The three blood banks in Franklin County are coordinated through a central offic
ID: 428480 • Letter: T
Question
The three blood banks in Franklin County are coordinated through a central office that facilitates blood delivery to four hospitals in the region. The cost to ship a standard container of blood from each bank to each hospital is shown in the table below. Also given are the biweekly number of containers available at each bank:
Bank 1 supply: 50
Bank 2 supply: 80
Bank 3 supply: 120
Further given are the biweekly number of containers of blood needed at each hospital:
Hospital 1 demand: 90
Hospital 2 demand: 70
Hospital 3 demand: 40
Hospital 4 demand: 50
Build an LP in Solver. Use the Simplex Method. How many shipments should be made biweekly from each bank to each hospital so that total shipment costs are minimized?
From/To Hospital 1 Hospital 2 Hospital 3 Hospital 4 Bank 1 8 9 11 16 Bank 2 12 7 5 8 Bank 3 14 10 6 7Explanation / Answer
TO
FROM
HOSPITAL
1
HOSPITAL
2
HOSPITAL
3
HOSPITAL
4
SUPPLY
BANK 1
8
9
11
16
50
BANK 2
12
7
5
8
80
BANK 3
14
10
6
7
120
DEMAND
90
70
40
50
250
Northwest Corner Method
TO
FROM
HOSPITAL
1
HOSPITAL
2
HOSPITAL
3
HOSPITAL
4
SUPPLY
BANK 1
50 8
X 9
X 11
X 16
50
BANK 2
40 12
40 7
X 5
X 8
80
BANK 3
X 14
30 10
40 6
50 7
120
DEMAND
90
70
40
50
250
Step 1: Bank 1 to Hospital 1: Supply is fully satisfied in this row and demand is reduced to 40.
Step 2: Bank 2 to Hospital 1: Demand is fully satisfied and supply in this row is reduced to 40.
Step 3: Bank 2 to Hospital 2: Supply is fully satisfied in this row and demand is reduced to 30.
Step 4: Bank 3 to Hospital 2: Demand is fully satisfied and supply in this row is reduced to 90.
Step 5: Bank 3 to Hospital 3: Demand is fully satisfied and supply in this row is reduced to 50.
Step 6: Bank 4 to Hospital 4: Demand and supply are satisfied.
Total Cost = {50 * 8} + {40 * 12} + {40 * 7} + {30 * 10} + {40 * 6} + {50 * 7} = $2,150
Minimum Cell Cost Method
TO
FROM
HOSPITAL
1
HOSPITAL
2
HOSPITAL
3
HOSPITAL
4
SUPPLY
BANK 1
50 8
X 9
X 11
X 16
50
BANK 2
X 12
40 7
40 5
X 8
80
BANK 3
40 14
30 10
X 6
50 7
120
DEMAND
90
70
40
50
250
Step 1: Bank 2 to Hospital 3(Has lowest unit cost of $5): Demand is fully satisfied and supply in this row is reduced to 40.
Step 2: Bank 2 to Hospital 2: Supply is fully satisfied in this row and demand is reduced to 30.
Step 3: Bank 3 to Hospital 4: Supply in this row is reduced to 70 and demand is fully satisfied
Step 4: Bank 1 to Hospital 1: Supply is fully satisfied in this row and demand is reduced to 40.
Step 5: Bank 3 to Hospital 2: Demand is fully satisfied and supply in this row is reduced to 40.
Step 6: Bank 3 to Hospital 1: Demand and supply are satisfied.
Total Cost = {50 * 8} + {40 * 14} + {40 * 7} + {30 * 10} + {40 * 5} + {50 * 7} = $2,090
Vogel’s Approximation Method
TO
FROM
HOSPITAL
1
HOSPITAL
2
HOSPITAL
3
HOSPITAL
4
SUPPLY
Row Difference
BANK 1
50 8
X 9
X 11
X 16
50
BANK 2
40 12
40 7
X 5
X 8
80
BANK 3
X 14
30 10
40 6
50 7
120
DEMAND
90
70
40
50
250
Column Difference
Total Cost = {50 * 8} + {40 * 12} + {40 * 7} + {30 * 10} + {40 * 6} + {50 * 7} = $2,050
* Vogel’s Method requires taking the two least unit costs for each row and column. These two values will be subtracted from each other to obtain a difference. The maximum value a row/column may have from the difference will be the area to place a value. The cell with the least unit cost from the row/column chosen as having the maximum difference will be where the values will be inputted.
Stepping-Stone Method
TO
FROM
HOSPITAL
1
HOSPITAL
2
HOSPITAL
3
HOSPITAL
4
SUPPLY
BANK 1
50 8
X 9
X 11
X 16
50
BANK 2
40 (-) 12
40(+) 7
X 5
X 8
80
BANK 3
X(+) 14
30(-) 10
40 6
50 7
120
DEMAND
90
70
40
50
250
Improvement Index: 7 – 10 + 14 – 12 = -1 à improves solution
TO
FROM
HOSPITAL
1
HOSPITAL
2
HOSPITAL
3
HOSPITAL
4
SUPPLY
BANK 1
8
50
9
X
11
X
16
X
50
BANK 2
12
10
7
70
5
X
8
X
80
BANK 3
14
30
10
X
6
40
7
50
120
DEMAND
90
70
40
50
250
Total Cost = {50 * 8} + {40 * 12} + {30 * 14} + {70 * 7} + {40 * 6} + {50 * 7} = $2,020àOptimal
If we try a different path,
TO
FROM
HOSPITAL
1
HOSPITAL
2
HOSPITAL
3
HOSPITAL
4
SUPPLY
BANK 1
50 (-) 8
X(+) 9
X 11
X 16
50
BANK 2
40 (+) 12
40 (-) 7
X 5
X 8
80
BANK 3
X 14
30 10
40 6
50 7
120
DEMAND
90
70
40
50
250
Improvement Index: 9 – 7 + 12 – 8 = 6 à worsens solution
TO
FROM
HOSPITAL
1
HOSPITAL
2
HOSPITAL
3
HOSPITAL
4
SUPPLY
BANK 1
10 8
40 9
X 11
X 16
50
BANK 2
80 12
X 7
X 5
X 8
80
BANK 3
X 14
30 10
40 6
50 7
120
DEMAND
90
70
40
50
250
Total Cost = {10 * 8} + {80 * 12} + {40 * 9} + {30 * 10} + {40 * 6} + {50 * 7} = $2,290
Another path could be,
TO
FROM
HOSPITAL
1
HOSPITAL
2
HOSPITAL
3
HOSPITAL
4
SUPPLY
BANK 1
50 8
X 9
X 11
X 16
50
BANK 2
40 12
40 (-) 7
X(+) 5
X 8
80
BANK 3
X 14
30 (+) 10
40(-) 6
50 7
120
DEMAND
90
70
40
50
250
Improvement Index: 5 -6 + 10 – 7 = 2 à worsens solution
TO
FROM
HOSPITAL
1
HOSPITAL
2
HOSPITAL
3
HOSPITAL
4
SUPPLY
BANK 1
50 8
X 9
X 11
X 16
50
BANK 2
40 12
X 7
40 5
X 8
80
BANK 3
X 14
70 10
X 6
50 7
120
DEMAND
90
70
40
50
250
Total Cost = {50 * 8} + {40 * 12} + {70 * 10} + {40 * 5} + {50 * 7} = $2,130
TO
FROM
HOSPITAL
1
HOSPITAL
2
HOSPITAL
3
HOSPITAL
4
SUPPLY
BANK 1
8
9
11
16
50
BANK 2
12
7
5
8
80
BANK 3
14
10
6
7
120
DEMAND
90
70
40
50
250
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