Question

This is your lucky day. You have just won a $20,000 prize. You are setting aside $8,000 for taxes and partying expenses, but you have decided to invest the other $12,000. Upon hearing this news, two different friends have offered you an opportunity to become a partner in two different entrepreneurial ventures, one planned by each friend. In both cases, this investment would involve expending some of your time next summer as well as putting up cash. Becoming a full partner in the first friend’s venture would require an investment of $10,000 and 400 hours, and your estimated profit (ignoring the value of your time) would be $9,000. The corresponding figures for the second friend’s venture are $8,000 and 500 hours, with an estimated profit to you of $9,000. However, both friends are flexible and would allow you to come in at any fraction of a full partnership you would like. If you choose a fraction of a full partnership, all the above figures given for a full partnership (money investment, time investment, and your profit) would be multiplied by this same fraction. Because you were looking for an interesting summer job anyway (maximum of 600 hours), you have decided to participate in one or both friends’ ventures in whichever combination would maximize your total estimated profit. You now need to solve the problem of finding the best combination.

(a) Formulate a linear programming model for this problem.

(b) Write each functional constraint in x2 = mx1 +b form and box the slope.

(c) Write the objective function in x2 = mx1 +b form and box the slope.

(d) Graph the feasible region.

(e) Graphically solve the problem by comparing the slopes of the functional constraint lines and the slope of the objective function line to determine which corner point feasible solution is optimal. Draw the objective function line through this CPF solution on your graph.

(f) Determine numerical values for the decision variables and objective function corresponding to this CPF solution.

Answer 1

(a) Linear programming model is formulated as below:

Let X1 be the fractional participation in first friend's venture

and, X2 be the fractional participation in second friend's venture

Total expected profit, Z = SUMPRODUCT of fractional participation in each venture multipled with profit in respective venture

Objective function: Maximize Z = 9000X1+9000X2

Total investment in both ventures = SUMPRODUCT of fractional participation in each venture multipled with investment required in respective venture

Total money that can be invested is $ 12000. Therefore, RHS (right hand side) of the constraint is 12000

Constraint for Total investment: 10000X1+8000X2 <= 12000

Total time investment = 400X1+500X2

Maximum available time is 600 hours. Therefore, RHS of the constraint is 600

Constraint for Total time investment: 400X1+500X2 <= 600

Maximum participation in each venture can be 100%. So, X1, X2 <= 1

Complete Linear program is following:

Maximize Z = 9000X1+9000X2

s.t.

10000X1+8000X2 <= 12000

400X1+500X2 <= 600

X1 <= 1

X2 <= 1

X1, X2 >= 0

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(b) For writing each function constraint in x2=mx1+b form, divide both sides of the constraint by the coefficient of X2 and then take X1 term to the RHS

Constraint 1 (C1): X2 = -(10000/8000)X1+12000/8000 or, X2 = -1.25X1+1.5 , slope coefficient of X1, which is (-1.25)

Constraint 2 (C2): X2 = -(400/500)X1+600/500 or, X2 = -0.8X1+1.2 , slope coefficient of X1, which is (-0.8)

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(c) Objective function is: Z=9000X1+9000X2

Dividing both sides by coefficient of X2 (9000) and take X term to the other side

X2 = -(9000/9000)X1+Z/9000 or,

X2 = -X1+Z' (let's say, Z/9000 = Z')

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(d) Feasible region is graphed as below:

Feasible region is the shaded region bounded by the corner points as shown and the origin (0,0)

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(e) Slope of objective function line is -1.

Comparing this with the slopes of constraints. Slope of constraint 1 is -125 and that of constraint 2 is -0.8.

So, Slope of objective function is between the slope of constraints 1 and 2

Hence, the objective function will pass through intersection of C1 and C2.

Optimal point is (0.667, 0.667)

Objective function is shown by thick red line on the graph is shown as below:

(f) Numerical values of the decision variables are:

X1 = 2/3 = 0.667

X2 = 2/3 = 0.667

Value of objective function = 9000*2/3 +9000*2/3 = $ 12,000