// Regarding the problem and answer, I do not understand why Z covers 1.3xm3 and 1.5xp3. When I draw a diagram about multiple-period investment, It was concluded that the Manufacturing section reached its fourth year on its second (Xm2) investment and that the Product Research reached its third year rather than its fourth year, considering that it was a three-year cycle. Could you explain what the problem is through the diagram? Thanks.

24. Ronald Thump is interested in expanding his firm. After careful consideration, he has determined three areas in which he might invest additional funds: (1) product research and development, (2) manufacturing operations improvements, and (3) advertising and sales promotion. He has $500,000 available for investment in the firm. He can invest in its advertising and sales promotion program every year, and each dollar invested in this manner is expected to yield a return of the amount invested plus 20% yearly. He can invest in manufacturing operations improvements every 2 years, with an expected return of the investment plus 30% (at the end of each 2-year period). An investment in product research and development would be for a 3-year period, with an expected return of the investment plus 50% (at the end of the 3-year period). To diversify the total initial investment, he wishes to include the requirement that at least $30,000 must be invested in the advertising and sales promotion program, at least $40,000 in manufacturing operations improve- ments, and at least $50,000 in product research and development initially (at the beginning of the first year). Ronald wants to know how much should be invested in each of the three alternatives, during each year of a 4-year period, to maximize the total ending cash value of the initial $500,000 investment. a. Formulate a linear programming model for this problem. b. Solve the model by using the computer. 24. a) Xj = dollar amount invested in alternative i in year j, where i =p (product research and development), m (manufacturing operations improvements), a (advertising and sales promotion) and j = 1,2,3,4 (denoting year): S; = slack, or uninvested funds in year j maximize Z= 54 + 1.2x04 + 1.3.xm3 + 1.5Xp3 subject to Xal> 30,000 Xm1 > 40,000 Api > 50,000 Xa1 + xm1 + Xp1 + $i = 500,000 Xa2 + Xm2 + Xp2 + S2 = S1 + 1.2x al Xa3 + Xm3 + X53 = S2 + 1.2.xq2 + 1.3xml Xa4 + X34 = S3 + 1.2xa3 + 1.3xm2 + 1.5xpl Xij, S;> 0