# linear programming

Suppose you have won \$6000 from OK Grand challenge promotion and you want to invest is. Upon hearing the news , your two differnt friends Mukanya and Mhofu offer you each an opportunity to become a partner in their different interpreneurial ventures. In both cases, this investiment involves committing some of your time next summer as well as putting up cash. Becoming a full time partner in Mukanya's ventures would require an investiment of \$ 5000 and 400 hours and your estimated profit (ignoring time value of money) would be \$4500. The corresponding figures for Mhofu's ventures are$4000 and 500 hours, with an estaimated profit to you of \$4500. However both friends are flexible and would allow you to come in at any fraction of a full partnership you would like; your share of profit would be proportional to this fraction. Because you were looking for an interesting job anyway (maximum 600 hours), you have decided to participate in one's friend's or both friend's ventures in which ever combination would maximise your total estimated profit. You now need to solve the problem of finding the best combination. (a) Formulate a linear programming model for this problem (b) Solve this problem graphically. What is your total estiamted profit (c)Suppose you wish to have at least 4/5 shares from Mukanya and at least 3/5 shares from Mhofu, would this feaseble. Eplain (d) perform a sensitivity analysis to determine the ranges of optimality for the profit contributions related to Mukanya and Mhofu Ventuers. (e)How much further profit would you enjoy for each additional hour at Mukanya's venture? - What have you tried? What is the problem you are facing with this question? – Daryl Sep 17 '12 at 10:37 If we define$p_1$and$p_2$to be the fraction of full partnership for Mukanya's venture and Mhofu's venture, respectively, then we can begin by stating the objective function which should be maximized and the constraints on our time, money, and the fractions themselves.$ \begin{array}{lrcl} \textrm{Maximize} & 4500p_1+4500p_2 & & &\textrm{objective function}\\ \textrm{subject to} & 5000p_1+4000p_2 & \leq & 6000 &\textrm{monetary constraint}\\ & 400p_1+500p_2 & \leq & 600 &\textrm{time constraint}\\ & p_1,p_2 & \geq & 0 &\textrm{non-negativity constraint} \end{array} \$ –  Max Sep 17 '12 at 18:54