The cost per pound of the nuts included in the Regular Mix, Deluxe Mix, and Holiday Mix is as follows. The cost per pound of the almonds is $1.25, of the Brazil Nuts is $.95, of the filberts is $.90, of the pecans is $1.20, and of the walnuts is $1.05. The cost per pound of the Regular Mix is $1.03, of the Deluxe Mix is $1.07, of the Holiday Mix is $1.10.

The optimal product mix and the total profit contribution are as follows. The optimal product mix is to produce 17500 pounds of the Regular Mix, 10625 pounds of the Deluxe Mix, and 5000 pounds of the Holiday Mix. The total profit contribution after subtracting the total cost per shipment of the various nuts in the mixes is $24925.

If an additional amount of nuts can be purchased to increase the profit contribution, the recommendation would be to purchase more almonds and walnuts. By looking at the constraint for the dual prices for almonds you will see for every one pound increase in almonds used, the objective function/profit contribution will increase at a rate of $8.50. And by looking at the dual prices for walnuts you will see for every one pound increase in walnuts used, the objective function/profit contribution will increase at a rate of $1.50.

If an additional 1000 pounds of almonds were offered at a price of $1000 from a suppler that overbought, we would recommend TJ’s to purchase the almonds. They would be purchasing the almonds at a price of $1.00 per pound, decreasing their cost by twenty-five cents per pound. Also they would increase their available pounds of almonds from 6000 to 7000, thus increasing their profit to $29,883.33 giving them an increase of $4,958.33.

If TJ’s does not satisfy all existing orders, a recommendation would be to increase the purchase of nuts for all of the mixes to reach an optimal solution to the objective function that would maximize the profit contribution. Also we would recommend to TJ’s not to fill all orders for the Holiday Mix.

TJ’s Inc., 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 types of nuts.

In preparation for the fall season, TJ’s has just purchased the following shipments of nuts at the prices shown:

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% filberts, 25% pecans, and 20% walnuts.

TJ’s accountant has analyzed the cost of packaging contribution per pound, and so forth, and has 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.

Customers orders already received are summarized below:

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

TJ’s is committed to using the available nuts to maximize profit over the fall season; nuts not used will be given to the Free Store. But even if it is not profitable to do so, TJ’s president has indicated that the orders already received must be satisfied.

To maximize TJ’s profit over the fall season, we must decide how many pounds of Regular Mix, Deluxe Mix, and Holiday Mix they should produce. We begin by declaring the decision variables, which represent the decision that TJ’s must make. The decision variables we used are as follows:

R = pounds of Regular Mix

D = pounds of Deluxe Mix

H = pounds of Holiday Mix

Using the data given in the problem, we must first obtain an objective function for maximizing the profit. The objective function that we chose to maximize is,

1.65R + 2.00D + 2.25H - 36450.

It is the total profit for each mix minus the total cost per shipment of the nuts, since these costs were not included in the profits.

There are eight necessary constraints for this model. They are necessary because of the customer orders already received and the limits placed on the available amount of nuts in the mixes.

Constraint 1 Pounds of Regular Mix produced > Orders of Regular Mix

Constraint 2 Pounds of Deluxe Mix produced > Orders of Deluxe Mix

Constraint 3 Pounds of Holiday Mix produced > Orders of Holiday Mix

Constraint 4 Total amount of almonds in all mixes < Amount of almonds available

Constraint 5 Total amount of Brazil nuts in all mixes < Amount of Brazil nuts available

Constraint 6 Total amount of filberts in all mixes < Amount of filberts available

Constraint 7 Total amount of pecans in all mixes < Amount of pecans available

Constraint 8 Total amount of walnuts in all mixes < Amount of walnuts available

Let us now develop mathematical statements of the constraints for the problem. The constraint for the pounds of Regular Mix produced can be written as follows:

R > 10000

The constraint for the pounds of Deluxe Mix produced can be written as follows:

D > 3000

The constraint for the pounds of Holiday Mix produced can be written as follows:

H > 5000

The constraint for the limit placed on the amount of almonds in all of the mixes can be written as follows:

.15R + .20D + .25H < 6000

The constraint for the limit placed on the amount of Brazil nuts in all of the mixes can be written as follows:

.15R + .20D + .15H < 7500

The constraint for the limit placed on the amount of filberts in all of the mixes can be written as follows:

.15R + .20D + .15H < 7500

The constraint for the limit placed on the amount of pecans in all of the mixes can be written as follows:

.10R + .20D + .25H < 6000

The constraint for the limit placed on the amount of walnuts in all of the mixes can be written as follows:

.25R + .20D + .20H < 7500

Combining all of the constraints with nonnegativity requirements enables us to write the complete linear programming model for the Product Mix Problem as follows:

R, D, H > 0

Using the software package The Management Scientist, to solve the Product Mix we came up with the following results.

The Objective Function Value shows the optimal solution to the problem will provide a maximum profit of $61375. The software package does not allow a constant in the objective function therefore we must subtract the $36450 from the original Objective Function Value after the software has solved the linear programming model. After the $36450, the total cost per shipment of the different nuts, is subtracted from the original Objective Function Value. The optimal solution to the problem will provide a maximum profit of $24935. The optimal values of the decision variables are given by R = 17500, D = 10625, and H = 5000. The optimal decision for TJ’s is to produce 17500 pounds of the Regular Mix, 10625 pounds of the Deluxe Mix, and 5000 pounds of the Holiday Mix.

The computer output information for the slack/surplus variable and the dual prices is restated below:

The pounds of Regular Mix constraint has a surplus of 7500 showing that the optimal solution exceeds the customer orders for Regular Mix by 7500 pounds. Also the pounds of Deluxe Mix constraint has a surplus of 7625 showing that the optimal solution exceeds the customer orders for Deluxe Mix by 7625 pounds. The surplus of zero associated with pounds of Holiday Mix is a result of this constraint being binding. The negative dual price indicates that increasing the customer orders for Holiday Mix from 5000 to 5001 pounds will actually decrease the profit contribution by $.18.

Looking at the dual prices for the different types of nuts available, the following recommendations would be suggested: to increase the pounds of almonds and walnuts purchased by TJ’s. By increasing the pounds of almonds purchased, TJ’s will increase their objective function/profit contribution at rate of $8.50 per pound of almonds. Also by increasing the pounds of walnuts purchased, TJ’s will increase their objective function/profit contribution at a rate of $1.50 per pound of walnuts. Although we would not recommend buying more Brazil Nuts, filberts, and pecans because there are unused amounts of each: 250 pounds of Brazil Nuts and filberts and 875 pounds of pecans. These slack variables would indicate, by looking at the dual prices, no profit is gained by purchasing more pounds of these types of nuts.

The computer output information for the sensitivity analysis on RIGHT HAND SIDE RANGES is restated below:

Several interpretations of these ranges are possible. Recalling that the dual price for almonds enabled us to conclude that for each additional pound purchased the profit would increase at a rate of $8.50. The range for the available amount of almonds shows that this statement is appropriate only up to 6583 pounds of almonds. Thus for each additional pound purchased the profit would increase at a rate of $8.50 only for 583 pounds. Increases above this level would not necessarily be beneficial. Also the dual price for walnuts enabled us to conclude that for each additional pound purchased the profit would increase at a rate of $1.50. The range for the available amount of walnuts shows that this statement is appropriate only up to 7750 pounds of walnuts. Thus for each additional pound purchased the profit would increase at a rate of $1.50 only for 250 pounds. Increases above this level would not necessarily be beneficial. Also note that the dual price of -.175 for the pounds of Holiday Mix suggested the desire to reduce the production/customer orders of this mix. The range of feasibility for this constraint shows that the customer orders could be reduced to zero and the value of reduction would be at the rate of $.18 per pound. Although keep in mind this software package for linear programming models considers only one change at a time. Thus these are only suggestions not necessarily changes that need to be made.

To maximize TJ’s profit over the fall season, we used a linear programming model to decide how many pounds of Regular Mix, Deluxe Mix, and Holiday Mix they should produce. We declared the decision variables as follows:

R = pounds of Regular Mix

D = pounds of Deluxe Mix

H = pounds of Holiday Mix

The objective function as:

1.65R + 2.00D + 2.25H + 36450.

And the constraints as:

R, D, H > 0 Nonnegativity Requirements

Using the software package, The Management Scientist, to solve the linear programming model we came up with following optimal decisions. The pounds produced should be 17500 of Regular Mix, 10625 of Deluxe Mix, and 5000 of Holiday Mix. Other suggestions were to purchase more almonds and walnuts and to reduce the pounds of Holiday Mix produced.