Linear Programming, Sensitivity Analysis
Case Problem..1 Product Mix
Par, Inc., is a small manufacturer of golf equipment and supplies whose management has decided
to move into the market for medium and high priced golf bags. Par’s distributor is enthusiastic
about the new product line and has agreed to buy all the golf bags Par produces over the next three
months.
After thorough investigation of the steps involved in manufacturing a golf bag, management
determined that each golf bag produced will require the following operations:
1. Cutting and dyeing the material
2. Sewing
3. Finishing (inserting umbrella holder, club separators, etc.)
4. Inspection and packaging
The director of manufacturing analyzed each of the operations and concluded that if the company
produces a medium priced standard model, each bag will require 7/10 hour in cutting and dyeing
department, ½ hour in the sewing department, 1 hour in finishing department and 1/10 hour in the
inspection and packaging department. The more expensive deluxe model will require 1 hour for
cutting and dyeing, 5/6 hour for sewing, 2/3 hour for finishing and ¼ hour for inspection and
packaging.
Par’s production is constrained by a limited number of hours available in each department. After
studying departmental workload projections, the director of manufacturing estimates that 630 hours
for cutting and dyeing, 600 hours for sewing, 708 hours for finishing and 135 hours for inspection
and packaging will be available for the production of golf bags during the next three months.
The accounting department analyzed the production data, assigned all relevant variable costs and
arrived at prices for both bags that will result in a profit contribution of $10 for every standard bag
and $9 for every deluxe bag produced. Determine the number of bags of each type should be
produced in order to maximize total profit contribution.
Suppose the management is also considering producing a light weight model designed specifically
for golfers who prefer to carry their bags. The design department estimates that each new
lightweight model will require 0.8 hours for cutting and dyeing, 1 hour for sewing, 1 hour for
finishing and 0.25 hours for inspection and packaging. Because of the unique capabilities designed
into the new model, Par’s management feels they will realize a profit contribution of $12.85 for
each lightweight model produced during the current production period. What is the effect of the
new product?
Linear Programming
Minimization Problems
1. Ashok Chemicals Company manufactures two chemicals A and B which are sold to the
manufacturers of soaps and detergents. On the basis of the next month’s demand, the
management has decided that the total production for chemicals A and B should be at least 350
kilograms. Moreover, a major customer’s order for 125 kilogram of product A must also be
supplied. Product A requires 2 hours of processing time per kilogram and product B requires
one hour of processing time per kilogram. For the coming month, 600 hours of processing time
are available.
The company wants to meet the above requirements at minimum total production cost. The
production costs are Rs.2 per kilogram for product A and Rs.3 per kilogram for product B.
Ashok Chemicals Company wants to determine its optimal product mix and the total minimum
cost relevant thereto.
2. Bluegrass Farms, located in Lexington, Kentucky has been experimenting with a special diet for
its race horses. The feed components available for the diet are a standard horse feed product, a
vitamin enriched oat product and a new vitamin and mineral feed additive. The nutritional
values in units per pound of the standard feed components are summarized below. For example,
each pound of the standard feed component contains 0.8 unit of ingredient A, 1 unit of
ingredient B and 0.1 unit of ingredient C. The minimum daily diet requirements for each horse
are three units of ingredient A, six units of ingredient B and four units of ingredient C. In
addition, to control the weight of the horses, the total daily feed for a horse should not exceed 6
pounds. Bluegrass Farms would like to determine the minimum-cost mix that will satisfy the
daily diet requirements.
Feed Component
Ingredient A
Ingredient B
Ingredient C
Cost per pound ($)
Standard
0.8
1.0
0.1
0.25
Enriched Oat
0.2
1.5
0.6
0.50
Additive
0.0
3.0
2.0
3.00
3. National Insurance Associates carries an investment portfolio of stocks, bonds and other
investment alternatives. Currently $200,000 of funds are available and must be considered for
new investment opportunities. The four stock options National is considering and the relevant
financial data are as follows:
Stock
Price per share ($)
Annual rate of return
Risk measure per dollar invested
A
100
0.12
0.10
B
50
0.08
0.07
C
80
0.06
0.05
D
40
0.10
0.08
The risk measure indicates the relative uncertainty associated with the stock in terms of its
realizing the projected annual return; higher values indicate greater risk. The risk measures are
provided by the firm’s top financial advisor.
National’s top management has stipulated the following investment guidelines: the annual rate
of return for the portfolio must be at least 9% and no one stock can account for more than 50%
of the total dollar investment.
a) Use linear programming to develop an investment portfolio that minimizes risk.
b) If the firm ignores risk and uses a maximum return-on-investment strategy, what is the
investment portfolio?
c) What is the dollar difference between the portfolio in parts (a) and (b)? Why might the
company prefer the solution in part (a)?
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% to 40% of the total portfolio value.
Similar percentages for the other two funds stipulate that between 20% to 50% of the total portfolio
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, William’s is currently forecasting annual yields of 18% for the growth fund, 12.5%
for the income fund and 7.5% fir the money market fund. Based on the information provided, how
should the new client be advised to allocate $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.
Managerial Report:
a. 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 change?
b. Assume that the client’s risk index could be increased to 0.55. How much would the yield
increase and how would the investment recommendation change?
c. 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%?
d. 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?
e. The asset allocation model you developed may be useful in modifying the portfolios for all 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?
Case Workforce Assignment:
McCormick Manufacturing Company produces two products with contributions to profit per
unit of $10 and $9, respectively. The labour requirements per unit produced and the total hours
of labour available from personnel assigned to each of four departments are shown in the table.
Assuming that the number of hours available in each department is fixed, find the optimal
solution.
Labour-Hours per Unit
Department
1
2
3
4
Product 1
0.65
0.45
1.00
0.15
Product 2
0.95
0.85
0.70
0.30
Total Hours Available
6500
6000
7000
1400
Suppose that McCormick has a cross-training program that enables some employees to be
transferred between departments. By taking advantage of the cross-training skills, a limited
number of employees and labour hours may be transferred as shown in the table below.
Cross-Training Transfers
Permitted to Department
From
Department
1
2
3
4
1
….
….
….
yes
2
yes
….
….
yes
3
yes
yes
….
….
4
….
yes
yes
….
Maximum Hours
Transferable
400
800
100
200
Row 1 of this table shows that some employees assigned to department 1 have cross-training
skills that permit them to be transferred to department 2 or 3. The right hand column shows that,
for the current production planning period, a maximum of 400 hours can be transferred from
department 1. Similar cross-training transfer capabilities and capacities are shown for
departments 2, 3, 4.
If the production manager has the flexibility to assign personnel to different departments,
reduced workforce idle time, improved workforce utilization, how will the solution improve?