4.
A product is produced by four factories A, B, C and D. Their production capacities are 50, 70, 30 and 50 units respectively. These factories supply the product to four stores, demands of which are 40, 35, 105, and 20 units respectively. Unit transport cost in rupees from each factory to each store is given in the table below.
Store
2
6
11
4
11
3
8
10
10
13
Factor 1
A
B
C
D
4
13
14
9
4
13
8
13
8
(a) Find an initial solution using Vogel's approximation method. (b) Find the optimum solution for the above problem using MODI method.
Answers
Answer:
the solution is optimal
Answer:
Calculate the shadow prices (also known as dual variables) using the following equations:
u1 + v1 = 5
u1 + v2 = 13
u2 + v2 = 8
u2 + v3
Step-by-step explanation:
(a) Using Vogel's Approximation method:
To apply Vogel's Approximation Method, we need to calculate the penalties for each row and column by subtracting the two smallest costs for each row and column, respectively.
Penalties for Rows:
Factory A: 13-4 = 9
Factory B: 13-8 = 5
Factory C: 8-4 = 4
Factory D: 13-9 = 4
Penalties for Columns:
Store 1: 6-2 = 4
Store 2: 10-4 = 6
Store 3: 13-11 = 2
Store 4: 11-8 = 3
The highest penalty is for Store 2, so we start by allocating as much as possible to that store from the factory with the lowest cost in its row.
Allocate 35 units from Factory C to Store 2 at a cost of 4 per unit.
Allocate 5 units from Factory D to Store 1 at a cost of 9 per unit.
Allocate 20 units from Factory A to Store 4 at a cost of 4 per unit.
Allocate 30 units from Factory B to Store 3 at a cost of 8 per unit.
Allocate 15 units from Factory D to Store 2 at a cost of 13 per unit.
Allocate 5 units from Factory A to Store 2 at a cost of 13 per unit.
Allocate 5 units from Factory A to Store 3 at a cost of 14 per unit.
Allocate 10 units from Factory D to Store 4 at a cost of 9 per unit.
The total cost of this initial solution is:
354 + 59 + 204 + 308 + 1513 + 513 + 514 + 109 = 1205 rupees.
(b) Using MODI Method:
Demand 40 35 105 20
We can see that the total supply (200) equals the total demand (200), which means that we have a balanced problem.
Demand 40 35 105 20
We have four allocations (A2, A3, B4, D1) that are not yet used to meet demand or supply. We need to calculate the opportunity costs for each of these allocations to see if there is a more efficient solution.
For this, we need to calculate the shadow prices (also known as dual variables) using the following equations:
u1 + v1 = 5
u1 + v2 = 13
u2 + v2 = 8
u2 + v3
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