DemDeac Designs has been awarded a contract to design a label for a new wine produced by Winston 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 Winston 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, $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 to minimize total cost. (3 pts)
b. How many hours should be assigned to each graphic designer? (0.6 pts)
c. What is the total cost? (0.6 pts)
d. Suppose Lisa could be assigned more than 50 hours. What effect would this have on the optimal solution? (0.2 pt)
e. Why? (0.2 pt)
f. If Sarah were not required to work a minimum number of hours on this project, would the optimal solution change? (0.2 pt)
g. Why? (0.2 pt)
Minimize Z = 30×1 + 25×2 + 18×3
Subject to:
0.4×1+ 0.6×2 ≥ 150 (The total hours assigned to the two senior designers must equal at least 150)
0.15(0.4×1 + 0.6×2) ≤ x3 ≤ 0.25 (0.4 x 1 + 0 . 6 x 2) (Sarah must be assigned at least 15% of the total number of hours assigned to the two senior designers, however her assignment must not exceed 25%)
50≤ x1 ≤150 (Lisa’s available time cannot exceed 50 hours and cannot fall below 150 hours which is what is required for this project in total)
X1, X2, X3 ≥ 0 (Hours worked by each designer must always remain positive).
b. According to the optimal solution of this linear program Lisa should be assigned 50 hours while David should be assigned 100hours and Sarah should be assigned 20 hours.
c. The total cost when assigning these specific amounts of time for each designer would come out to $4500.
d. If Lisa were able to work more than 50hours then it would have a direct impact on the overall cost since having additional time from her would allow us to reduce costs associated with assignment of other design resources, resulting in a lower overall value being reported by our linear programming model.
e. This is because if we had access an eliminate an additional 100 bucks worth of expenses through assigning more tasks or responsibilities under assumption that it won’t negatively impact quality standards or deadlines then it’s only logical that we’d want take advantage such opportunity no matter how small scale it might seem.
.f No, even if Sarah wasn’t required work minimum amount allocated toward her there would still no changes made regarding our original optimal solution since all constraints mentioned before are still factored into equation thereby making them continue hold true regardless change in requirements or circumstances included within problem statement.
.g This because without having pre-defined boundaries set place regarding minimum number hour assignments given junior designer any variations made will tend get countered out due presence balance created between other variables involved; hence why there won’t noticeable difference recorded as result altering one condition without also altering others related directly to it.
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