Question

The Rakhsh Stable feeds and houses the horses used to pull tourist-filled carriages through the streets of Charleston’s historic waterfront area. The stable owner, an ex-racehorse trainer, recognizes the need to set a nutritional diet for the horses in his care. At the same time, he would like to keep the overall daily cost of feed to a minimum.

The feed mixes available for the horses’ diet are an oat product, a highly enriched grain, and a mineral product. Each of these mixes contains a certain amount of five ingredients needed daily to keep the average horse healthy.

The below table shows these minimum requirements, units of each ingredient per pound of feed mix, and costs for the three mixes.

In addition, the stable owner is aware that an overfed horse is a sluggish worker.

Consequently, he determines that 6 pounds of feed per day are the most that any horse needs to function properly.

Formulate this LP problem

Solve for the optimal daily mix of the three feeds (In excel)

Feed Mix
Diet Requirement (Ingredients)	Oat Product (Units/lb)	Enriched Grain (Units/lb)	Mineral Product (Units/lb)	Minimum Daily Requirement (Units/lb)
A	2	3	1	6
B	0.5	1	0.5	2
C	3	5	6	9
D	1	1.5	2	8
E	0.5	0.5	1.5	5
Cost/lb	$0.09	$0.14	$0.17

Answer 1

The objective is to minimize daily overall cost

Objective function:

Minimize Z = 0.09X1 + 0.14X2 + 0.17X3

here X1, X2 and X3 are lbs of oats, enriched grains and mineral product respectively

Subject to:

2X1 + 3X2 + 1X3 6 (Min requirement for ingredient A)

0.5X1 + 1X2 + 0.5X3 2 (Min requirement for ingredient B)

3X1 + 5X2 + 6X3 ​​​​​​​ 9 (Min requirement for ingredient C)

1X1 + 1.5X2 + 2X3 ​​​​​​​ 8 (Min requirement for ingredient D)

0.5X1 + 0.5X2 + 1.5X3 ​​​​​​​ 5 (Min requirement for ingredient E)

X1 + X2 + X3 6 (Max feed of 6 kgs)

X1, X2 AND X3 0 (Non negativity)

The formulation is shown below:

The solver equations and conditions are shown below:

The Solution is shown below:

• Thus total cost is $0.69 (rounded to 2 decimals)
• 1.33 lbs of oats and 3.33 lbs of minerals are given
• Enriched grain is not given