Question

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 as follows:

                       Warehouse

Plant           1                       2            Capacity

1                4                        7               450

2                8                         5              600

3                5                         6               380

                            Customer

Warehouse          1                        2                            3                             4

1                            6                       4                            8                             4

2                           3                         6                            7                             7

Demand                300                     300                         300                        400

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?

Answer 1

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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