The three blood banks in Franklin Country are coordinated th

The three blood banks in Franklin Country 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 biweekly number of containers available at each bank and the biweekly number of containers of blood needed at each hospital. How many shipments should be made biweekly from each blood bank to each hospital so that total shipment costs are minimized?

Thanks a lot

Hospital Supply LI 2 3 4 Bank 1 Bank 2 Bank 3 Demand $8 12 14 90 $9 7 10 70 $11 5 6 40 $16 50 80 120 50

Solution

Decision Variable:

X_ij be the number of containers to be transported from bank i to hospital j,

Where,

i = 1,2,3 for three banks

j = 1,2,3,4 for four banks hospitals

Objective Function:

Objective is to minimize the total transportation cost:

Min. Z = $8X₁₁ + $9X₁₂ + $11X₁₃ + $16X₁₄ + $12X₂₁ + $7X₂₂ + $5X₂₃ + $8X₂₄

+ $14X₃₁+ $10X₃₂ + $6X₃₃ + $7X₃₄

Subject to:

Supply Constraint:

Bank 1: X₁₁ + X₁₂ + X₁₃ + X₁₄ = 50

Bank 2: X₂₁ + X₂₂ + X₂₃ + X₂₄ = 80

Bank 3: X₃₁ + X₃₂ + X₃₃ + X₂₄ = 120

Demand constraint:

Hospital 1: X₁₁ + X₂₁ + X₃₁ <= 90

Hospital 2: X₁₂ + X₂₂ + X₃₂ <= 70

Hospital 3: X₁₃ + X₂₃ + X₃₃ <= 40

Hospital 4: X₁₄ + X₂₄ + X₃₄ <= 50

Nonnegative Constraint: X_ij>= 0

Part 2:

The allocation X12 (Bank 1 and hospital 2) in the optimal solution is zero, and reduced cost is 6. It means if the allocation is done in between bank 1 and hospital 2, the cost of transportation should be reduced by $6. It means if the cost is reduced beyond 9-6 = $3 and below the optimal solution will consider the allocation a basis variable unless it will not change the optimal solution of the problem. As the proposed reduced cost is $6, which is greater than allowable minimum, the solution will not change. The optimal objective function value will remain same.