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Comparative study of Multi-objective Unbalanced Transportation Problems with and without Budgetary Constraints

Fegade M. R.¹, Jadhav V.A.², Doke D. M.³, Irpenwar B. M.⁴

{¹Research Scholar, ²Head} Dept. of Statistics, Comp. Sci. and IT, N.E.S. Science College, Nanded, Maharashtra, INDIA.

³Head, Department of Mathematics and Statistics, M. L. Dahanukar College, Vile Parle (East), Mumbai, Maharashtra, INDIA.

⁴Head, Department of Statistics, Vivekanand College, Kolhapur, Maharashtra, INDIA.

Corresponding Addresses:

¹[email protected], ²[email protected], ³[email protected], ⁴[email protected]

Abstract: A new method is proposed for finding an optimal solution for unbalanced transportation problems with budgetary constraints where demand and budget are imprecise. Further it is compared with the situation in which budgetary constraints are not known. The proposed method enables the decision makers to choose the optimal distribution according to their budget. Finally, with the help of numerical example, the proposed method is illustrated.

Keywords: Transportation problem, Optimal solution, Budget, Fuzzy logic.

Introduction

Research Methodology

Preliminaries:

                We need the following definitions of the basic arithmetic operators and partial ordering on closed bounded intervals which can be found in [4].

Let D = { , a b and a and b are in R} denote the set of all closed bounded intervals on the real line R.

Definition 1: Let  and   be in D. Then,

          ;

          ,     

where  and   

Consider the following unbalanced transportation problem with budgetary constraints:

(A) Find the values of x_ij, i = 1,2,…m; j = =1,2,…n  such that the following conditions are satisfied:

                               (1)

                (2)

                                      (3)

,   i=1,2,…m and   j=1,2,…n                (4)

Where c_ij is the cost of shipping one unit from supply point i to the demand point j;

a_i is the supply at supply point i; is the imprecise demand at demand point j;

x_ij is the number of units shipped from supply point i to demand point j and [z₁,z₂] is the imprecise budget.

The unbalanced transportation problem related to the problem (p) is given below:

(IP)    Minimize  

                 Subject to

,    i = 1,2,…m  and  j = 1,2,…n and integers.

Now, we need the following condition which finds a relation between optimal solutions of an interval transportation problem and a pair of induced transportation problems.

 If the set { for all i and j} is an optimal solution of the upper bound transportation problem (UP) of (IP) where

(UP)     Minimize

subject to

, i= 1,2,…m and j = 1,2,…n and are integers               

And the set { for all i and j} is an optimal solution of the lower bound transportation problem (LP) of the problem (IP) where

(LP)    Minimize

Subject to

, i=1,2,…m and j=1,2,…n      and are integers,

Then the set of intervals { , for all i and j} is an optimal solution of the problem (IP) provided   _, for all i and j.

3. Interval-Point Method

We, now propose a new method namely, interval-point method for finding an optimal solution to the problem (P).

The interval-point method proceeds as follows:

Step 1: Construct an unbalanced interval transportation problem (IP) related to the given UTPBC.

Step  2:  Construct  the  upper  bound  unbalanced  transportation  problem  (UP)  of  the problem (IP) and solve the problem (UP) by the zero point method [3]. Let { y , for

all i and j} be an optimal solution to the problem (UP).

 Step  3:  Construct  the  lower  bound  unbalanced  transportation  problem  (LP)  of  the given the [problem (IP) and solve the problem (LP) with the upper bound constraints , for all i and j by the zero point method [3]. Let { for all i and j} be the optimal solution to the problem (LP) with , for all i and j.

Step 4: The optimal solution to the problem (IP) is{ [ ],for all i and j } by the optimal objective values of the problem (IP) is  

Step 5: Let  be the given budget cost for transportation cost of the given UTPBC. Now, we are write Z as in the form, for some , . This implies

Here, we considered two cases,

Case 1: Values of Z is known i.e. fixed or given.

Case 2: Value of Z is not known.

Step 6: Compute the values of decision variables

 ,  

where µ is as in step 5(including 5.1 and 5.2).

Step 7: The optimal solution to the given UTPBC is 

  for the given budget is Z.

Numerical Example

The interval-point method for solving unbalanced transportation problem with budgetary constraints is illustrated by the following example.

Example.  There are four warehouses i.e. source from where the food grains are supplied to three different destinations i.e. demand stations. C_ij’s are the cost coefficients expressed in rupee per metric ton and a_i’s , l_j’s and u_j’s are expressed in lakhs of metric ton. The transportation matrix is given in the following table:

Determine an optimal distribution plan to transport the items from the source points to the destination points for the budget Rs.100.

Now, the interval transportation problem (IP) to the given problem is given below:

Now, the upper bound problem of (IP), (UP) of the problem (IP)   is given below:

Now, using the zero point method, the optimal solution to the problem (UB) is:

 = 10, =4,  = 4,  = 8 ,  and = 4.

Now, the lower bound problem of interval transportation problem with the upper bounded (LB) constraints is:

With , i= 1, 2,…,m and j = 1,2,…n.

Now, using the zero point method, the optimal solution to the problem (LP) is

 = 7, =4,  = 4,  = 2 ,  and = 3.

Now, as in the step 3., we consider the optimal solution to the problem (IP) is [x°₁₃,y°₁₃] = [7,10], [x°₂₂, y°₂₂] = [4,4], [x°₃₁, y°₃₁] = [4,4], [x°₄₁,y°₄₁] = [2,8] and [x°₄₂,y°₄₂] = [3,4] and also, the minimum interval transportation cost is [71,110].

Now, as in step 4., we have the total transportation cost as Z = 71 + 39 µ where Z is given budget. This implies that

Now, as in step 5. And the step 6., we have 

 x₂₂=4; x₃₁=4;

Case 1:

Given Z = 100. Then, the optimal solution to the given UTPBC for Z = 100 are:

 x₁₃ = 9.23, x₂₂ =4, x₃₁ = 4, x₄₁ = 6.46  and x₄₂ = 3.74.

The total number of units transported = 27.43 tons.

Case 2:

We get, different values of Z for µ (  ) is given as,

Conclusion

In this paper, we have considered the unbalanced transportation problem with and without budgetary constraints cases. The proposed without budgetary constraints method is useful for finding an optimal solution for decreasing or reducing the cost and finding a compromise solution. It is very easy to understand and easy to compute. Also, enables the decision makers to optimize the economical activities and make the correct managerial decisions depending on their financial position. The proposed method enables the decision makers to evaluate the economical activities and make self satisfied managerial decisions when they are handling a variety of unbalanced transportation problems with budgetary constraints.

References

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