Question

A farm family owns 125 acres of land and has $40,000 in funds available for investment. Its members can produce a total of 3,500 person-hours worth of labor during the winter months (mid- October to mid-May and 4 O person oursduring the summer f any of these person-hours are not needed younger membe soft family wil uset to wor o aneighbo o r m . S5.00/hour, during the winter months and $6.00/hour during the summer. Cash income may be obtained from three crops and two types of livestock: dairy cows and laying hens. No investment funds are nceded for the crops. However, cach cowll reqire an investment outlay of S1,200, and each hen will cost $9. These outlays should be considered as a one time capital investment and should not be included in computing the annual profitability of the farm. Each cow wil require 1.5 acres of land, 100 person-hours of work from November to April, and 50 person-hours from May to October. Each cow will produce a net annual cash income of $1,000 for the family. The coresponding figures for cach hen are: no acrcage, 0.6 pcrson-hours from November to April, 0.3 person-hours from May to October, and an annual nct cash income of $5. The chicken house can accommodate a maximum of 3,000 hens, and the size of the barn limits the herd to a maximum of 32 cows Estimated person-hours and income per acre planted in each of the three crops are Soybeans Corn Oats Winter person-hours Summer person-hours Net annual cash inco 500 750 350 20 50 35 10 75 40 The family wishes to determine how much acreage should be planted in each of the crops and how many cows and hens should be kept to maximize its net cash income. Formulate the linear g model for this problem.

Answer 1

In order to Formulate the linear Programming model, we need to find the decision variables, constraints and objective functions. A Decision variables in the Problem Is = amount of area allocated for soyabean LC = amount of area allocated for corn LO = amount of area allocated for cat C = number of areas purchased H = number of hens purchased E = Excess person - hours in the summer B. constraints in the Following linear programming model. Ls + lc + lo + 1.5 c < = 125 acres 1200c + 9H < = 40000 ( < = 32, H < = 3000 (since the maximum values of ben and cow are given in the problem) labor limitation for winters 20ls + 35lc + 10lo + 100 c + 0.6 H + w = 350 u labor limitation for summers 50sl + 75lc + 40lo + 150c + 0.9 H + S = 4000

Answer 2

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Answer 3

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