Transportation Problem

Dr . K. VANITHEESWARI ,Ph .D.,

Assistant professor,

PG and Research department of Commerce

C.P.A . College, Bodinayakanur

Dr . K. VANITHEESWARI ,Ph .D., Assistant professor, PG and Research department of Commerce C.P.A . College, Bodinayakanu

ASSIGNMENT

• • The transportation problem is a special type of linear programming problem, where the objective is to minimize the cost of distributing a product from a number of sources to a number of destinations. •Transportation deals with the transportation of a commodity (single product) from ‘m’ sources (origin or supply or capacity capacity centres) centres) to ‘n’ destinations destinations (sinks or demand or requirement requirement centres) centres). • It is assumed that, level of supply of each source and the amount of demand at each destination areknown . The unit transportation cost of commodity from each source to each destination are known. • The objective is to determine the amount to be shifted from each source to each destination such thatthe total transportation cost is minimum.

Dr . K. VANITHEESWARI ,Ph .D., Assistant professor, PG and Research department of Commerce C.P.A . College, Bodinayakanu

TYPES OF TRANSPORTATION PROBLEM

• Balanced Transportation Problem: where the total supply equals to the total demand • Unbalanced Transportation Problem: where the total supply is not equal to the total demand Optimality condition: m+n-1 no of allocation

Dr . K. VANITHEESWARI ,Ph .D., Assistant professor, PG and Research department of Commerce C.P.A . College, Bodinayakanu

METHODS

• Phase-1 obtains the initial basic feasible solution : 1. North West Corner Cell Method. 2. Least Call Cell Method. 3. Vogel’s Approximation Method (VAM )

• Phase-2 obtains obtains the optimal optimal basic solution solution : 1.modified distribution method 2.  stepping-stone method 

Dr . K. VANITHEESWARI ,Ph .D., Assistant professor, PG and Research department of Commerce C.P.A . College, Bodinayakanu

PROBLEM 1: Obtain initial feasible solution for the following (i) North West Corner rule.

Dr . K. VANITHEESWARI ,Ph .D., Assistant professor, PG and Research department of Commerce C.P.A . College, Bodinayakanu

(ii) Least Cost Method

Dr . K. VANITHEESWARI ,Ph .D., Assistant professor, PG and Research department of Commerce C.P.A . College, Bodinayakanu

. (iii) VAM Method

Dr . K. VANITHEESWARI ,Ph .D., Assistant professor, PG and Research department of Commerce C.P.A . College, Bodinayakanu

ASSIGNMENT PROBLEM

• An assignment problem is a particular case of transportation problem. The objective is to assign a number of resources to an equal number of activities . So as to minimize total cost or maximize total profit of allocation. The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different. Suppose that we have n jobs to be performed on m machines (one job to one machine). Our objective is to assign the jobs to the machines at the minimum cost (or maximum profit). Under the assumption that each machine can perform each job but with varying degree of efficiencies

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Dr . K. VANITHEESWARI ,Ph .D., Assistant professor, PG and Research department of Commerce C.P.A . College, Bodinayakanu

Hungarian method

Dr . K. VANITHEESWARI ,Ph .D., Assistant professor, PG and Research department of Commerce C.P.A . College, Bodinayakanu

Here the number of rows and columns are equal.

∴  The given assignment problem is balanced. Now let us find the solution.

Step 1: Select a smallest element in each row and subtract this from all the elements in its row.

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Look for at least one zero in each row and each column .Other wise go to step 2.�Step 2: Select the smallest element in each column and subtract this from all the elements in its column.�

Dr . K. VANITHEESWARI ,Ph .D., Assistant professor, PG and Research department of Commerce C.P.A . College, Bodinayakanu

• Since each row and column contains atleast one zero, assignments can be made.
• Step 3 (Assignment):
• Examine the rows with exactly one zero. First three rows contain more than one zero. Go to row D. There is exactly one zero. Mark that zero by  (i.e) job D is assigned to machine I. Mark other zeros in its column by × .
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• Step 4: Now examine the columns with exactly one zero. Already there is an assignment in column I. Go to the column II. There is exactly one zero. Mark that zero by . Mark other zeros in its rowby × .
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• Column III contains more than one zero. Therefore proceed to Column IV, there is exactly one zero. Mark that zero by . Mark other zeros in its row by × .
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• Step 5: Again examine the rows. Row B contains exactly one zero. Mark that zero by .
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Dr . K. VANITHEESWARI ,Ph .D., Assistant professor, PG and Research department of Commerce C.P.A . College, Bodinayakanu

Dr . K. VANITHEESWARI ,Ph .D., Assistant professor, PG and Research department of Commerce C.P.A . College, Bodinayakanu

Thus all the four assignments have been made. The optimal assignment schedule and total cost is

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