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# Linear programming problem:Optimal product mix

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Develop the objective function and constraints required for the problem. Determine optimal product mix and profit contribution. Please note that the profit contribution per pound of \$1.65 for the Regular Mix, \$2.00 for the Premier Mix, and \$2.25 for the Holiday Mix are the coefficients to use in the objective function. Respond to questions 1,2 and 3 below.

Case Problem

JG's Incorporated makes 3 nut mixes for sale to grocery chains located in the Southwest. The 3 mixes, referred to as the Regular mix, the Premier mix, and the Holiday Mix, are made by mixing different percentages of 5 types of nuts. In preparation for the autumn season, JG's Incorporated has just purchased the following shipments of nuts at the prices shown:

Type of Nut Shipment Amount (pounds) Cost per Shipment (\$)

Almond 6000 7500

Brazil 7500 7125

Filbert 7500 6750

Pecan 6000 7200

Walnut 7500 7875

The Regular Mix consists of 15% almonds, 25% Brazil nuts, 25% filberts, 10% pecans, and 25% walnuts. The Premier Mix consists of 20% of each type of nut, and the Holiday Mix consists of 25% almonds, 15% Brazil nuts, 15% filberts, 25% pecans, and 20% walnuts.

JG's Incorporated accountant analyzed the cost of packaging materials, sales price per pound, and so forth and determined that the profit contribution per pound is \$1.65 for the Regular Mix, \$2.00 for the Premier Mix, and \$2.25 for the Holiday Mix. These figures do not include the cost of specific types of nuts in the different mixes because that cost can vary greatly in the commodity markets. Customer orders already received are summarized here:

Type of Mix Orders (pounds)
Regular 10,000
Premier 3,000
Holiday 5,000

Because demand is running high, it is expected that JG's will receive many more orders than can be satisfied.

JG's Incorporated is committed to using the available nuts to maximize profit over the autumn season; nuts not used will be given to a local charity. Even if it is not profitable to do so, JG's president indicated that the orders already received must be satisfied.

A. Perform an analysis of JG's product-mix problem, and prepare a report for JG's president that summarizes the findings. Be sure to include the information and analysis on the following:

1. The optimal product mix and the total profit contribution
2. Recommendations regarding how the total profit contribution can be increased if additional quantities of nuts can be purchased
3. Recommendation as to whether JG's should purchase an additional 1000 pounds of almonds for \$1000 from a supplier who overbought.

https://brainmass.com/math/linear-programming/linear-programming-problem-optimal-product-mix-126747

#### Solution Summary

This posting contains solution to following Linear programming problem on product mix.

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## Linear Programming: Operations Research

See the attached file.

Consider the following problem:
A supermarket store manager needs to determine how much to stock of two brands of soda: PC and CC, for Super Bowl Sunday. The store needs to tell the suppliers how many "units" are available and the suppliers then stock the store with their product. Units consist of different flavors and sizes depending on what the suppliers think will sell best. Fractions of units are OK. The store's makes a profit margin of 15 cents for each unit of PC and 10 cents for each unit of CC. In the soda aisle, there is a maximum of 36 linear feet of shelf space: each unit of PC takes up 9 linear feet of shelf space and each unit of CC takes up 4 linear feet of shelf space. In the end displays, there is a 20 linear feet of display space available: each unit of PC takes up 4 linear feet of display space and each unit of CC takes up 4 linear feet of display space. In other words, 1 unit of PC will require 9 feet in the soda aisle plus 4 feet in the end display area.

Problem 1
Set up the problem. What is the objective function? What are the decision variables? What are the constraints?
Problem 2
Solve the problem graphically. Identify all corner point (basic feasible) solutions.

Problem 3
Solve the problem again, this time using the Simplex Method. What is the optimal solution(s)?
In addition to the display area constraints, the manager must meet certain sales quotas from his district manager. The store manager gets 1 point for each unit of PC sold and 4 points for each unit of CC - he must meet a quota of 14 points.

What the district manager doesn't know is that the store manager has a side deal going with the supplier of PC. If the store sells at least half as much PC as CC, the store manager gets tickets to next year's Super Bowl.

Given these two additional constraints, how many units of each brand should the store manager stock?

Problem 4
Solve the problem with these additional two constraints, using either the LINDO or LINGO software. Using the LINDO or LINGO output (show the output) only and without solving the problem again, answer the following two questions.

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