Quantitative Models For Decision Makers
Group Case Study
Exclusive Summary
● Case Problem 1: working Balancing
● Case Problem 2: Investment Strategy
● Reference: Excel File
Chapter 2 -- Case Problem 1 Working Balancing
Digital Imaging (DI) produces color printers for both the professional and
consumer markets. The DI consumer division recently introduced two new
color printers. The DI-910 model can produce a 4" 3 6" borderless color print in
approximately 37 seconds. The more sophisticated and faster DI-950 can even
produce a 13" 3 19" borderless color print. Financial projections show profit
contributions of $42 for each DI-910 and $87 for each DI-950. The printers are
assembled, tested, and packaged at DI’s plant located in New Bern, North
Carolina. This plant is highly automated and uses two manufacturing lines to
produce the printers. Line 1 performs the assembly operation with times of 3
minutes per DI-910 printer and 6 minutes per DI-950 printer. Line 2 performs
both the testing and packaging operations with times of 4 minutes per DI-910
printer and 2 minutes per DI-950 printer. The shorter time for the DI-950 printer
is a result of its faster print speed. Both manufacturing lines are in operation
one 8-hour shift per day.
Q1
1.
The recommended number of units of each printer to
produce to maximize the total contribution to profit for an
8-hour shift. What reasons might management have for
not implementing your recommendation?
Based on the optimal solution we got, which
is D1=0, D2=80, the value of optimal
solution is $6960.
Management would not implement this
solution because there is no DI-910 model
would be produced.
Q2
2.Suppose that management also states that the number of DI-910
printers produced must be at least as great as the number of DI950 units produced. Assuming that the objective is to maximize the
total contribution to profit for an 8-hour shift, how many units of
each printer should be produced?
The optimal solution is D1=D2=53,
and the value of the optimal
solution is $6880.
The company will produce 53 units
of DI-910 and 53 units DI-950.
Q3
3. Does the solution you developed in part (2) balance the total time spent on line 1 and the total time spent on line 2? Why
might this balance or lack of it be a concern to management?
Time spent on line 1 = 3D1+6D2= 3(53)+6(53)=480’
Time spent on line 2 = 4D1+2D2= 4(53)+2(53)=320’
So our solution doesn’t balance the total time spent on Line 1 and time spent on Line 2.
This situation may caused by inappropriate workload that assigned to two lines. The
management need to fix this problem to improve the performance of production.
Q4
4. Management requested an expansion of the model in part (2) that would provide a better balance between the total time
on line 1 and the total time on line 2. Management wants to limit the difference between the total time on line 1 and the
total time on line 2 to 30 minutes or less. If the objective is still to maximize the total contribution to profit, how many units
of each printer should be produced? What effect does this workload balancing have on total profit in part (2)?
We assume, T1=total time spent on line 1, and T2=total time spent on line 2
So, T2-30=<T1<=T2+30
Also we know, T1=3D1+6D2, and T2=4D1+2D2
3D1+6D2>=4D1+2D2-30
3D1+6D2<=4D1+2D2+30
Thus, -D1+4D2>=-30, -D1+4D2<=30
Based on excel, we obtain the optimal solution, D1=97, D2=32, and the value of the
optimal solution is $6815.
So this workload balancing made the total profit decreased by $6880-$6815=$65.
Q5
5.Suppose that in part (1) management specified the objective
of maximizing the total number of printers produced each shift
rather than total profit contribution. With this objective, how
many units of each printer should be produced per shift? What
effect does this objective have on total profit and workload
balancing?
The optimal solution is D1=107,D2=27.
The total profit is 42*107+87*27=$6800
Total production=107+27=134
Compared to part 4, 134-97-32=5 printers
But, it has reduced profit by 6815-6800=$15.
Also, this solution results in perfect workload balancing
because the total time spent on each line is 480 mins.
Chapter 3—Case Problem 2 Investment Strategy
J. D. Williams, Inc., is an investment advisory firm that manages more than $120 million in 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,
an 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 example, Williams just contracted with a new client who has $800,000 to invest. Based on an
evaluation of the client’s risk tolerance, Williams assigned a maximum risk index of 0.05 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 where the weights are the fraction of the client’s portfolio
invested in each of the funds. Additionally, Williams is currently forecasting annual yields of 18% for the growth fund, 12.5% for the
income fund, and 7.5% for the money market fund. Based on the information provided, how should the new client be advised to
allocate the $800,000 among the growth, income, and money market funds? Develop a linear programming model that will provide the
maximum yield for the portfolio. Use your model to develop a managerial report.
Q1
1.Recommend how much of the $800,000 should be invested in each of the three funds. What is
the annual yield you anticipate for the investment recommendation?
Q2
2.Assume that the client’s risk index could be increased to 0.055. How much would the yield
increase and how would the investment recommendation change?
Increasing 0.05 to 0.55 the annual yield on investment would increase by 4,667, so income would
have been 93946+4667=98614, annual yield would be 98614/800000=12.3%
Q3
3.Refer again to the original situation where the client’s risk index was assessed to be 0.05. How
would your investment recommendation change if the annual yield for the growth fund were
revised downward to 16% or even to 14%?
The allowable decrease in the sensitivity is 0.03, up to 15% there will be no change in the optimal
solution. So, at 16% there will be no change.
But at 14% it will change and the optimal solution will be
Q4
4.Assume that the client expressed some concern about having too much money in the growth
fund. How would the original recommendation change if the amount invested in the growth fund is
not allowed to exceed the amount invested in the income fund?
In this case, an extra constraint will add up into the system as G<=I or , G-I<=0
The result is as following
Q5
5.The asset allocation model you developed may be useful in modifying the portfolios for all of the
firm’s clients whenever the anticipated yields for the three funds are periodically revised. What is
your recommendation as to whether use of this model is possible?
The company can use the above asset allocation model for modifying the portfolios of its different
clients. however , this model would be only applicable if all the clients meet the same criteria
depicted for the current client in terms of objective and constraints. This model cannot be applied
for rest of the situations.