directed graph where N = {1, 2, 3………, n} is a vertex set, and A = {(i, j): i ≠ j]} is an arc set. Vertex '1' denotes depots at which 'm' identical vehicles are based with uniform capacity 'Q' are known and the remaining vertices of N represent (n-1) cities.
Every city i have a requirement Qi which is known .The value of m is given. Every arc (i, j) is associated a positive distance cij. (For the sake of simplicity, the terms "distances," "travel times," and "travel costs" will be used interchangeably.) The VRP consists of designing a set of total least cost vehicle routes in such a way that every route starts and ends at the depot. Every city of N -{1} is visited exactly once by exactly one vehicle and every city is associated with a positive demand Qi. The total demand of any vehicle route will not exceed the vehicle capacity Q. The aim of the problem is to find feasible solution which meets the above conditions such that the total cost/distance is minimum. A 0-1 programming formulation of the problem of routing to minimize the cost subject to vehicle capacity constraint is given below. Constraint (1) represents the objective function i.e., a fixed fleet of delivery vehicles of uniform capacity must service to known customer demands of a single commodity with the minimum transportation cost.

MATHEMATICAL FORMULATION
Constraint (2) and (3) represents that each city is visited exactly only once, except the home city.
Constraint (4) and (5) represents that 'm' vehicles starts at home city visits cities according to the requirements and returns to home city.
Constraint (6) represents that the number of connectivity's on supplying the required demands from the home city.
Constraint (7) represents that a vehicle travels from i th city to j th city is denoted by 1 otherwise 0.
Constraint (8) represents that the minimum number of vehicles.
In the sequel we developed a Lexi-Search algorithm using "Pattern Recognition Technique" to solve this problem which takes care of simple combinatorial structure of the problem.

SOLUTION PROCEDURE
In the above figure-1, for the feasible solution we observe that 10 ordered pairs are taken along with the values from the cost/distance matrix for this numerical example in Table- Table- Table -4, which is an infeasible solution. Table   There  .

Definition of a Word
Let SN = (1,2,…) be the set of indices, D be an array of corresponding distances of the ordered pairs and Cumulative sums of elements in D is represented as an array DC. Let arrays R, C be respectively, the row, column indices of the ordered pairs. Let Lk = {a1, a2, -----, ak}, ai ∈ SN be an ordered sequence of k indices from SN. The pattern represented by the ordered pairs whose indices are given by Lk is independent of the order of ai in the sequence. Hence for uniqueness the indices are arranged in the increasing order such that ai < ai+1, i = 1, 2, ----, k-1.
The set SN is defined as the "Alphabet- Table" with alphabetic order as (1, 2, ----, n 2 ) and the ordered sequence Lk is defined as a "word" of length k. A word Lk is called a "sensible word". If ai < ai+1, for i =1, 2, ----, k-1 and if this condition is not met it is called a "insensible word". A word Lk is said to be feasible if the corresponding pattern X is feasible and same is with the case of infeasible and partial feasible pattern. A Partial word Lk is said to be feasible if the block of words represented by Lk has at least one feasible word or, equivalently the partial pattern represented by Lk should not have any inconsistency.
In the partial word Lk any of the letters in SN can occupy the first place. Since the words of length greater than n-1 are necessarily infeasible, as any feasible pattern can have only n unit entries in it. Lk is called a partial word if k < n-1, and it is a full length word if k = n-1, or simply a word. A partial word Lk represents, a block of words with Lk as a leader i.e. as its first k letters.
A leader is said to be feasible, if the block of word, defined by it has at least one feasible word.

A recursive algorithm is developed for checking the feasibility of a partial word. A leader
Lk is said to be feasible if the block of words defined by it contains at least one feasible word.
Let Lk+1= (α1, α2…αk, αk+1) given that Lk is a feasible partial word. We will introduce some more notations which are useful in the sequel.  Table -6, given below. 1. LB (Lk) < VT. Then we check whether Lk is feasible or not. If it is feasible we proceed to consider a partial word of order (k+1), which represents a sub block of the block of words represented by Lk. If Lk is not feasible then consider the next partial word of order by taking another letter which succeeds ak in the k th position. If all the words of order 'k' are exhausted then we consider the next partial word of order (k-1).
2. LB (Lk) > VT. In this case we reject the partial word Lk. We reject the block of word with Lk as leader as not having optimum feasible solution and also reject all partial words of order 'k' that succeeds Lk. Now we are in a position to develop a Lexi-Search algorithm to find an optimal feasible word. 2) LB (Lk) ≥VT. In this case we reject the partial word Lk. We reject the block of word with Lk as leader as not having optimum feasible solution and also reject all partial words of order 'k' that succeeds Lk. Now we are in a position to develop a Lexi-Search algorithm to find an optimal feasible word.
The current value of VT at the end of the search is the value of the optimal word. At the end if VT= ∞, it indicates that there is no feasible allotment.

EXPERIMENTAL RESULTS
The following  for the search of the optimal solution is moderately less.

COMPARISON DETAILS
We implemented Lexi Search Algorithm (LSA) using Pattern Recognition Technique with C language for this model. We tested the proposed algorithm by different set of problems and compared the computational results with the published vehicle routing problem by awarded thesis of Madhu Mohan Reddy (2015). Then Table-11 shows that the comparative results of different sizes. In the following table microseconds are represented by zero. The graphical representation of the CPU run time for the two models presented in the above 5 instances is given below. In the Graph-1, X axis taken the SN and Y axis taken the values of CPU run time for the published and proposed models.

GRAPH-1
From the above Graph-1, series 2 represent that CPU run time for getting optimal solution by proposed model and series 1 represent that CPU run time for searching the optimal solution by the published model. Also the proposed model takes less time than published model for giving the solution.

CONCLUSION
In this paper, we have presented an exact algorithm called Lexi-Search algorithm using pattern recognition technique to solve "Solving the Capacitated Vehicle Routing Problem with Lexi-Search Approach". First the model is formulated into a zero-one programming problem. A Lexi-Search Algorithm using Pattern Recognition Technique is developed for getting an optimal solution. The problem is discussed with suitable numerical illustration. We have programmed the proposed algorithm using C-language. The computational details are reported. As an observation the CPU run time is fairly less for higher values to the parameters of the problem to obtain optimal solutions.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.