David Palma EGR 546 By David Palma Page 92 problem 3 Problem statement A bank is attempting to determine where its assets should be invested during the current year At present 500 000 00 is available for investment in bonds home loans auto loans and personal loans The annual rate of return on each type of investment is known to be bonds 10 home loans 16 auto loans 13 personal loans 20 To ensure that the banks portfolio is not to risky the bank s investment manager has placed the following three restrictions on the bank s portfolio 1 The amount invested in personal loans cannot exceed the amount invested in bonds 2 The amount invested in home loans cannot exceed the amount invested in auto loans 3 No more than 25 of the total amount invested may be in personal loans The bank s objective is to maximize the annual return on its investment portfolio Formulate an LP that will be enabling the bank to meet this goal Table 1 Summary of input data Amount invested in a group B HL AL PL Bond Home Loans Auto Loans Personal Loans ARR 10 16 13 20 The decision variables must satisfy the following 4 constrains Constrain 1 All investments must be a maximum of 500 00 00 B HL AL PL 500 000 Constrain 2 bonds P B 0 Constrain 3 The amount invested in personal loans cannot exceed the amount invested in The amount invested in home loans cannot exceed the amount invested in auto loans HL AL 0 Constrain 4 No more than 25 of the total amount invested may be in personal loans x4 0 25 x1 x2 x3 x4 NOTICE THE LOGIC HERE 25 B 25 HL 25 AL 75 PL 0 Objective function 1 David Palma EGR 546 MaxZ 10 B 16 HL 13 AL 20 PL Table 2 Input Data WinQSB Original Problem Primal Table 3 Solution WinQSB Primal The following tables and graphs show the sensitivity analysis over the objective function variable coefficients and the right hand side values of the constraints Sensitivity Analysis Table for OF coefficient for bonds 2 David Palma EGR 546 Sensitivity Analysis Graph for OF coefficient for bonds 3 David Palma EGR 546 Sensitivity Analysis Table for RHS of C1 All investments must be a maximum of 500 000 Sensitivity Analysis Graph for RHS of C1 All investments must be a maximum of 500 000 The following is the report to a hypothetical manager 4 David Palma EGR 546 Dear hypothetical Manager Per your request the following is the result of my research on this problem In order to find out the best purchasing plan which minimizes the overall cost of weekly food purchases I used a simple linear programming approach and with help of a user friendly software WinQSB I came up with the followings We should buy 125 000 in bonds to obtain a total contribution of 12 500 125 000 in home loans to obtain a total contribution of 20 000 125 000 in auto loans to obtain a total contribution of 16 250 and 125 000 in personal loans to obtain a total contribution of 25 000 The total revenue will be 73 750 In order to find out the sensitivity of the above mentioned plan on uncertainties over the costs of different investments and also the required minimum and maximum amount of each investment I used the same software and here is the analysis If the annual rate of return in bonds is lower than 9 or higher than 16 we will need to recalculate the purchasing of each investment to find a new optimal revenue If the annual rate of return in home loans is lower than 13 or higher than 17 we will need to recalculate the purchasing of each investment to find a new optimal revenue If the annual rate of return in auto loans is lower than 4 or higher than 14 we will need to recalculate the purchasing of each investment to find a new optimal revenue If the annual rate of return in personal loans is lower than 19 we will need to recalculate the purchasing of each investment to find a new optimal revenue Constrain 1 5 David Palma EGR 546 We will invest the 500 000 available for the project For every dollar we add we will have 1475 increase in or profit 6