bellamore
21.06.2021 •
Mathematics
The Pitts Barbecue Company makes three kinds of barbecue sauce: Extra Hot, Hot, and Mild. Pitts’ vice president of marketing estimates that the company can sell 8,000 cases of its Extra Hot sauce plus 10 extra cases for every dollar it spends promoting this sauce; 10,000 cases of Hot sauce plus 8 extra cases for every dollar spent promoting this sauce; and 12,000 cases of its Mild sauce plus 5 extra cases for every dollar spent promoting this sauce. Although each barbecue sauce sells for $10 per case, the cost of producing the different types of sauce varies. It costs the company $6 to produce a case of Extra Hot sauce, $5.50 to produce a case of Hot sauce, and $5.25 to produce a case of Mild sauce. The president of the company wants to make sure the company manufactures at least the minimum amounts of each sauce that the marketing vice president thinks the company can sell. A budget of $25,000 total has been approved for promoting these items of which at least $5,000 must be spent advertising each item. How many cases of each type of sauce should be made, and how do you suggest that the company allocate the promotional budget to maximize profits?
a. Formulate an LP model for this problem.
b. Create a spreadsheet model for this problem, and solve it using Solver.
c. What is the optimal solution?
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Ответ:
Case of Extra hot sauce 8000 cases
Case of Hot Sauce 10000 cases
Case of Mild Sauce 12000 + 2000 = 14000
z´(max) = 143500 $
Step-by-step explanation:
Profit of each product:
sale price $ cost $ Profit $
Case of Extra hot sauce 10 6 4
Case of Hot Sauce 10 5.5 4.5
Case of Mild Sauce 10 5.25 4.75
The President of the company already orders the minimum of each case of sauce as follows:
Case of Extra hot sauce 8000
Case of Hot Sauce 10000
Case of Mild Sauce 12000
Let´s call
x₁ quantity of cases of Case of Extra hot sauce over 8000 cases
x₂ quantity of cases of Case of hot sauce over 10000 cases
x₃ quantity of cases of Case of mild sauce over 12000 cases
Then Objective function will be:
z = 4*x₁ + 4.5*x₂ + 4.75*x₃ to maximize
Promoting budget constraint:
each dollar investing in promoting x₁ will become 10*x₁ ( cases)
each dollar investing in promoting x₂ will become 8*x₂(cases )
each dollar investing in promoting x₃ will become 5*x₃ ( cases)
Total investment in promotion is 10000 $ again here we need by sure to invest 5000 $ in promotion for each type of sauce, leaving only extra 10000 $, then:
Constraint due to promotional budge:
10*x₁ + 8*x₂ + 5*x₃ ≤ 10000
General constraints:
x₁ ≥ 0 x₂ ≥0 x₃ ≥ 0 all integers
Then the model is:
z = 4*x₁ + 4.5*x₂ + 4.75*x₃ to maximize
Subject to:
10*x₁ + 8*x₂ + 5*x₃ ≤ 10000
x₁ ≥ 0 x₂ ≥0 x₃ ≥ 0 all integers
After 6 iterations using an on-line solver we got optimal solution:
x₁ = x₂ = 0 x₃ = 2000
z(max) = 9500 $
Then as previous comments, the production will be:
Case of Extra hot sauce 8000 + 0 = 8000
Case of Hot Sauce 10000 + 0 = 10000
Case of Mild Sauce 12000 + 2000 = 14000
The whole profits :
z´ = 4*8000 + 4.5*10000 + 4.75 ( 12000 + 2000)
z´(max) = 32000 + 45000 + 66500
z´(max) = 143500 $
Aditional comments:
If we have n products at the same price it looks obvious that we manufacture the one wich the lowest cost. In this problem, the sales price is the same for the three sauces, and the production cost is the lowest for the Mild sauce, therefore as a consequence, it will be more profitable to manufacture Mild sauce.
Ответ:
.45* 12.6 = 5.67