Question

Al Rifai roastery is preparing for the holiday season. The owner must decide how many bags of standard mix and how many bags of deluxe mix of Peanut/Raisin to put up.

The standard mix has 1/2 pound raisins and 1/2 pound peanuts per bag whereas the deluxe mix has 2/3 pound raisins and 1/3 pound peanuts. The shop has 90 pounds of raisins and 60 pounds of peanuts to work with.

Peanuts cost $0.60 per pound and raisins cost $1.50 per pound. The standard mix will sell for $2.55 for a one-pound bag, and the deluxe mix will sell for $2.90 for a one-pound bag.

The owner estimates that no more than 110 bags of one type can be sold.

1. (a) If the goal is to maximize profit, how many bags of each type should be prepared? (Solve it graphically)

2. (b) What is the expected profit?

3. (c) Repeat a) and b) by solving using Excel Solver

4. (d) Write 3 interpretations that you can conclude from the Sensitivity Analysis (with

   numbers and discussion)

Answer 1

Decision variables:

Let,

S = number of bags (1 pound) produced for standard mix

D = number of bags (1 pound) produced for deluxe mix

	Standard Mix	Deluxe Mix	Pound Available	Cost per pound
Raisin	1/2	2/3	90	$0.60
Peanut	1/2	1/3	60	$1.50

Determining profit per pound for each bag:

	Standard Mix	Deluxe Mix	Cost per pound
Raisin	½*.60 = $0.30	2/3*0.60 = $0.40	$0.60
Peanut	½*1.50 = $0.75	2/3*1.50 = $0.50	$1.50
Cost per bag	$1.05	$0.90
Selling price per bag	$2.25	$2.90
Profit per bag	2.25-1.05 = $1.20	2.90- 0.90 = $2.00

Objective function:

Objective is to maximize the profit of total production:

Maximize A = $1.20 x S + $2.00 x D

Subject To:

Raisin availability:

Raisin required <= raisin available

½*S + 2/3*D <= 90

Peanut availability

Peanut required <= peanut available

½*S + 1/3*D <= 60

Maximum Demand is of 110 bags for each product:

S<= 110

D <= 110

Non-negativity Constraint:

S, D >= 0

Excle Model:

Excel Formulas:

Optimal solution:

S = 33

D = 110

Total profit = $255

The company should produce 33 bags of standard mix and 110 bags of deluxe mix to maximize the profit by $255.

Semsitivity Report:

According to Sensitivity report,

1. The raisin availability constraint is binding constraint and the shadow price of the constraint is 2.10. It means if additional pound of the raisin is made available the profit will increase by $2.10 per additional unit without changing the optimality of problem.

2. The peanut availability constraint is non-binding constraint and thus, even if peanuts additional pound is made available, it will not change the optimal mix and profit.

3. For the profit coefficient of standard bags, the allowable increase is 0.45, it means if the profit per bag of the standard mix is increased by more than $0.45, the product mix will change.