31. Optimal Production. Video-Scapes, Inc. (VSI) is a rapidly growing landscaping service in Oakland,
California that specializes in merging videos of one’s home and property with changeable computer-generated
graphics of various landscape details. VSI offers budget landscaping designs for $1,500 and deluxe designs for
$3,000. Both services use scarce computing, creative and consulting resources. Each budget design requires 4
hours of computer time, 2 hours of creative time, and 4 hours of consulting time with the client. Each deluxe
design requires 6 hours of computer time, 6 hours of creative time, and 4 hours of consulting time. VSI
currently has 60 hours of computer time, 42 hours of creative time, and 36 hours of consulting time available on
a weekly basis. What output mix would be optimal if VSI wishes to maximize total sales revenue?
A.
Using the equality form of the constraint conditions, set up and interpret the linear program that VSI might use to determine an optimal
weekly service mix.
B.
Solve for and interpret all solution values.
32. Optimal Lending. Penny Lane is a senior loan officer with Citrus National Bank in Tampa, Florida. Lane
has both corporate and personal lending customers. On average, the profit contribution margin or interest rate
spread is 1.75% on corporate loans and 2.25% on personal loans. This return difference reflects the fact that
personal loans tend to be riskier than corporate loans. Lane seeks to maximize the total dollar profit contribution
earned, subject to a variety of restrictions on her lending practices. In order to limit default risk, Lane must
restrict personal loans to no more than 40% of total loans outstanding. Similarly, to ensure adequate
diversification against business cycle risk, corporate lending cannot exceed 80% of loaned funds. To maintain
good customer relations by serving the basic needs of the local business community, Lane has decided to extend
at least 40% of her total credit authorization to corporate customers on an ongoing basis. Finally, Lane cannot
exceed her current total credit authorization of $50 million.
A.
Using the inequality form of the constraint conditions, set up and interpret the linear programming problem Lane would use to
determine the optimal dollar amount of credit to extend to corporate (C) and personal (P) lending customers. Also formulate the LP
problem using the equality form of the constraint conditions.
B.
Use a graph to determine the optimal solution, and check your solution algebraically. Fully interpret solution values.
Lane’s goal is to maximize the profit contribution earned on loans to corporate (C) plus personal (P) lending customers, subject to a
33. Optimal Production. Ozark Telephone, Inc. (OTI) is a small telephone company offering local dial-tone
service to its franchised areas in rural southeastern Missouri. A new office park development site is being
planned within OTI’s territory and John Sample, a network engineer, has to maximize the conversation capacity
per line under cost and technology constraints using both traditional copper-wire lines and new fiber-optic lines.
OTI wants to gradually move into the all-digital communication environment possible with fiber-optics, so a
company policy has been adopted specifying that at least 3 fiber-optic lines be employed for every 2 copper
lines on new installations. To minimize the need to quickly retrain its linemen, OTI wants at least 30% of new
telephone lines installed to be copper. No existing telephone facilities run to the development site, and OTI
must use its own facilities to carry the traffic (it cannot lease capacity from any other local telephone company).
Finally, current costs and technologies dictate that 1 fiber line can carry the equivalent of 5 copper lines at the
same cost to OTI. That is, if one copper line can carry one telephone conversation, fiber optic lines can carry
five conversations at no cost penalty. Sample’s objective is to maximize the capacity per line of the
transmissions facilities being built to carry traffic to/from the office park.
A.
Using the inequality form of the constraint conditions, set up and interpret the linear programming problem Sample would use to
determine the optimal percentage of copper and fiber-optic lines. Also formulate the problem using the equality form of the constraint
conditions.
B.
With a graph, determine the optimal solution; check your solution algebraically. Fully interpret solution values.
C.
Holding all else equal, how much would the capacity of fiber optic lines have to fall to alter the optimal construction mix determined in
part B?
D.
Calculate the opportunity cost, measured in terms of conversation capacity per line, of OTI’s 30% copper line constraint.
34. Cost Minimization. Environmental Products, Inc. manufactures surfactants used in fertilizers and
herbicides (surfactants minimize runoff). Environmental Products uses two manufacturing processes, each
having advantages. The Hidex process is more expensive ($0.50 per gallon of Hidex) but produces more
surfactant and fewer emissions of the pollutant toxide per gallon used. The Lodex process is less efficient in
producing surficants but, is less expensive ($0.25 per gallon of Lodex). The Lodex also has the advantage of
producing fewer pollutants per gallon used in the production process.
To satisfy contracts with its customers, Environmental Products must produce at least 240 gallons of surfactants
per hour. Each gallon of Hidex used in the production process generates 4 gallons of surfactant and 10 parts per
million (ppm) of toxide. Each gallon of Lodex produces 3 gallons of surfactant and 6 ppm of toxide. The EPA
limits toxide emissions to 600 ppm per hour. In addition, municipal regulations limit emissions of solids to 180
ppm per hour, and the Hidex process produces solids at a rate of 2 ppm per gallon used and the Lodex process
produces 3 ppm per gallon used.
A.
Set up and interpret the linear program Environmental Products would use to minimize hourly costs.
B.
Calculate, graph, and interpret all relevant solution values.
C.
Holding all else equal, how much would the price of Lodex have to rise before Hidex would be used exclusively to produce surfactants?
Explain.
35. Profit Maximization. Samantha Spade & Associates, Ltd. is a small architectural firm located in Portland,
Oregon, specializing in the preparation of multi-family residential housing complex, R, and commercial retails,
C, architectural designs. Prevailing prices in the market are $10,000 for residential housing designs and $25,000
for commercial retail designs.
Six architects run the firm, and work a 50-hour workweek, 50 weeks per year. They are assisted by six drafting
personnel and two secretaries, all of whom work a typical 40-hour workweek, 50 weeks per year. The firm must
decide how to target its promotional efforts so as to best use its resources during the coming year. Based on
previous experience, the firm expects that an average of 150 hours of architect and 100 hours of drafting time
will be required for each residential housing complex design, whereas commercial retail design will require an
average of 250 architect hours and 200 drafting hours. Fifty hours of secretarial time will also be required for
each architectural design. In addition, variable computer and other processing costs are expected to average
$1,000 per residential design and $1,500 per commercial retail design.
A.
Set up the linear programming problem the firm would use to determine the profit-maximizing output levels for residential and
commercial designs. Show both the inequality and equality forms of the constraint conditions.
B.
Completely solve and interpret the solution values for the linear programming problem.
C.
Calculate maximum possible net profits per year for the firm assuming that architects draw a salary of $100,000 per year, drafting
personnel earn $65,000 per year, secretaries are paid $20 per hour, and fixed overhead (including promotion and other expenses)
averages $50,000.
D.
After considering the above data, one senior architect recommended reducing one drafting personnel to part-time status (adjusting
salary accordingly) while retaining the rest of the current staff full-time. What are net profits per year under this suggestion?
