AN EFFICIENT ALGORITHM FOR
VARIANT BULK TRANSPORTATION
PROBLEM
SOBHAN BABU.K*
Department of Mathematics, University College of Engineering, JNTUKAKINADA KAKINADA, ANDHRA PRADESH-533003, INDIA
sobhanjntu@gmail.com
SUNDARA MURTHY.M
Professor,Sri Venkateswara University, Tirupati Tirupati, Chittor(Dt), Andhra Pradesh, INDIA
profmurthy@gmail.com
Abstract :
A large number of real-world planning problems called Combinatorial Optimization Problems share the following properties: They are Optimization Problems, are easy to state, and have a finite but usually very large number of feasible solutions. Lexi-Search is by far the mostly used tool for solving large scale NP-hard Combinatorial Optimization problems. Lexi-Search is, however, an algorithm paradigm, which has to be filled out for each specific problem type, and numerous choices for each of the components exist. Even then, principles for the design of efficient Lexi-Search algorithms have emerged over the years. Although Lexi-Search methods are among the most widely used techniques for solving hard problems, it is still a challenge to make these methods smarter. The motivation of the calculation of the lower bounds is based on ideas frequently used in solving problems. Computationally, the algorithm extended the size of problem and find better solution.
Keywords: Bulk Transportation Problem, Lexi-Search, Pattern Recognition.
1. Introduction
Bulk Transportation problem has been generalized in various directions. In this paper we study a problem called “Three Dimensional Variant Multi Commodity Bulk Transportation Problem”. To discuss a problem with I = {1,2,. . . ,m} set of m sources, J ={1,2, . . .,n} set of n warehouses, K = {1,2, . . .,l} set of l quantities(commodities) to be supplied. r(i,j,k) is the bulk supply of the kth commodity from the ith source to the jthwarehouse whose bulk cost is C(i,jk). D(j,k) be the requirement of the jth warehouse of the kth commodity, B(i,k) is the availability of the kth commodity at the ith source. The commodities requirement at the jth warehouses should be supplied from the ith sources subject to its capacities. The problem is to minimize the cost of allocates of the requirement available at the sources with least cost and satisfying the constraints.
In the sequel we developed a lexi-search algorithm based on the “Pattern Recognition Technique” to solve this problem which takes care of simple combinatorial structure of the problem.
3. Lexi-Search Algorithms for the VBTP:
The name Lexicographic-search or Lexi-search method implies that the search is made for an optimal solution in a systematic way, just as one search for meaning of a word in a dictionary. When the process of feasibility checking of a partial word becomes difficult, though lower bound computation is easy, Pattern Recognition Technique (Sundara Murthy, 1979) can be used. Lexi-Search algorithms, in general, require less memory, due to the existence of Lexicographic order of partial words. If Pattern Recognition Technique is used, the dimension requirement of the problem can be reduced, since it reduces to the two-dimensional cost array into a linear and the problem can be reduced to a linear form of finding an optimal word of length n (Sundara Murthy, 1979) and hence reduces computational work in getting an optimal solution.
4 The Algorithm:
The name Lexicographic-search or Lexi-search method implies that the search is made for an optimal solution in a systematic way, just as one search for meaning of a word in a dictionary. When the process of feasibility checking of a partial word becomes difficult, though lower bound computation is easy, Pattern Recognition Technique (Sundara Murthy, 1979) can be used. Lexi-Search algorithms, in general, require less memory, due to the existence of Lexicographic order of partial words. If Pattern Recognition Technique is used, the dimension requirement of the problem can be reduced, since it reduces to the two-dimensional cost array into a linear and the problem can be reduced to a linear form of finding an optimal word of length n (Sundara Murthy, 1979) and hence reduces computational work in getting an optimal solution. The concepts and notations involved in the Lexi-Search are briefly described below.
4.1 Pattern:
An indicator matrix X, associated with an appropriate assignment of tasks to the agents is defined as a Pattern. A Pattern is said to be feasible, if X is feasible. Each pattern X can also be represented by the set of all ordered triples {(i, j, k)}, for which X (i, j, k) =1. In general, there will be m*n*k ordered pairs in a matrix X (m, n, k).
4.2 Alphabet – Table & Word:
The value of the (partial) word Lk, V(Lk) is recursively defined as V(Lk) = V(Lk-1) + BD (ak) with V(L0) =0, where BD is the cost array arranged such that, BD(ak) < BD(ak+1). V (Lk) and the value of the pattern X, will be the same, since X is the (partial) pattern represented by Lk.
4.4 Lower Bound of a Partial Word:
A lower bound LB (Lk), for the values of the blocks of words represented by Lk = (a1, a2. . . ak) can be defined as follows:
LB = LV (I) + EC (I+NM1) – EC (I)
It can be seen that LB (Lk) is the value of the complete word, which is obtained by concatenating the first m0 –k letters to the partial word Lk.
4.5 Feasibility criterion of a partial word:
A feasibility criterion is 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.
IR be an array where IR (i) = 1, i I = (1, 2, - - -, m) represents that source i assigned to the warehouse j, otherwise IR (i) = 0
IC be an array where IC (j) = 1, j J = (1, 2, - - - , n) represents that warehouse j is required from some source i, otherwise IC (j) = 0
IT be an array where IT (k) = 1, k K = (1, 2, - - -, l) represents that warehouse which gets of its requirement from any source at a point of commodity k.
LW be an array, where LW (i), is the letter in ith position of a word. Then for a given partial word Lk = ( a1, a2, - - - -, ak)
The values of the arrays IC, IT, LW are as follows
IC (C (ai)) = 1, i = 1, 2, - - - , k and IC (j) = 0 for other elements of j IT (T (ai)) = 1, i = 1, 2, - - - , k and IT (j) = 0 for other elements of j LW (i) = i, i = 1, 2, - - - - -, k, and LW (j) = 0, for other elements of j.
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), CA = C (ap+1) and
TA = T (ap+1).
Algorithm-1:
STEP 1: IX = 0
IS (IC (CA) = 0) IF YES GO TO 2 IF NO GO TO 5
STEP 2: IS (IT (TA) + 1 < TX (TA) ) IF YES GOTO 3 IF NO GOTO 5
STEP 3: IS ( (RX (RA) + DR (CA) ) < SC (RA)) IF YES GO TO 4 IF NO GO TO 5
STEP 4: I X = 1 GO TO 5 STEP 5: STOP
We start with the partial word L1 = (a1) = (1) . A partial word Lp is constructed as Lp = Lp-1 * ( p). Where * indicates chain formulation. We will calculate the values of V (Lp) and LB (Lp) simultaneously. Then two situations arises one for branching and other for continuing the search.
2. LB (Lp) > VT. In this case we reject the partial word Lp. We reject the block of word with Lp as leader as not having optimum feasible solution and also reject all partial words of order p that succeeds Lp.
Now we are in a position to develop a Lexi-search algorithm to find an optimal feasible word.
Algorithm- 2: (Lexi - Search algorithm)
STEP 1: Initialization
The arrays SN, D, DC, R, C, T, SC, DR, N, are made available. IC, IT, RX, TX, TC, L, SW, V, LB initialized to zero. The values I = 1, J = 0.
STEP 2: J = J + 1
GOTO 3 STEP 3: L (I) = J JA = J + N - I
V (I) = V (I - 1) + D (J)
LB (I) = V (I) + DC (JA) - DC (J) GOTO 4
STEP 4: IS (LB (I) > VT) IF YES GO TO 10 IF NO GO TO 5
STEP 5: RA = R (J) CA = C (J) TA = T (J) GOTO 6
STEP 6: Check the feasibility of L (I) (using algorithm 1) IS (IX = 1) IF YES GO TO 7
IF NO GO TO 2
STEP 7: IS (I = N) IF YES GOTO 9 IF NO GO TO 8
STEP 8: L (I) = J IC (CA) = 1
RX (RA) = RX (RA) + DR (CA) IT (TA) = IT (TA) + 1
I = I + 1
IF (I < N) IF YES GO TO 2 IF NO GO TO 12
STEP 9: L (I) = J, L (I) is a full length word and is feasible VT = V (I), Record L (I) and VT
GOTO 10
STEP 10: IS (I = 1) IF YES GO TO 12 IF NO GO TO 11
STEP 11: I = I -1 J = L (I) CA = C (J) RA = IR (J) TA = T (J) IC (CA) = 0
RX (RA) = R X (RA) - DR (CA) IT (TA) = IT (TA) - 1
In this section, computational experiments were conducted based on the above algorithm. The experiments were conducted on a Pentium 4 computer with 512 MB RAM memory and 1.8 GHz speed and the elements of the cost matrix C are real numbers randomly selected from [0,1]. In many instance, the Lexi-Search method was more effective than Branch and Bound. Using 100 random samples for each number of cities, the experimental results showed the Lexi-Search method gave better results than Branch and Bound’s method especially for large size city (See Figure 1).For the number of cities smaller than 20, the Lexi-Search method and Branch and Bound’s method have the same cost. When the number of city increases, the Lexi-Search method works better than Branch and Bound’s method. Most of the time for the number of cities of more than 100, the Lexi-Search method gives better solution.
Figure1: The mean of cost in 100 sample
To determine the confidence percentage between the Branch and Bound method and the Lexi-Search method, the null and alternative hypotheses have been defined as follows (μ is the mean of population cost):
H0: μB&B ≤ μLEXI
H1: μB&B > μLEXI
The F-test was used to recognize the equality between sample variances and then T-test was used to accept or reject the hypotheses based on the F-test results (Figure2 and Table1)
Number of cities Variance equality Confidence for reject H0
5 Yes 52% 10 Yes 70% 20 Yes 58% 50 Yes 99% 100 Yes 99.90% 150 No 99.90%
Table 1: T-test and F-test results
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Figure 2: Comparing the Confidence’s Percentage between the number of cities
6 Conclusion:
The Lexi-Search algorithm presented in this paper, incorporating Pattern Recognition technique is tested. The same problem sets have been tested with the Branch and Bound algorithm. Lexi-Search algorithm is faster than Branch and Bound in most of the cases. Even with the restriction imposed, the Lexi-Search algorithm takes reasonably less time. Further it is observed that with the modification of the sort procedure while arranging the alphabet table, the Lexi-Search algorithm is becoming more efficient. On the whole, it is felt that Lexi-Search algorithm is faster than the Branch and Bound algorithm.
Acknowledgments
The authors are very much thankful to the referees for giving their useful comments and suggestions.
References
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