Question

Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of

vessels: A type A vessel has 60 deluxe cabins and 160 standard cabins, whereas a type B vessel has 80 deluxe

cabins and 120 standard cabins. Under a charter agreement with the Odyssey Travel Agency, Deluxe River

Cruise sis to provide Odyssey with a minimum of 360 deluxe and 680 standard cabins for their 15-day cruise

in May. It costs $44000 to operate a type A vessel and $54000 to operate a type B vessel for that period.

i.) How many of each type of vessel should be used to keep the operating costs to a minimum?

ii.) Find the range of values that the cost of operating a type A vessel can assume without changing the

optimal solution.

iii.) Find the range of values that the requirement for deluxe cabins can assume.

iv.) Find the shadow price for the requirement for deluxe cabins.

Answer 1

Let the Deluxe River Cruises provides of type A fleet and   of type A fleet to Odyssey

	Type A	Type B	Available
Delux	60	80	360
Satndard	160	120	680
Cost	44000	54000

The objective of the problem is to minimize the total cost. The objective function is defined as,

The constraints of the problem are,

Deluxe cabin constraint,

Standard cabin constraint,

and the non negativity constraints are,

The LP is formulated as,

subject to

Now, the LP is solved using the excel solver by following these steps,

Step 1: Write the decision variable with value zero. The screenshot is shown below

Step 2: Write the objective function equation while taking the decision variable value. The screenshot is shown below,

Step 3: Write the constraints equation while taking the decision variable value and write the right side value of the constraint

The screenshot for constraint deluxe cabin is shown below,

The screenshot for constraint standard cabin is shown below,

The screenshot for non-negativity constrains shown below,

Step 4: (If you have not install the solver excel follow, FILE > Options > Add-ins > Manage: select ExcelAdd-ins > Go then tick Solver Add-in > OK)

DATA > Solver > OK. The screenshot is shown below,

Step 5:

Set Objective: Select objective value,

To: select Min

Subject to the Constraints > Add > in Cell Reference select constraint value and in Constraint: select right hand side value of constraint and select the >= inequality.

Tick Make Unconstrained Variables Non-Negative

Select a Solving Method: Simplex LP

then click Solve. The screenshot is shown below,

Step 6: Select Reports > Answer, Sensitivity then Ok

The result is obtained. The screenshots are shown below,

The Answer Report

The sensitivity report

i.) How many of each type of vessel should be used to keep the operating costs to a minimum?

Type A: X1 = 2

Type B: X2 = 3

ii.) Find the range of values that the cost of operating a type A vessel can assume without changing the optimal solution.

From the sensitivity report

	Allowable Increase	Allowable Decrease
X1	28000	3500

iii.) Find the range of values that the requirement for deluxe cabins can assume.

From the Answer report

	Final value	Allowable Increase	Allowable Decrease
Deluxe cabin	360	93.33333333	105

The range of values are,

	Max	Min
Delux cabin	453.3333333	255

iv.) Find the shadow price for the requirement for deluxe cabins.

From the Answer report

	Shadow price
Delux cabin	600