Question

In: Operations Management

The distribution system for the Herman Company consists of three plants, two warehouses, and four customers....

The distribution system for the Herman Company consists of three plants, two warehouses, and four customers. Plant capacities and shipping costs per unit (in $) from each plant to each warehouse are shown below along with customer demand and shipping costs per unit (in $) from each warehouse to each customer

Formulate the linear programming model to minimize the cost of shipping for this transshipment problem.

A-at the optimal solution how much is shipped from Plant 3 to Warehouse 1?

B-what is the range of optimality of coefficient of cost from Plant 3 to Warehouse 1 and what does this mean?

C-what is the range of feasibility for the supply amount for Plant 2 and what does it mean?

d-what is the range of feasibility for the demand amount for customer 2 and what does it mean?

Warehouse 1

Warehouse 2

Plant Capacity

Plant 1

4

7

450

Plant 2

8

5

600

Plant 3

5

6

380

Customer 1

Customer 2

Customer 3

Customer4

Warehouse 1

6

4

8

4

Warehouse 2

3

6

7

7

Demand

300

300

300

400

Solutions

Expert Solution

Let the units shipped from Plant 1 to Warehouse 1 be Xpw11, Plant 1 to Warehouse 2 be Xpw12 and so on. Hence, we get decision variables as Xpw11, Xpw12, Xpw21, Xpw22, Xpw31, Xpw32, Xwc11, Xwc12, Xwc13, Xwc14, Xwc21, Xwc22, Xwc23, Xwc24

Total cost = 4*Xpw11 + 7*Xpw12 + 8*Xpw21 + 5*Xpw22 + 5*Xpw31 + 6*Xpw32 + 6*Xwc11 + 4*Xwc12 + 8*Xwc13 + 4*Xwc14 + 3*Xwc21 + 6*Xwc22 + 7*Xwc23 + 7*Xwc24

We have to minimize this cost

Total Capacity = 450 + 600 + 380 = 1430

Total Demand = 300 + 300 + 300 + 400 = 1300

Total Demand < Total capacity. hence, we will get <= constraint for capacity

We get capacity constraints as:

Xpw11 + Xpw12 <= 450

Xpw21 + Xpw22 <= 600

Xpw31 + Xpw32 <= 380

We get Demand Constraints as:

Xwc11 + Xwc21 <= 300

Xwc12 + Xwc22 <= 300

Xwc13 + Xwc23 <= 300

Xwc14 + Xwc24 <= 400

We solve the given probelm in Excel using Excel solver as shown below. We further generate Sensitivity Reprot to anwer questions B to D.

The above table in the form of formulas along with Excel solver extract is shown below for better understanding and reference:

As seen from above, the units shipped from Plant 3 to warehouse 1 = 250 units

We solve further questions based on the sensitivity report shown below:

b. Under table for Variable cells,

Plant 3 to warehouse 1 conditions are given against Cell B5

Against Cell B5, the allowable increase for plant 3 to warehouse 1 = 0

Allowable decrease for plant 3 to warehouse 1 = 1

Hence, Range = 5 + 0 = 5 to 5 - 1 = 4

Hence, the range of optimality of coefficient of cost from Plant 3 to Warehouse 1 is 4 to 5. This means that out of this range, if there is a change in the cost per unit of shipping, the optimal solution will change.

c. Under the table for constraints,

Plant 2 supply conditions are given against Cell D4.

Against Cell D4, the allowable increase for plant 2 supply = 0

Allowable decrease for plant 2 supply  = 130

Hence, range of feasibility = 600 + 0 = 600, 600 - 130 = 470.

If the capacity of Plant 2 changes out of this range, the optimal solution is changed.

d. Under the table for constraints,

Customer 2 demand conditions are given against Cell M11.

Against Cell M11, the allowable increase for Customer 2 demand = 130

Allowable decrease for Customer 2 demand  = 250

Hence, range of feasibility = 300 + 130 = 430, 300 - 250 = 50.

If the Demand for Customer 2 changes out of this range i.e more than 430 or less than 50, the optimal solution is changed.

_______________________________________________________________________________________

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