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The Robinsons are planning a wedding and reception for their daughter, Rachel. They estimate they need at least four servings (a glass of wine or bottle of beer) for each of the 400 guests. A bottle of wine contains five glasses. They estimate that more than 50% of the guests will prefer wine to beer. A bottle of wine costs $6, and a bottle of beer costs $1. The Robinsons have budgeted $1,500 for wine and beer. The caterer advised that typically 8% of the wine and 15% of the beer will be left over. The Robinsons want to minimize their waste (cost from unused wine and beer). How many bottles of wine and beer should the Robinsons order?

(a) Formulate a linear programming model for this problem.

(b) Graphically illustrate the feasible area and identify all the possible extreme points.

(c) Solve the model – find the optimal solution points and the optimal objective function value.

Answer :

a. A linear programming model for this problem is 6x + y ≤ 1500, 5x + y ≥ 1600, 5x - 1.5y ≥ 0, x ≥ 0, y ≥ 0.

b. A graph of the feasible area with all the possible extreme points (1125/7, 3750/7), (250, 0), (48, 160), and (80, 0) is shown in the picture below.

c. The optimal solution point is (80, 0) and the optimal objective function value is $6.4.

Part a.

Based on the information provided above, the variables for the problem can be defined as follows;

Let the variable x represent the number of bottles of wine.

Let the variable y represent the number of bottles of beer.

Since a bottle of wine costs $6 and a bottle of beer costs $1, and the Robinsons have budgeted $1,500 for wine and beer, an inequality that defines this constraint is given by;

6x + y ≤ 1500

Since the Robinsons are planning on 400 guests at the reception, and they estimate that they need at least four servings, we have;

Total servings = 400 × 4 = 1600 servings.

5x + y ≥ 400

Also, they estimated that more than 50% guests will prefer wine to beer. Therefore, this constraint is given by;

5x ≥ (1 + 50/100)y

5x ≥ 1.5y

5x - 1.5y ≥ 0

Lastly, the caterer advised them that typically 8% of the wine and 15% of the beer will be left over. Therefore, the objective function for the minimization is given by;

Z = 0.08x + 0.15y

Non-negativity constraints;

x ≥ 0

y ≥ 0

Part b.

By critically observing the graph of the constraints above, the vertices include;

(1125/7, 3750/7), (250, 0), (48, 160), and (80, 0)

Part c.

Next, we would find the minimum cost as follows;

Z = 0.08(1125/7) + 0.15(3750/7) = $93.21.

Z = 0.08(250) + 0.15(0) = $20.

Z = 0.08(48) + 0.15(160) = $27.84.

Z = 0.08(80) + 0.15(0) = $6.4.

In conclusion, the optimal solution point is (80, 0) and the optimal objective function value is $6.4.

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