Industrial Designs has been awarded a contract to design a label for a new wine produced by Lake View Winery. The company estimates that 150 hours will be required to complete the project. The firm’s three graphic designers available for assignment to this project are Lisa, a senior designer and team leader; David, a senior designer; and Sarah, a junior designer. Because Lisa has worked on several projects for Lake View Winery, management specified that Lisa must be assigned at least 40% of the total number of hours assigned to the two senior designers. To provide label designing experience for Sarah, the junior designer must be assigned at least 15% of the total project time. However, the number of hours assigned to Sarah must not exceed 25% of the total number of hours assigned to the two senior designers. Due to other project commitments, Lisa has a maximum of 50 hours available to work on this project. Hourly wage rates are $30 for Lisa, $15 for David, and $18 for Sarah.Formulate a linear program that can be used to determine the number of hours each graphic designer should be assigned to the project to minimize total cost (in dollars). (Assume L is the number of hours Lisa is assigned to the project, D is the number of hours David is assigned to the project, and S is the number of hours Sarah is assigned to the project.)

[tex]1.X1\geq 0.4\left ( X1+X2 \right )\\2.X3\geq 0.15\left ( X1+X2+X3 \right )\\3.X1+X2+X3\leq 150\\4.X3\geq 0.25\left ( X1+X2 \right )\\5.X1\leq 50\\6.X1,X2,X3\geq 0[/tex]

b) Total cost=[tex]30 \times 48+15\times72+18\times30[/tex] = $3180.

c) As the dual price for constraint five is zero hence additional work hours for Lisa won't change the optimum solution.

Step-by-step explanation:

Step 1:-

a)

Let's take

X1 to be the number of hours assigned to Lisa

X2 to be the number of hours assigned to David

X3 to be the number of hours assigned to Sarah.

The objective function is to attenuate the entire cost of the project by deciding an optimum number of hours for every person. the target function is given by -

Minimize Z =30 X1 +25 X2+18 X3

subject to following constraints

[tex]1.X1\geq 0.4\left ( X1+X2 \right )\\2.X3\geq 0.15\left ( X1+X2+X3 \right )\\3.X1+X2+X3\leq 150\\4.X3\geq 0.25\left ( X1+X2 \right )\\5.X1\leq 50\\6.X1,X2,X3\geq 0[/tex]

Constraints and explanation:

1. Lisa must be assigned a minimum of 40% of the entire number of hours assigned to the 2 senior designers.

2. Sarah must be assigned a minimum of 15% of the entire project time.

3. The corporate estimates that 150 hours are going to be required to finish the project.

4. The number of hours assigned to Sarah must not exceed 25% of the entire number of hours assigned to the 2 senior designers.

5. Lisa features a maximum of fifty hours available to figure on this project.

6. Non-negative condition.

Step 2:-

b)

From the above equations, we get

The number of hours assigned to Lisa is 48 hours

The number of hours assigned to David 72 hours

The number of hours assigned to Sarah 30 hours.

Total cost=[tex]30 \times 48+15\times72+18\times30[/tex] = $3180.

Step 3:-

c)

As the dual price for constraint five is zero hence additional work hours for Lisa won't change the optimum solution.