The Natural Fertilizer Company makes three types of fertilizer: 20-5-5 for lawns

Question

The Natural Fertilizer Company makes three types of fertilizer: 20-5-5 for lawns, 10-15-10 for gardens, and 5-5-5 for trees. The numbers in each case refer to the percentage by weight of nitrates, phosphates, and potash, respectively. The fertilizer is packed in 100-pound sacks. The company has the following amounts of raw materials on hand: 7 tons of nitrates, 4 tons of phosphates, and 3 tons of potash. The profit per 100-pound sack of lawn fertilizer is dollar 6, the profit per sack of garden fertilizer is dollar 4, and the profit per sack of tree fertilizer is dollar 3. The company has a contract to supply at least 1 ton of garden fertilizer. How much of each fertilizer should the company produce to maximize profit question

Explanation / Answer

Decision Variables

Let X, Y, and Z represents the quantities of the three fertilizers: 20-5-5 for lawns, 10-15-10 for gardens, and 5-5-5 for trees respectively. Quantities are in terms of 100-pound sacks. X= 1 means 100-pound sack of lawn fertilizer having 20 pounds of nitrate, 5pounds of phosphate and 5pounds of potash and 70 pounds of other materials. (note total sum of percentages is not equal to 100)

Objective Function

As stated in the question Company wants to maximize profits, therefore the objective function is to Maximize P:

Maximize P = 6X + 4Y + 3Z

Constraints:

Constraints are in terms of availability of raw materials required for producing the fertilizers and also the minimum quantity of garden fertilizer (Y). It is considered that 1 ton = 2000 pounds.

20X + 10Y + 5Z <=(less than equal) 14000 (availability of 7 tons of nitrates)

5X + 15Y+ 5Z <= (less than equal) 8000 (availability of 4 tons phosphates)

5X + 10Y + 5Z <=(less than equal) 6000 ( availability of 3 tons of potash)

Y >= (greater than equal) 20 (At least 1 ton of garden fertilizer) 20 sacks of 100-pound makes it 2000 pounds=1ton

Non-negativity constraints X, Y, and Z >= 0

Solution of the above linear programming problem using excel solver is as follows:

The maximum profit is $5,159

Number of lawn fertilizers sacks is 533 as this gives the maximum profir per unit of $6

Number of garden fertilizer sacks is 20 to meet the contracted demand

Number of tree fertilizer sacks is 627 because it takes the minimum quantities of the inputs.

Binding constraint is availability of potash.

Values of Dec.Var. 533 20 627 Decision Variables X Y Z Objective-Profit 6 4 3 5159 Constraints SumProd. sign RHS Nitrate 20 10 5 13995 "<= 14000 Phosphate 5 15 5 6100 "<= 8000 Potash 5 10 5 6000 "<= 6000 Garden fertilizer 0 1 0 20 ">= 20

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