Trans-Shipment Problems: Meaning and Characteristics | Operations Research

After reading this article you will learn about:- 1. Meaning of Trans-Shipment Problems 2. Main Characteristics of Trans-Shipment Problems 3. Solution.

Meaning of Trans-Shipment Problems:

In a transportation problem shipment of commodity takes place among sources and destinations. But instead of direct shipments to destination, the commodity can be transported to a particular destination through one or more intermediate or trans-shipment points. Each of these points in turn supply to other points.

Thus, when the shipments pass from destination to destination and from source to source, we have a trans-shipment problem. Four sources and three destinations problem is shown diagrammatically in figure (a) & (b).

Main Characteristics of Trans-Shipment Problems:

(i) In a T.P. number of sources and destinations are m and n resp., but in a trans­shipment problem, we have m + n sources and destination

(ii) If S₁ is the i-th source and D_j is the j^th destination then commodity may move along the route

S_i → D_i →D_j S_i → S_j → D_j → S_i →D_i:→ S_j → D_j, or in various other ways

In such case transportation cost from S_i to S_j is not equal to transportation cost than S_j to S_i

(iii) To solve these problems, firstly, we have to convert them into transportation problem and then we can solve it in usual manner.

(iv) The basic feasible solution contains 2m + 2n -1 basic variables. If we omit the variables appearing in the (m + n) diagonal ceris, we are left with m + n-1 basic variables.

Solution of Trans-shipment Problems:

The solution procedure is as follows. If there are m sources and n destinations, then size of the transportation matrix will be (m + n) x (m + n) instead of mx n. If the total no of units transported from all sources to all destination is n, then the given supply of each source and demand at each destination are added to No.

The demand at each source and supply at each destination are set equal to n. The problem, now can be solved by MODI method in a similar way as in the transportation problem.

In the final solution, ignore units transported from point to itself i.e., diagonal cells, as they do not have any physical significance as no actual transportation takes place.

Example of Trans-shipment Problems:

Consider a firm having two factories, the firm is to ship its product from the factories to three retail stores. The no. of units available at factories X and Y are 200 and 300, resp. while those demanded at A,B and Care 100,150 and 250, resp. Rather than shipping directly from factories to retail stores, it is asked to investigate the possibility of trans-shipment. The transportation cost (in rupees) per units is given in the table.

Solution:

In the trans-shipment problem each source acts as a destination and each destination as a source. Since quantity available at retail stores is zero and demand at each factory is zero, thus to make it a trans-shipment problem a factious supply and demand quantity termed as buffer stock is added to both supply and demand at all the points. This buffer stock is ∑a_i = ∑b_i = 500.

Modified problem is given in following Table:

1. To find initial basic feasible solution:

The initial solution to this trans-shipment problem obtained by Vogel’s approximation method is given in following table.

Initial Solution:

2. To check optimality:

For this we apply MODI Method for this we calculate u_j, v_j & Δ_ij.

Using following relations:

Since all Δ_ij ≥ 0, 1 this the initial solution is optimal. This solution shown in following table:

C_ij = c_i – v_j for occupied cell

Δ_ij = C_ij – (u_i + v_j) for unoccupied

Since all Δ_ij ≥ 0, thus the initial solution is optimal.

This solution shown in following Table:

Thus the optimal cost of transportation is

= (7×100)+(8x 100)+(4x 50)x(3 x250)

= 700+800+200+750

= Rs. 2450