problem1)

Production manager of Koulder Refrigerators must decide how many refrigerators to produce in each of next four months to meet demand at the lowest overall cost.  There is a limited capacity in each month although this will increase in month 3.  Due to new contract, costs are expected to increase.  Relevant information is provided in the table below.

Month    Capacity    Demand    Cost of production
1               140          110         $80 per unit
2               140          150         $85 per unit
3               160          130         $90 per unit
4               160          140         $95 per unit

Each item that is left at the end of the month and carried over to the next month incurs the carrying cost equal to 10% of the unit cost in that month (e.g. anything left in inventory at the end of month one incurs an $8 cost).  Management wants to have at least 30 units left at the end of month four to meet any unexpected demand at that time.  Linear program has been developed to help with this.  However, this may or may not be totally correct.  You must verify that it is the correct formulation before solving the problem.  If it is not correct, make any necessary changes to the linear program before solving it on the computer.

X1 = number of units produced in month 1;  X2 = number of units produced in month 2; 
X3 = number of units produced in month 3; X4 = number of units produced in month 4;
N1 = number of units left at end of month 1; N2 = number of units left at end of month 2;
N3 = number of units left at end of month 3; N4 = number of units left at end of month 4

Minimize cost = 80X1 + 85X2 + 90X3 + 95X4+ 8N1 + 8.5N2 + 9N3+ 9.5N34
X1< 140
X2< 140
X3< 160
X4< 160
X1 = 110 + N1
X2 + N1 = 150 + N2
X3 + N2 = 130 + N3
X4 + N3 = 140 + N4
N4>30
All variables > 0

problem2)

An investment advisory firm manages funds for its numerous clients. The company uses an asset allocation model that recommends the portion of each client’s portfolio to be invested in a growth stock fund, and income fund, and a money market fund. To maintain diversity in each client’s portfolio, the firm places limits on the percentage of each portfolio that may be invested in each of the three funds. General guidelines indicate that the amount invested in the growth fund must be between 20% and 40% of the total portfolio value. Similar percentages for the other two funds stipulate that between 20% and 50% of the total portfolio value must be in the income fund and at least 30% of the total portfolio value must be in the money market fund.
    In addition, the company attempts to assess the risk tolerance of each client and adjust the portfolio to meet the needs of the individual investor. For ex, Williams just contracted with a new client who has $300,000 to invest, and all of it must be invested. Based on an evaluation of the client’s risk tolerance, Williams assigned a maximum risk index of 0.06 for the client. The firm’s risk indicators show the risk of the growth fund at 0.10, the income fund at 0.07, and the money market fund at 0.01. An overall portfolio risk index is computed as a weighted average of the risk rating for the three funds.  The average risk of the portfolio would be the total risk divided by the total investment as shown here:

    (average risk) = (total risk)/(total investment).

For ex, to find out risk of a $100,000 portfolio with 50,000 in the growth fund, 30,000 in the income fund, and 20,000 in the money market fund, total risk would be 0.10(50,000) + 0.07(30,000) + 0.01(20,000) = 7,300; the average risk would be  7,300/100,000 = 0.073.  (NOTE: When putting the risk measure into the linear program, it is better to work with the total risk rather than the average risk to avoid round-off errors.)
    Additionally, Williams is currently forecasting annual yields of 8% for the growth fund, 6% for the income fund, and 2% for the money market fund. Based on the information provided, how must the new client be advised to allocate the $300,000 among the growth, income, and money market funds?  Linear program has been developed to help with this.  However, this may or may not be totally correct.  You must verify that it is the correct formulation before solving the problem.  If it is not correct, make any necessary changes to the linear program before solving it on the computer.

G = dollars invested in the growth fund
I = dollars invested in the income fund
M= dollars invested in the money market fund

Maximizeyield (return) = 0.08G + 0.06I + 0.02M

Subject to:

G + I + M = 300000                         Total investment
0.10G + 0.07I + 0.01M <18000        Total risk (based on 6% average risk)
G >60000                                       Minimum in growth fund
G < 120000                                    Maximum in growth fund
I >60000                                        Minimum in income fund
I < 150000                                     Maximum in income fund
M >90000                                      Minimum in money market fund
G, I, M > 0

The equality for the total investment constraint simplifies the other constraints.  While the total amount invested is G + I + M, this can be replaced by 300000 due to the equality condition.  For ex, 20% of (G + I + M) becomes 20% of $300,000 or simply $60,000. If this were a less-than-or-equal-to constraint, this could not be done and that constraint would be G > 0.20(G + I + M).