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Unformatted text preview: SELE test SELE test GENERAL LINEAR PROGRAMMING NOTATION :3! Southern Oil Company produces two grades of gasoline: regular and premium. The
profit contributions are $0.30 per litre for regular gasoline and $0.50 per litre for
premium gasoline. Each litre of regular gasoline contains 0.3 litre of grade A crude oil
and each litre of premium gasoline contains 0.6 litre of grade A crude oil. For the next
production period, Southern has 18,000 litres of grade A crude oil available. The refinery used to produce the gasolines has a production capacity of 50,000 litres for
the next production period. Southern Oil’s distributors indicated that demand for the premium gasoline for the next production period will be at most 20,000 litres. Formulate a linear programming model that can be used to determine the number
of litres of regular gasoline and the number of litres of premium gasoline that
should be produced in order to maximize total profit contribution. 3: What is the optimal solution? What are the values and interpretations of the slack variables? d. What are the binding constraints? Does the following linear programme involve infeasibility, unbounded and/or _
alternative optimal solutions? Explain. Max 4x1 +8X2
S.t.
2X1 +2X2 S10
1X1 +1X2 Z 8
x1, x2 2 0 Does the following linear programme involve infeasibility, unbounded and/or
alternative optimal solutions? Explain. .3.ch
Edith'3? Max 1x1+1x2
SI.
8X1 + 6X2 2 24
4x1 + 6x2 :3 12
2x2 2 4 X19X220 34 The manager of a small independent grocery store is trying to determine the best use of
her shelf space for soft drinks. The store carries national and generic brands and
currently has 200 square metres of shelf space available. The manager wants to allocate
at least 60% of the space to the national brands and, regardless of the profitability.
allocate at least 10% of the space to the generic brands. How many square metres of
space should the manager allocate to the national brands and the generic brands if The national brands are more profitable than the generic brands? Both brands are equally profitable? The generic brands are more profitable than the national brands? :35 Discuss what happens to the M&D Chemicals problem (see Section 2.5) if the cost
per litre for product A is increased to €3.00 per litre. What would you recommend? Explain. tire For the M&D Chemicals problem in Section 2.5, discuss the effect of management’s
requiring total production of 500 litres for the two products. List two or three actions Identify all the extreme points of the feasible region. AN INTRODUCTION TO LINEAR PROGRAMMING b. Suppose RMC discovers a way to increase the profit of its solvent base to €60
per kilo. Does this profit increase change the optimal solution? If so, how?
0. Suppose the profit for the solvent base is €50 per kilo. What is the optimal solution now? Comment on any special characteristics that may exist with this profit for the
solvent base. 38 Reconsider the HMO situation in Problem 16. Suppose that management adds the requirements that at least 30 kilos of fuel additive and at least 15 kilos of solvent base
must be produced. a. Graph the constraints for this revised RMC problem. What happens to the feasible
region? Explain. b. If no feasible solutions can be found, explain what is needed to produce 30 kilos
of fuel additive and 15 kilos of solvent base. 39 PharmaPlus operates a chain of 30 pharmacies. The pharmacies are staffed by
licensed pharmacists and pharmacy technicians. The company currently employs 85
fulltime equivalent pharmacists (combination of full time and part time) and 175 full
time equivalent technicians. Each spring management reviews current staffing levels
and makes hiring plans for the year. A recent forecast of the prescription load for the
next year shows that at least 250 fulltime equivalent employees (pharmacists and
technicians) will be required to staff the pharmacies. The personnel department
expects 10 pharmacists and 30 technicians to leave over the next year. To
accommodate the expected attrition and prepare for future growth, management
stated that at least 15 new pharmacists must be hired. In addition, PharmaPlus’s new
service quality guidelines specify no more than two technicians per licensed pharmacist. The average salary for licensed pharmacists is €40 per hour and the
average salary for technicians is €10 per hour. 3. Determine a minimumcost staffing plan for PharmaPlus. How many pharmacists
and technicians are needed?
b. Given current staffing levels and expected attrition, how many new hires (if any) must be made to reach the level recommended in part (a)? What will be the impact
on the payroll? 40 Expedition Outfitters manufactures a variety of speciality clothing for hiking, skiing
and mountain climbing. They decided to begin production on two new parkas
designed for use in extremely cold weather. The names selected for the two models
are the Mount Everest Parka and the Fiocky Mountain Parka. Their manufacturing
plant has 120 hours of cutting time and 120 hours of sewing time available for
producing these two parkas. Each Mount Everest Parka requires 30 minutes of
cutting time and 45 minutes of sewing time, and each Flocky Mountain Parka requires
20 minutes of cutting time and 15 minutes of sewing time. The labour and material
cost is €150 for each Mount Everest Parka and €50 for each Flocky Mountain Parka,
and the retail prices through the firm’s internet catalogue are €250 for the Mount
Everest Parka and €200 for the Flocky Mountain Parka. Because management
believes that the Mount Everest Parka is a unique coat that will enhance the image of
the firm, they have Specified that at least 20% of the total production must consist of
this model. Assuming that Expedition Outfitters can sell as many coats of each type as they can produce, how many units of each model should they manufacture to
maximize the total profit contribution? 41 English Motors, Ltd. (EML), developed a new allwheeldrive sports utility vehicle. As part of the marketing campaign, EML developed a video sales presentation to send to
both owners of current EML fourwheeldrive vehicles as well as to owners of four wheeldrive sports utility vehicles offered by competitors; EML refers to these two GENERAL LINEAR PROGRAMMING NOTATION target markets as the current customer market and the new customer market.
Individuals who receive the new promotion video will also receive a coupon for a test
drive of the new EML model for one weekend. A key factor in the success of the new
promotion is the response rate, the percentage of individuals who receive the new
promotion and test drive the new model. EML estimates that the response rate for the
current customer market is 25% and the response rate for the new customer market is
20%. For the customers who test drive the new model the sales rate is the percentage
of individuals who make a purchase. Marketing research studies indicate that the
sales rate is 12% for the current customer market and 20% for the new customer
market. The cost for each promotion, excluding the test drive costs, are £4 for each
promotion sent to the current customer market and £6 for each promotion sent to the
new customer market. Management also specified that a minimum of 30,000 current
customers should test drive the new model and a minimum of 10,000 new customers
should test drive the new model. In addition, the number of current customers who
test drive the new vehicle must be at least twice the number of new customers who
test drive the new vehicle. If the marketing budget, excluding test drive costs, is
£1,200,000, how many promotions should be sent to each group of customers in
order to maximize total sales? Creative Sports Design (CSD)manufactures a standardsize tennis racket and an
oversize tennis racket. The firm’s rackets are extremely light due to the use of a
magnesiumgraphite alloy that was invented by the firm’s founder. Each standardsize
racket uses 0.125 kilograms of the alloy and each oversize racket uses 0.4 kilograms;
over the next twoweek production period only 80 kilograms of the alloy are available.
Each standardsize racket uses 10 minutes of manufacturing time and each oversize
racket uses 12 minutes. The profit contributions are €1 0 for each standardsize racket
and €15 for each oversize racket, and 40 hours of manufacturing time are available
each week. Management specified that at least 20% of the total production must be
the standardsize racket. How many tennis rackets of each type should CSD
manufacture over the next two weeks to maximize the total profit contribution?
Assume that because of the unique nature of their products, CSD can sell as many rackets as they can produce. riff Management of High Tech Services (HTS)would like to develop a model that will help
allocate their technicians” time between service calls to regular contract customers
and new customers. A maximum of 80 hours of technician time is available over the
twoweek planning period. To satisfy cash flow requirements, at least €800 in revenue
(per technician) must be generated during the twoweek period. Technician time for
regular customers generates €25 per hour. However, technician time for new
customers only generates an average of €8 per hour because in many cases a new
customer contact does not provide billable services. To ensure that new customer
contacts are being maintained, the technician time spent on new customer contacts
must be at least 60% of the time spent on regular customer contacts. Given these
revenue and policy requirements, HTS would like to determine how to allocate
technician time between regular customers and new customers so that the total
number of customers contacted during the twoweek period will be maximized.
Technicians require an average of 50 minutes for each regular customer contact and 1 hour for each new customer contact. a. DeveIOp a linear programming model that will enable HTS to allocate technician
time between regular customers and new customers.
1.1. Find the optimal solution. Jackson Hole Manufacturing is a small manufacturer of plastic products used in the
automotive and computer industries. One of its major contracts is with a large 82 CHAPTER 2 AN INTRODUCTION TO LINEAR PROGRAMMING computer company and involves the production of plastic printer cases for the
computer company’s portable printers. The printer cases are produced on two
injection molding machines. The M100 machine has a production capacity of 25
printer cases per hour, and the M200 machine has a production capacity of 40 cases
per hour. Both machines use the same chemical material to produce the printer
cases; the M100 uses 40 kilos of the raw material per hour and the M200 uses 50
kilos per hour. The computer company asked Jackson Hole to produce as many of
the cases during the upcoming week as possible and has said that it will pay €18 for
each case Jackson Hole can deliver. However, next week is a regularly scheduled
holiday period for most of Jackson Hole's production employees; during this time,
annual maintenance is performed for all equipment in the plant. Because of the
downtime for maintenance. the M100 will be available for no more than 15 hours. and
the M200 will be available for no more than 10 hours. However, because of the high
setup cost involved with both machines, management has a requirement that. if production is scheduled on either machine, the machine must be Operated for at least
5 hours. The supplier of the chemical material used in the production process informed Jackson Hole that a maximum of 1000 kilos of the chemical material will be
available for next week’s production; the cost for this raw material is as per kilo. In addition to the raw material cost, Jackson Hole estimates that the hourly cost of
operating the M100 and the M200 are €50 and €75. respectively. a. Formulate a linear programming model that can be used to maximize the contribution to profit. b. Find the optimal solution. CASE PROBLEM 1 Workload Balancing igital Imaging (DI) produces photo printers for
both the professional and consumer markets. The DI consumer division recently introduced two
photo printers that provide colour prints rivalling
those produced by a professional processing lab.
The Dl—910 model can produce a 10 cm x 15 cm
borderless print in approximately 37 seconds. The
more sephisticated and faster Dl—950 can even pro
duce a 35 cm x 50 cm borderless print. Financial
projections show profit contributions of $42 for each
Dl910 and $87 for each Dl950. The printers are assembled, tested, and pack
aged at Dl’s plant located in Shanghai. 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 Dl950 printer. Line 2
performs both the testing and packaging Operations.
Times are 4 minutes per Dl910 printer and 2
minutes per Dl950 printer. The shorter time for the
Dl—950 printer is a result of its faster print speed.
Both manufacturing lines are in operation one 8—hour
shift per day. Managerial Report Perform an analysis for Digital Imaging in order to
determine how many units of each printer to produce.
Prepare a report to Dl’s president presenting your find
ings and recommendations. Include (but do not limit
your discussion to) a consideration of the following: 1 The recommended number of units of each
printer to produce to maximize the total
contribution to profit for an 8hour shift. What
reasons might management have for not
implementing your recommendation? 2 Suppose that management also states that the
number of Dl91 0 printers produced must be at least as great as the number of Dl950 units produced.
Assuming that the objective is to maximize the total
contribution to profit for an 8hour shift, how many
units of each printer should be produced? 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? 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)? GENERAL LINEAR PROGRAMMING NOTATION 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? For each solution that you develop include a copy of
your linear programming model and graphical solu
tion in the appendix to your report. CASE PROBLEM 2 Production Strategy B etter Fitness, Inc. (BFI), manufactures exercise
equipment at its plant in Munich, Germany. It
recently designed two universal weight machines
for the home exercise market. Both machines use
BFl—patented technology that provides the user with
an extremely wide range of motion capability for
each type of exercise performed. Until now, such
capabilities have been available only on expensive
weight machines used primarily by physical thera—
pists. At a recent trade show, demonstrations of the
machines resulted in significant dealer interest. In
fact, the number of orders that BFI received at the
trade show far exceeded its manufacturing capabil
ities for the current production period. As a result,
management decided to begin production of the two
machines. The two machines, which BFI named the
BodyPlus 100 and the BodyPlus 200, require differ
ent amounts of resources to produce. The BodyPlus 100 consists of a frame unit, a press
station and a pecdec station. Each frame produced
uses 4 hours of machining and welding time and 2
hours of painting and finishing time. Each press sta
tion requires 2 hours of machining and welding time
and 1 hour of painting and finishing time, and each
pec—dec station uses 2 hours of machining and weld
ing time and 2 hours of painting and finishing time. In
addition, 2 hours are spent assembling, testing and
packaging each BodyPlus 100. The raw material costs
are €450 for each frame, €300 for each press station
and €250 for each pecdec station; packaging costs are estimated to be €50 per unit.
The BodyPlus 200 consists of a frame unit, a press station, a pec—dec station and a legpress
station. Each frame produced uses 5 hours of machining and welding time and 4 hours of painting
and finishing time. Each press station requires 3
hours machining and welding time and 2 hours of
painting and finishing time, each pecdec station
uses 2 hours of machining and welding time and 2
hours of painting and finishing time, and each leg
press station requires 2 hours of machining and
welding time and 2 hours of painting and finishing
time. In addition, 2 hours are spent assembling, test
ing and packaging each BodyPlus 200. The raw
material costs are €650 for each frame, €400 for
each press station, €250 for each pecdec station
and €200 for each legpress station; packaging
costs are estimated to be €75 per unit. For the next production period, management
estimates that 600 hours of machining and welding time, 450 hours of painting and finishing time and 140 hours of assembly, testing and packaging time
will be available. Current labour costs are €20 per
hour for machining and welding time, €15 per hour
for painting and finishing time and €12 per hour for
assembly, testing and packaging time. The market
in which the two machines must compete suggests
a retail price of €2400 for the BodyPlus 100 and
€3500 for the BodyPlus 200, although some flexibil
ity may be available to BFl because of the unique
capabilities of the new machines. Authorized BFl
dealers can purchase machines for 70 per cent of the suggested retail price. BFl's president believes that the unique capabil
ities of the BodyPlus 200 can help position BFI as
one of the leaders in highend exercise equipment.
Consequently, he has stated that the number of units
of the BodyPlus 200 produced must be at least 25 per cent of the total production. 83 ...
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