If total supply ≠ total demand, the problem is called unbalanced T.P. To obtain feasible solution, the unbalanced problem should be converted to balanced problem by introducing dummy source or dummy destination ,whichever is required. Suppose, (supply=) Σ_{ai} > Σ_{bj }( = demand). Then add one dummy destination with demand = (Σ_{ai} - Σ_{bj }) with either zero transport costs or some penalties , if they are given. Suppose (supply =) Σ_{ai} - Σ_{bj }(= demand). Then add one dummy source with supply= = { Σ_{bj - }Σ_{ai}) with either zero transportation costs or some penalties ,if they are given.

After making it balanced the mathematical formulation is similar to the balanced T.P.

**Example 8.** A company wants to supply materials from three plants to three new projects. Project I requires 50 truck loads, project II requires 40 truck loads and project III requires 60 truck loads. Supply capacities for the plants P1, P2 and P3 are 30, 55 and 45 truck loads. The table of transportation costs are given below :

Determine the optimal distribution.

**Solution.** Here total supplies = 130 and total requirements = 150. The given problem is unbalanced T.P. To make it balanced consider a dummy plants with supply capacity o f 20 truck loads and zero transportation costs to the three projects . Then the balanced T.P. is image

Using YA M , we obtain the initial BFS as given below :

To find optimal solution let us apply uv-method.

** **

For non-basic cells, c_{ij}= u_{i} + v_{j}- c_{ij}

C_{21} =- 6,c_{22} = - 7, c_{32} = - 2, c_{33} =- 1, c_{41} = - 3, c_{43} = 2,

Since c_{43 }is only positive value assign an unknown quantity 9 in (4, 3 ) cell. Identify a loop and subtract and add 9 to the comer points of the loop which is shown above.

Select θ= min. (5, 20) = 5 so that the cell ( 1, 3) leaves the basis and the cell (4, 3) enters into the basis.

**Iteration 2.**

For non-basic cells, we obtain

C_{13} =- 2, c_{21} = - 4, C_{22} =- 5, c_{32} = - 2, c_{33 }=- 3, c_{41 }= - 3

Since C_{ij}< 0, the current solution is optimal. Thus the optimal solution is

Supply 15 truck lo ads from P_{1} to I, 25 truck lo ads from P 1 to II, 55 truck lo ads from P 2 to III, 45 truck lo ads from P 3 to I. Demands o f 15 truck lo ads for II an d 5 truck lo ads for III will remain unsatisfied.

**PROBLEMS**

- There are three sources which store a given product. The sources supply these products to four dealers. The capacities of the sources and the demands of the dealers are given. Capacities S1 = 150, S2 = 40, S3 = 80, Demands D1 = 90, D2 = 70, D3 =50, D4 = 60. The cost matrix is given as follows:
- 2. Find the minimum cost of T.P. There are three factories F1, F2, F3 situated in different areas with supply capacities as 200, 400 and 350 units respectively. The items are shipped to five markets Ml, M2, M3, M4 and M5 with demands as 150, 120, 230, 200, 250 units respectively. The cost matrix is given as follows :
Determine the optimal shipping cost and shipping patterns.

3. Find the initial basic feasible solution to the following T.P. using (a) NWC, (b) LCM, and (c) YAM :

4.Solve the following transportation problem :

(Supply from S2 to D3 and S4 to D1 are restricted)

5. A transportation problem for which the costs, origin and availabilities, destinations and requirements are given below :

Check whether the following basic feasible solution x

_{1 }= 20, x_{13}= 20, x_{21}= 10, x_{22 }= 50, and x_{31 }= 10 is optimal. If not, find an optimal solution.6. Goods have to be transported from sources Si' S2 and S3 to destinations Di' D2 and D3. The T.P. cost per unit capacities of the sources and- requirements of the destinations are given in the following table :

Determine a T.P. schedule so that the cost is minimized.

7. Four products are produced in four machines and their profit margins are given by the table below :

Find a suitable production plan of products in machines so that the profit is maximized while the capacities and requirements are met.

8. Identical products are produced in four factories and sent to four warehouses for delivery to the customers. The costs of transportation, capacities and demands are given as below :

Find the optimal schedule of delivery for minimization of cost of transportation. Is there any alternative solution ? If yes, then find it.

9. Starting with LCM initial BFS, find the optimal solution to the following T.P. problem :

10. A company manufacturing air coolers has two plants located at Mumbai and Kolkata with a weekly capacity of 200 units and I 00 units respectively. The company supplies air coolers to its 4 showrooms situated at Ranchi, Delhi, Lucknow and Kanpur which have a demand of 75, 100, 100 and 30 units respectively. The cost per unit (in Rs.) is shown in the following table :

Ranchi Delhi Lucknow Kanpur

Mumbai Kolkata

90 50

90 70

100 130

100 85

Plan the production programmes so as to minimize the total cost of transportation.