Question

Fred Perkins has found that adding goats to the product line of his traditional family farm has turned out to be profitable – the demand for goats in ethnic markets has increased dramatically in the past few years. Fred now wants to ensure that he is providing the best possible diet to his animals at the lowest possible cost. Goats are fed a mix of three types of feed – corn, silage and alfalfa, all of which Fred grows himself. He knows that the most profitable use for his crops is to feed them to his goats, but he grows much more of each than he could possibly use for his own herd and is able to sell the balance to other farmers for additional profit.

The nutritional makeup of the feeds, the selling price (foregone revenue) for each type of feed and the minimum daily requirements of the nutritional ingredients (per goat) are shown in the table below.

Nutritional Ingredient	Grams per kg of corn	Grams per kg of silage	Grams per kg of alfalfa	Minimum daily requirement (gms)
Carbohydrates	90	20	40	200
Protein	30	80	60	180
Vitamins	10	20	60	150
Cost/Kg	$0.84	$0.72	$0.60

Fred’s herd consists of 100 goats.

Formulate Fred’s problem as a linear programming problem. Clearly define your decision variables as a first step.

Enter the problem in Excel and use Solver to get the optimal solution. How much of each feed type will Fred divert from sales each day to feed his herd? How much revenue does he give up each day?

Question 6

Refer to the previous question and the answer and sensitivity reports generated by Solver. Whenever possible, answer the following questions by referring to the reports (rather than adjusting and re-solving the problem).

Fred has been offered a carbohydrate supplement by a national drug company representative. The supplement comes in liquid form and costs $0.23 per litre. Each litre added to the feed will reduce the daily requirement for carbohydrates by 60 grams per goat. Should Fred purchase the supplement? If yes, by how much will his daily profit increase?

A recent decision by the federal government to mandate a minimum requirement for ethanol in gasoline has resulted in an increase in demand and a consequent price increase for corn. (Ethanol is made principally from corn.) It looks as if the price increase will take corn to $1.02 per kilogram. Will this change Fred’s allocation of feed to his animals? How will it affect the daily profit that he sees from his herd?

How would you interpret the shadow price for vitamins in your solution?

Answer 1

Let the decision variables are,

X1 = grams per kg of corn are used,

X2 = grams per kg of silage are used and

X3 = grams per kg of alfalfa are used

The objective of the LP is to minimize the cost per kg of diet used hence the objective function is defined as,

And the constraints are defined as,

Corn constraint

Protein constraint

Vitamins constraint

And the non-negativity constraints are,

Now the LP is solved using the excel solver by following these steps,

Step 1: Write the decision variable with value zero. The screenshot is shown below,

Step 2: Write the objective function equation while taking the decision variable value. The screenshot is shown below,

Step 3: Write the constraints equation while taking the decision variable value and write the right side value of the constraint

The screenshot for constraint corn is shown below,

The screenshot for constraint silage is shown below,

The screenshot for constraint alfhalfa is shown below,

The screenshot for nonnegativity of X1, X2, and X3 is shown below,

Step 4: (If you have not install the solver excel follow, FILE > Options > Add-ins > Manage: select ExcelAdd-ins > Go then tick Solver Add-in > OK)

DATA > Solver > OK. The screenshot is shown below,

Step 5:

Set Objective: Select objective value,

To: select Min

Subject to the Constraints > Add > in Cell Reference select constraint value and in Constraint: select right hand side value of constraint and select the >= inequality.

Tick Make Unconstraint Variables Non-Negative

Select a Solving Method: Simplex LP

then click Solve. The screenshot is shown below,

Step 6: Select Reports > Ansawer, Sensitivity and Limits. then Ok

The result is obtained. The screenshots are shown below,

The Answer Report

The Sensitivity Report

The Limit Report

6)

From the sensitivity analysis report,

The allowable decrease for carbohydrate = 80 and the shadow price = 0.007714286

Adding of drug suppliment will decrease the requirement of cabohydrate by 60 which is within the allowable range hence the adding of suppliment will decrease the price by,

which is greater than the cost of the suppliment hence the cost will be minimized.

Hence Fred should purchase the supliment.

The reduced cost for decision variable corn is zero within the allowable increase upto 0.51 which means within this allowable range the cost will be unaffected..

Hence change the corn price $0.85 to $1.02, which is (1.02 - 0.85 =) $0.17 increase in price and this price is within the allowable increase of cost price. Hence it will no affect the optimize cost value