22. 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 graphics 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, Sarah 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, $25 for David, and $18 for Sarah.
a. Formulate a linear program that can be used to determine the number of hours each graphic designer should be assigned to the project in order to minimize total cast.
b. How many hours should each graphic designer be assigned to the project? What is the total cost?
c. Suppose Lisa could be assigned more than 50 hours. What effect would this have on the optimal solution? Explain.
d. If Sarah were not required to work a minimum number of hours on this project, would the optimal solution change? Explain.
a. To formulate a linear program, we need to define decision variables, objective function, and constraints:
Decision Variables:
Let x1 represent the number of hours assigned to Lisa.
Let x2 represent the number of hours assigned to David.
Let x3 represent the number of hours assigned to Sarah.
Objective Function:
Minimize the total cost, which can be expressed as:
Cost = 30x1 + 25x2 + 18x3
Constraints:
Lisa must be assigned at least 40% of the total hours assigned to the senior designers:
x1 >= 0.4(x1 + x2)
Sarah must be assigned at least 15% of the total hours:
x3 >= 0.15(x1 + x2 + x3)
Sarah's hours must not exceed 25% of the total hours assigned to the senior designers:
x3 <= 0.25(x1 + x2)
The total number of hours assigned to the project is 150:
x1 + x2 + x3 = 150
Non-negativity Constraints:
x1, x2, x3 >= 0
b. We need to solve this linear program using a solver to find the optimal solution. The solver will give us the values for x1, x2, and x3, which represent the number of hours each graphic designer should be assigned to the project. The total cost can be calculated using the objective function.
c. If Lisa could be assigned more than 50 hours, it would have an impact on the optimal solution. The constraint x1 <= 50 would no longer be applicable, and Lisa could potentially be assigned more hours to minimize the total cost.
d. If Sarah were not required to work a minimum number of hours on this project, the optimal solution might change. The constraint x3 >= 0.15(x1 + x2 + x3) ensures that Sarah gets a minimum share of the project time. Without this constraint, the optimal solution may allocate fewer hours to Sarah and potentially increase the total cost.
a. To formulate the linear program, let's define the decision variables:
Let x1 represent the number of hours assigned to Lisa (senior designer and team leader).
Let x2 represent the number of hours assigned to David (senior designer).
Let x3 represent the number of hours assigned to Sarah (junior designer).
Now, let's set up the objective function and the constraints:
Objective function: Minimize the total cost
The total cost is calculated by multiplying the number of hours with the corresponding hourly wage rate for each designer. Thus, the objective function can be formulated as:
Minimize Z = 30*x1 + 25*x2 + 18*x3
Constraints:
1. Lisa must be assigned at least 40% of the total number of hours assigned to the two senior designers:
x1 >= 0.4*(x1 + x2)
2. Sarah must be assigned at least 15% of the total project time:
x3 >= 0.15*(x1 + x2 + x3)
3. The number of hours assigned to Sarah must not exceed 25% of the total number of hours assigned to the two senior designers:
x3 <= 0.25*(x1 + x2)
4. Lisa has a maximum of 50 hours available to work on this project:
x1 <= 50
5. Non-negativity constraints for all variables:
x1, x2, x3 >= 0
b. To find the solution, you can use linear programming techniques such as the simplex method or software tools like Excel Solver. By solving the linear program, you will obtain the optimal values of the decision variables and the corresponding total cost.
c. If Lisa could be assigned more than 50 hours, it would change the feasible region of the problem. This change in constraints may lead to a different optimal solution if it allows for a more cost-effective allocation of the hours among the designers.
d. If Sarah were not required to work a minimum number of hours on this project, it would change the constraints of the problem. This change may result in a different optimal solution as Sarah's minimum requirement could be redistributed among the other designers, potentially leading to a more cost-effective allocation of the hours.