3.S1. Farm Management Dwight and Hattie have run the family farm for over 30 yea

Question

3.S1. Farm Management Dwight and Hattie have run the family farm for over 30 years. They are currently planning the mix of crops to plant on their 120-acre farm for the upcoming season. The table gives the labor-hours and fertilizer required per acre, as well as the total expected profit per acre for each of the potential crops under consideration. Dwight, most 6,500 total hours during the upcoming season. They have 200 tons of fertilizer available. What mix of crops should be planted to maximize the family's total profit? a. Formulate and solve a linear programming model for this Hattie, and their children can work at problem in a spreadsheet. b. Formulate this same model algebraically.

Explanation / Answer

1. This is a resource allocation problem. The activity here is planting the crops and limited resources are land, labour and fertilizers. We will first build the following spreadsheet. Oats Wheat Corn Profit per acre $500 $600 $950 Total Resources Resources used per acre Available Land (acres) 1 1 1 120 Labour (hours) 50 60 105 6500 Fertilizers (tons) 1.5 2 4 200 Now, to take the decision regarding how many acres of each crop to plant, we will add one more row heading acres palnted Oats Wheat Corn Profit per acre $500 $600 $950 Total Available Resources Resources used per acre Land (acres) 1 1 1 120 Labour (hours) 50 60 105 6500 Fertilizers (tons) 1.5 2 4 200 Total Profit Acres planted 1 1 1 $2,050 Total Profit = $500 x 1 acre + $600 x 1 acre + $950 x 1 acre Now, the limited resources are land, labour and fertilizers, we will now add one column for total used Oats Wheat Corn Profit per acre $500 $600 $950 Total Used Total Available Resources Resources used per acre Land (acres) 1 1 1 3 120 Labour (hours) 50 60 105 215 6500 Fertilizers (tons) 1.5 2 4 7.5 200 Total Profit Acres planted 1 1 1 $2,050 Now to maximise the profit we will take the decision to plant each crop Oats Wheat Corn Profit per acre $500 $600 $950 Total Used Total Available Resources Resources used per acre Land (acres) 1 1 1 120 120 Labour (hours) 50 60 105 6400 6500 Fertilizers (tons) 1.5 2 4 200 200 Total Profit Acres planted 80 40 0 $64,000 Thus, oats should be planted on 80 acres, wheat on 40 acres while not planting corn will result in total profit of $64,000 2. Algebraical model Decision Variables x1 = how many acres of land to be used to plant oats X2 = how many acres of land to be used to plant wheat x3 = How many acres of land to be used to pant corn Z = Profit Objective function Max Z = $500X1 + $600X2 + $950X3 Limited resources X1 + X2 + X3 ? 120 acres 50X1 + 60X2 + 105X3 ? 6500 hours 1.5X1 + 2X2 +4X3 ? 200 tons Result $64000 = $500 x 80acres + $600 x 40acres + $950 x 0acres

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