Case 2:
Ch3 – Case Problem 1 (page 131)
Case Problem 1 Product Mix
TJ Inc.’s makes three nut mixes for sale to grocery chains located in the Southeast. The three mixes, referred to as the Regular Mix, the Deluxe Mix, and the Holiday Mix, are made by mixing different percentages of five types of nuts.
In preparation for the fall season, TJ Inc.’s 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 Deluxe 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.
An accountant at TJ, Inc. 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 Deluxe 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 |
| Deluxe | 3,000 |
| Holiday | 5,000 |
Because demand is running high, it is expected that TJ, Inc. will receive many more orders than can be satisfied.
TJ, Inc. is committed to using the available nuts to maximize profit over the fall season; nuts not used will be given to a local charity. Even if it is not profitable to do so, TJ. Inc. president indicated that the orders already received must be satisfied.
Managerial Report
Perform an analysis of the TJ Inc.’s product-mix problem, and prepare a report for the president of TJ, Inc. that summarizes your findings. Be sure to include information and analysis on the following:
1. Please formulate a linear programming model of optimal product mix and the total profit contribution. (Please provide both SOLVER and LINGO output)
2. Please write a report to describe the sensitivity report regarding optimal solution, objective function values, slack variables, binding/nonbinding, and shadow price.
Note that the cost of the five shipments of nuts is a sunk (not a relevant) cost and should not affect the decision. However, this information may be useful to management in future pricing and purchasing decisions.
