I mainly need help on what the constraints are and what the variables are. Also

Question

I mainly need help on what the constraints are and what the variables are. Also the objective and the solution as well. I try to set up my spreadsheets like this:

If it could also be done like that so it is easy for me to understand that would be amazing!

Industrial Designs has been awarded a contract to design a label for a new wine produced by Lake View Winery. The Co estimates that 150 hours will be required to complete the project. The firm's three graphics designers available for assisgment to this project ar 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 maximim 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 cost.
(b) How many hours should each graphic designer be assigned to the project? What is the total cost?

Explanation / Answer

Let the the number of hours assigned to Lisa, David and Sarah be l,d,s respectively.

Developing the constraints, we have -

l+d+s = 150     -(1)                             #total hours

l <= 50     -(2)                                     #no. of hours available with Lisa

l >= 0.40*(d+s)

or l >= 0.40*(150-l)

i.e. l >= 42.857 -(3)    #company mandated hour limit for Lisa

s >= 0.15 *(l+d+S)

or s >= 0.15*(150)

i.e. s >= 22.5     -(4)                           #min no. of hours for sarah

s <= 0.25*(l+d)

or s <= 0.25*(150-s)

i.e. s <= 30      -(5)                             #max no. of hours for sarah

Total cost = 30l + 25d + 18s

Because, Sarah's per hour wage is minimum, we can assign her complete 30 hours to work out Minimum project cost

then, l+d = 120

If Lisa's wages are maximum, therefore min. hours can be assigned to Lisa within permissible limit

so, l = 60/1.4

Hence, d = 120 -(60/1.4) = 108/1.4

Therefore, total cost is = 30*60/1.4 + 25*108/1.4 + 18*30 = 1285.71 + 1928.57 + 540 = 3754.28$

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