A STUDY ON VEHICLE ROUTING PROBLEM WITH INTER LOADING FACILITIES

In this article, we discuss variant of the multi-depot vehicle routing problem where depots can act as intermediate replenishment facilities along the route of a vehicle. This problem is a generalization of the Vehicle Routing Problem (VRP). In the sequel we developed a Lexi-Search algorithm based “Pattern Recognition Technique” to solve this problem which takes care of simple combinatorial structure of the problem and computational results are reported.In this paper, we consider a real-world problem with a loaded vehicle. The vehicle is available for pickup and delivery service. Single-commodity cargo, that is, same cargo type, is present at the customer sites. The vehicle is able to carry out a Pickup and delivery service of such cargo among the customers. By starting and ending at the origin depot, the vehicle follows a route without having a predefined sequence of Pickup and delivery


INTRODUCTION
As energy costs increases, driver shortages continue and hours of service regulations get tighter, transportation providers are motivated to use the capacity available to them as efficiently as possible. Excess capacity that is not utilized can result in declined service requests or additional costs, as vehicles that would otherwise not be necessary must be dispatched. Trucking firms are eager to eliminate any such occurrences, whether through finding loads for the backhaul portion of a route or consolidating loads from multiple customers onto a common vehicle. Improving capacity utilization through these methods is rather common. However, a more subtle extension of load consolidation is to allow multiple vehicles to service the same load. Splitting loads such that the delivery of certain loads is completed in multiple trips rather than one trip results in opportunities for a reduction in cost and the number of vehicles used. While this generally requires additional visits to a load's origin and destination, it may eliminate a dedicated trip to deliver the load by apportioning that load to other vehicles with excess capacity. Several studies have shown the benefit of split deliveries for the Vehicle Routing Problem (VRP), Dror et al. In the year 2002 Gendreau, Laporte and Portvin [1], in their article presented mataheuristics for the capacitated VRP and proposed six main types of mataheuristics that have been applied to the VRP. In the year 2006 Hollis, Forbes and Douglas [2] presented a new multi-depot combined vehicle and crew scheduling algorithm and uses it in conjunction with a heauristic vehicle routing algorithm to solve the intra-city mail distribution problem faced by Australia post.
Baldacci, Hadjiconstantinou and Mingozzi [3] in the year 2003, in their article described a new integer programming formulation for the travelling salesman problem with mixed deliveries and collections based on a two commodity network flow approach. Anju Gupta, Vanita Verma and Puri [4] in the year 1995, in their research article, studied special type of bicriteriabulk transportation problem involving the trading-off cost against time. In the year 2007, Borovska, Lazarova and Bahudejla [5], in their paper, discussed different strategies for parallel genetic computation of optimization problems. Abuali, Wainwright and Schoenefeld [6], in the year 1995, in their research article, described a new encoding scheme for the representation of spanning trees and this new encoding scheme is based on the factorization of the determinant of the in-degree matrix of the original graph. In the year 2006, Herer,Tzur and Yucesan [7], in their article, consider a supply chain which consists of several retailers and one supplier and demonstrated that how the values of the order-up to quantities can be calculated using a sample-path-based optimization procedure. In 2003, Vijayalakshmi [8], in her paper, introduced Lexiseaarch approach to the travelling salesman problem. In the year 2012, K.Chendra Sekhar [9], in his paper, depicted a problem recognition LexiSearch approach to travelling salesman problem with additional constraints. Zakir Hussain Ahmed [10], in the year 2011, first modified an existing Lexisearch algorithm by incorporating good upper and lower bounds to obtain exact optimal solution to the problem and then presented adata guided Lexisearch algorithm. Shalini Arora [11], in 2017, in their research article, studied time minimizing assignment problem also known as the bottleneck assignment problem and proposed a lexisearch approach to find an optimal feasible assignment of the facilities so as to minimize total time of completion of n' out of n jobs. In 2011, Amit Kumar, Amarpreet Kaurand Anila Gupta [12], in their paper, proposed new methods for solving Fuzzy transportation problems with some additional transshipments.
Kek, Chen and Meng [13], in 2008, in their research article, proposed two new distance-constrained capacitated vehicle rooting problems to investigate for the first time and study potential benefits in flexibly assigning start and end depots. In 2006,Archetti,Hertz and Speranza [14] described a tabu search algorithm for the vehicle rooting problem with split deliveries and in their discussion the insertion of a customer into a route is done by means of the cheapest insertion method. This special case of the VRP is most commonly referred to as the Split Delivery Vehicle Routing Problem (SDVRP), which occurs when a destination may be serviced by multiple vehicles. The PDPSL is a more complex problem than the SDVRP, primarily because the available capacity of the vehicle changes each time a load is picked up or delivered for the PDPSL, without the vehicle ever returning to a depot. The SDVRP load planning is done with the same fixed capacity prior to a vehicle leaving the depot. With the PDPSL, searching through each instance of available capacity to determine where to insert a split load is significantly more difficult, as all loads that are to be picked up or delivered in between the pickup and delivery of the load to be inserted must be accounted for in evaluating the capacity. Despite the differences, some of the approaches used for solving the SDVRP are applicable to the PDPSL, such as determining how to divide a load that is to be split and finding routing improvements. The PDP is a generalization of the Traveling Salesman Problem (TSP), making it NP-hard in the strong sense. This has focused most current research on heuristic methods.
In this paper, we consider a real-world problem with a loaded vehicle. The vehicle is available for pickup and delivery service. Single-commodity cargo, that is, same cargo type, is present at the customer sites. The vehicle is able to carry out a Pickup and delivery service of such cargo among the customers. By starting and ending at the origin depot, the vehicle follows a route without having a predefined sequence of Pickup and delivery services. This problem is called a one-commodity vehicle routing pickup and delivery problem (1-VRPPD). The main feature of 1-VRPPD is that delivery customers can be served with a cargo gathered from Pickup customers.
This problem was first defined by Hernández-Pérez and Salazar-González (2004) as a one-commodity Pickup and delivery traveling salesman problem (1-PDTSP), which is the same as 1-VRPPD, when the number of vehicles is equal to 1. In general, the VRP problem and its variations are NP-hard. 1-VRPPD is NP-hard problem in the strong sense it specializes VRP.
The main VRP feature that differs from traveling salesman problem (TSP) is vehicle capacity constraint.
In this article, we discuss variant of the multi-depot vehicle routing problem where depots can act as intermediate replenishment facilities along the route of a vehicle. This problem is a generalization of the Vehicle Routing Problem (VRP). In the sequel we developed a Lexi-Search algorithm based "Pattern Recognition Technique" to solve this problem which takes care of simple combinatorial structure of the problem and computational results are reported.

PROBLEM DESCRIPTION
In this paper we consider 'Vehicle Routing Problem with Inter Loading Facilities'. There are some cities/stations available. Among them some of the cities act as sources including head quarter and remaining cities act as destinations. All sources have some availability of goods/commodity/load and all destinations have the requirements of goods. The vehicle starts from the head quarter with given load capacity and supply requirements of some destinations. If all the destination requirements are satisfied then the vehicle comeback to the head quarter city.
Suppose the load/goods of a vehicle is low while supplying the destinations, then there is a facility that the vehicle fill the sufficient load by visiting the near source station and supply the destinations. Here the vehicle need not visit all source cities while supplying the destination requirements, but all the destinations must be satisfied. The availability of goods at the source cities are always greater than or equal to vehicle capacity. The aim of the problem is to find minimum distance/cost for satisfy the all destinations subject to the above conditions. Let N be the set of n stations defined as N= {1, 2, 3, 4,……,., n} and here the city '1' taken as the home city/head quarter city. Among them let S be the set of k sources including city 1 and defined as S= {α1, α2, α3 ,…., αk}. Let N1be the set of n-k destinations defined as N1= {β1, β2, β3 ,….,βn-k}. Let the requirement of destination j є N1 is DR (j) and the capacity of the source i є S is SC (i). Let the vehicle load capacity be 'α'.The vehicle starts its tour from the home city (say 1) and come back to it after supplying the requirement of all n-k destinations. The vehicle 4469 VEHICLE ROUTING PROBLEM WITH INTER LOADING FACILITIES may or may not visit all the ksource stations in its tour. While supplying the destination requirements, if the load of a vehicle is less than the requirement of destination requirements then the vehicle must visit the nearest source city to filling the load up to its capacity and supply the destination requirements. Hence the objective of the problem is to find a minimum total distance in its tour while completing all the destination requirements subject to the conditions. For this we developed an algorithm called as Lexi-Search algorithm using pattern recognition Technique.

MATHEMATICAL FORMATION
Main results.
Subject to the constraints: , ∈ , 1 , 2 , 3 , … … … … . . , , be the sequence of cities in the tour The equation (1) describes the objective function of the problem i.e., to minimize the total distance/cost subjected to the constraints. The constraint (2) represents that the trip includes m cities. The constraint (3) takes care of the restriction of availability and requirement of the product between sources and destinations, i.e., the sum of the available capacities at sources is more than the sum of the demands at destinations of a product. The constraint (4) assumes that the demand at the destination from source to source must be lesser than or equal to the vehicle capacity. The constraint (5) denotes that if the vehicle is traversed from i to j its value is 1 otherwise it is 0.

NUMERICAL FORMULATION
The concepts and the algorithms will be illustrated by a suitable numerical example. In which we have taken number of cities as n=9 (N = {1, 2, 3 is the requirement of a product at the destinations. Then the distance/cost matrix D is given below in  Suppose D (4, 7) = 2 means that the distance between the cities 4 and 7is 2. Moreover, SC and

FEASIBLE SOLUTION
Feasible solution is a solution, which satisfies all the constraints in the problem. The constraints are discussed in the mathematical formulation. Consider an ordered pair set {(1, 2), (2, 6), (6,8), (8,9), (9,3), (3,4), (4, 7), (7, 5), (5,1)} represents a feasible solution. In the following figure-1, the values in the ellipse denote name of the destination city and the values in the parenthesis of ellipse denotes the requirement of the destination city. The rectangle box represents sources/pickup points and its available capacity indicated in the respective parenthesis. The values along the arcs indicate the distance between the connected cities.

SOLUTION PROCEDURE
In the above figure-1, for the feasible solution we observe that 9 ordered pairs are taken along with values from distance matrix for numerical example in Table-1. The 9 ordered pairs are selected such that we they represent a feasible solution in figure-1. So, the problem is that we selected9 ordered pairs from distance matrix order [9×9] along with values such that the total distance is minimum to represents a feasible solution. The number of ordered pairs will be >6, here it is 9, which satisfies the supply to the 6 cities, this 9 is not fixed, it can vary. For this selection of 9 ordered pairs we arrange all the ordered pairs with increasing order and call this formation as alphabet table and we developed an algorithm for the selection along with checking for the feasibility.

INFEASIBLE SOLUTION
Infeasible solution is a solution which does not satisfies all the constraints in the problem. Further the vehicle comeback to head quarter city from city 5.This is again a contradiction that the vehicle has completed its tour without supplying all destinations' requirements. Finally, the vehicle started its trip from the source city 8and reached the destination city 9, there, it has supplied its requirement from the vehicle; again, it has reached another destination city 3 and = 0 + 0 + 1 + 1 + 2 + 3 + 4 +9 + 12 Z = 32 units.

Definition of a Pattern
An indicator two-dimensional array which is associated with an assignment is called a 'pattern'. A Pattern is said to be feasible if X is a solution.  Table-3 Table -3, which is an infeasible solution. . The cost assigned is the single array, for convenience it is also named as D.   (4,9).Then D(21)=D(4,9)= 13 and DC(21)=129.

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. 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.  Where DC (ak) = ∑ ( ) =1 . It can be seen that LB (Lk) is the value of the complete word, which is obtained by concatenating the first (n-k) letters of SN to the partial word Lk.

8.6.Feasibility criterion of a Partial Word
An algorithm was developed, in order to check the feasibility of a partial word Lk+1 = (a1, a2, ----ak, ak+1) given that Lk is a feasible word. We will introduce some more notations which will be useful in the sequel.   Table -5   1  2  3  4  5  6  7  8  9   L  1  2  3  5  8 --- The recursive algorithm for checking the feasibility of a partial word Lp is given as follows in the algorithm first we equate IX = 0. At the end if IX = 1 then the partial word is feasible, otherwise it is infeasible. For this algorithm we have RA = R (ap+1) and CA = C (ap+1) We start with the partial word L1 = (a1) = (1). A partial word Lk is constructed as Lk = Lk-1 * (p).
Where * indicates chain formulation. We will calculate the values of V (Lk) and LB (Lk) simultaneously. Then two situations arise one for branching and other for continuing the search.

SEARCH TABLE
The working details of getting an optimal word using the above algorithm for the illustrative numerical example is given in the Table-6.The columns (1), (2), (3), (4), (5), (6), (7), (8) and (9) gives the letters in the first, second, third, fourth, fifth, sixth, seventh, eighth and ninth positions of a word respectively. The next two columns V and LB are indicate the value and lower bound of the respective partial word. The column R and C gives the row and column indices of the letter.
The last column gives the remarks regarding the acceptability of the partial words (i.e., if a partial word is feasible word, then accept the letter otherwise reject the letter) and here A indicates the acceptance and R for rejectance of the letter in the respective position.  1, 2, 4, 5, 7, 8, 10 15, 17). Then the following figure-3represents the optimal solution to the problem.

EXPERIMENTAL RESULTS
The following table shows that the computational results for proposed Lexi-Search algorithm using pattern recognition technique. We presented computer program for this algorithm in C language, and it is verified by the system COMPAQ dx2280 MT. We ensure this algorithm by trying a set of problems for different sizes. We took different random numbers as  Table -9 are given below. For each instance, five to seven data sets are tested. It is seen that the time required for the search of the optimal solution is fairly less. In the following In the above Table-

CONCLUSION
In the above conversation, we have presented an exact algorithm called Lexi-Search algorithm using pattern recognition technique to solve "Vehicle Routing Problem with Inter Loading Facilities". 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.