The Freezer Fresh Company freezes fresh vegetables at three plants and then ship
ID: 469970 • Letter: T
Question
The Freezer Fresh Company freezes fresh vegetables at three plants and then ships them to four warehouses from which grocery stores are supplied. The following information is known about the company.
Plant
Capacity
Warehouse
Demand
1
100
A
60
2
100
B
85
3
100
C
90
D
65
Costs
To
A
B
C
D
From
1
7
10
9
4
2
3
5
10
3
3
2
6
9
4
Using the SHORT-CUT method, find the OPTIMAL solution and TOTAL COST. Must use Short-cut and show steps.
Plant
Capacity
Warehouse
Demand
1
100
A
60
2
100
B
85
3
100
C
90
D
65
Explanation / Answer
Initial Feasible Solution by Vogel’s approximation Method (VAM)
FromTo
A
B
C
D
Supply
Penalty
1
35
65
100
7
10
9
4
3
2
1
2
15
100
3
85
5
10
3
0
2
5
3
60
40
100
2
6
9
4
2
4
3
Demand
60
85
90
65
Penalty
1
1
0
1
1
1
0
1
0
Checking optimality by Short-cut or Modified Distribution or Dual variable method:
Detemine Dual Variable values
Assign u1, u2, and u3 to rows and v1, v2, v3, and v4 to the columns of the cost table.
v1 =
v2 =
v3 =
v4 =
u1 =
7
10
9
4
u2 =
3
5
10
3
u3 =
2
6
9
4
The bold values in Table represent the allocated cells. Now, calculate the ui and vj values for allocated cells with the help of dual relation ui + vj = cij. First, assume one of the dual variable v3 = 0 (column with maximum allocation in a matrix). The rest of the values are calculated as:
Calculate the opportunity cost values for unallocated cells.
The calculations are:
d11 = c11 (u1 + v1) = 7 (9 – 7) = 5
d21 = c21 (u2 + v1) = 3 (10 –7) = 0
d12 = c12 (u1 + v2) = 10 – (9 –5) = 6
d24 = c24 (u2 + v4) = 3 – (10 –5) = -2
d32 = c32 (u3 + v2) = 6 – (9 – 5) = 2
d34 = c34 (u3 + v4) = 4 – (9 –5) = 0
Obtaining Revised Allocations of TP
As per the calculations, most negative value is for d24, that is, plant 2 and Warehouse D is an incoming variable. There is opportunity to reduce the cost by allocating at P2WD cell. Now, improve the allocation of IBF by allocating + units at cell P2WD. Closed loop is drawn from cell P2WD, as shown in following Table, the closed loop path is P2WD-P2WC-P1WC-P1WD-P2WD. Assign + and – units alternatively to the corners of the loop.
FromTo
A
B
C
D
Supply
1
35
65
100
7
10
+
9
-
4
2
85
15
100
3
5
-
10
+
3
3
60
40
100
2
6
9
4
Demand
60
85
90
65
Here, = minimum (15, 65) = 15 units. Thus, new allocation for the cells in loop will be:
The improved transportation table with improved allocation is shown below:
FromTo
A
B
C
D
Supply
1
50
50
100
7
10
9
4
2
85
15
100
3
5
10
3
3
60
40
100
2
6
9
4
Demand
60
85
90
65
Again, the improved solution is tested for optimality.
Assign u1, u2, and u3 to rows and v1, v2, v3, and v4 to the columns of the cost table.
v1 =
v2 =
v3 =
v4 =
u1 =
7
10
9
4
u2 =
3
5
10
3
u3 =
2
6
9
4
The bold values in Table represent the allocated cells. Now, calculate the ui and vj values for allocated cells with the help of dual relation ui + vj = cij. First, assume one of the dual variable v3 = 0 (column with maximum allocation in a matrix). The rest of the values are calculated as:
Now, calculate the opportunity cost values for unallocated cells.
The calculations are:
d11 = c11 (u1 + v1) = 7 (9 –7) = 5
d12 = c12 (u1 + v2) = 10 – (9 – 3) = 4
d23 = c23 (u2 + v3) = 10 (8 + 0) = 2
d21 = c21 (u2 + v1) = 3 – (8 – 7) = 2
d32 = c32 (u3 + v2) = 6 – (9 – 3) = 0
d34 = c34 (u3 + v4) = 4 – (9 –5) = 0
All the values of dij are greater than or equal to 0, thus optimal solution is obtained and is as follows:
FromTo
A
B
C
D
Supply
1
50
50
100
7
10
9
4
2
85
15
100
3
5
10
3
3
60
40
100
2
6
9
4
Demand
60
85
90
65
Calculate Total cost
Plant
Warehouse
Quantity
Cost
Total Cost
1
C
50
9
450
1
D
50
4
200
2
B
85
5
425
2
D
15
3
45
3
A
60
2
120
3
C
40
9
360
Total Cost
1600
Total Cost of Transportation allocation = $1,600
FromTo
A
B
C
D
Supply
Penalty
1
35
65
100
7
10
9
4
3
2
1
2
15
100
3
85
5
10
3
0
2
5
3
60
40
100
2
6
9
4
2
4
3
Demand
60
85
90
65
Penalty
1
1
0
1
1
1
0
1
0
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