Question

QUESTION TWO A petroleum company has three major oil fields and five oil refineries. The shipping costs from the fields to the refineries, fields are as shown in the table Refinery Capacity Refinery Field Production Barrels per day 20,000 25,000 30,000 Barrels per day 10,000 13,000 13,000 16,000 18,000 2 Field 1 2 A 400 350 370 Cost per Barrel (K) B C 300 330 350 380 300 400 D 380 320 350 E 360 350 340 (1) (ii) Determine the optimum scheme using the North West Corner rule Determine the optimum scheme using the Least Cost Method

Answer 1

Total production from three fields = 2000+25000+30000 = 75000 barrels

Total Refinery capacity = 10000+13000+13000+16000+18000 = 70000 barrels

Total production is more than Refinery capacity (demand) Therefore, a dummy demand node is created to make it a balanced transportation problem

(i) North West Corner rule

Basic Feasible (IBF) Solution using North-West Corner Method In NWC method, we start with the Northwest Corner cell and allocate the maximum quantity possible and then move on to either the next cell to the right or below, depending upon the remaining demand or supply in that particular row or column. The process is repeated until all the demand is satisfied. Field E Refinery C D 330 380 Dummy Du A 400 Supply B 300 20000 10000 10000 350 380 320 25000 3000 13000 9000 370 300 400 340 350 7000 30000 18000 5000 Demand 10000 13000 13000 16000 18000 5000 Total Cost = 24440000

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(ii) Least Cost Method

In this method, start with the lowest cost cell and assign the maximum quantity possible based on the demand and supply for corresponding cell. After allocation, strike out the corresponding row or column. Again identify the lowest cost of the remaining cells and allocate the maximum quantity and strike out the corresponding row or column for which demand or supply is fully exhausted. Repeat the process, until the demand is completely satisfied.

Refinery C D Field A B E Dummy Supply 400 300 330 20000 13000 7000 350 350 380 320 350 25000 9000 O 10 16000 O 370 300 340 w 400 6000 30000 1000 0 18000 5000 Demand 10000 10000 13000 13000 16000 18000 Total Cost = 23370000

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(c) Vogel's Approximation Method

Vogel's Approximation Method This method also takes costs into account in allocation. Five steps are involved in applying this heuristic: Step 1: Determine the difference between the lowest two cells in all rows and columns, including dummies. Step 2: Identify the row or column with the largest difference. Ties may be broken arbitrarily. Step 3: Allocate as much as possible to the lowest-cost cell in the row or column with the highest difference. If two or more differences are equal, allocate as much as possible to the lowest-cost cell in these rows or columns. Step 4: Stop the process if all row and column requirements are met. If not, go to the next step. Step 5: Recalculate the differences between the two lowest cells remaining in all rows and columns. Any row and column with zero supply or demand should not be used in calculating further differences. Then go to Step 2. Supply penalty: iteration 1 2 3 4 300 30 60 - 5 - 6 - 20000 320 30 30 30 300 25000 300 40 40 40 10 30 30000 Refinery Field A B C D E Dummy 1 400 300 330 380 360 0 1 107000 13000 000 350 350 380320 3500 4000 0 0 160000 5000 370300 400 350 340 3 6000 60000 180000 Demand 10000 13000 13000 16000 18000 5000 penalty iteration 1 20 0 5030100 iteration 2 20 0 50 30 10 iteration 3 20 0 iteration 4 20 50 - 30 10 iteration 5 20 - iteration 6 20 L - L - 0 1919 Total Cost = 23050000

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(d)

Comparing the three methods,

Vogel's Approximation Method (VAM) gives the lowest total cost.

Therefore, VAM is the best method for this problem.