DMOP: Sample problems – Deterministic Optimization
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DMOP: Sample problems – Deterministic Optimization
1.
FreeFlight Tours
FreeFlight Tours is planning a charter trip to a major resort in Hawaii. The eight-day-sevennight package will include round-trip air transportation, surface transportation, hotel, meals,
and selected tour options. The charter trip is restricted to 200 persons, and past experience
indicates that there will be no problem finding 200 participants. What the company must
do is determine the number of deluxe, standard, and economy tour packages to offer in this
charter. These three plans each differ according to seating and service for the air flight,
quality of accommodations, meal plans, and tour options. Table 1 summarizes proposed
prices for the three packages and corresponding expenses for FreeFlight Tours. The company
has chartered a jet airliner for a flat fee of $20,000.
Tour
Hotel
Plan
Price Costs
Meals
Deluxe
$1,000
$300
$475
Standard
$700
$220
$250
Economy
$650
$190
$220
Table 1: The three plans for FreeFlight Tours.
In planning the trip, certain considerations must be taken into account:
1. At least 20 packages must be of the deluxe type.
2. At least 70 but no more than 140 of the packages must be of the standard type.
3. At least 60 packages must of the economy type.
4. The airliner allows for no more than 60 deluxe packages.
5. The hotel requests a guarantee that at least 120 of the tourists be on deluxe or standard
packages.
FreeFlight Tours wishes to determine the number of packages of each type to offer so as
to maximize total profit. Formulate an LP model for this problem.
2.
The Nuclear Problem
South Central Utilities has just announced the August 1st opening of its second nuclear
generator at its Baton Rouge, Louisiana, nuclear power plant. Its personnel department has
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DMOP: Sample problems – Deterministic Optimization
been directed to determine how many nuclear technicians need to be hired and trained over
the remainder of the year.
The plant currently employs 350 fully trained technicians and projects the following
manpower needs:
Month
Labor Needed(in hours)
August
40,000
September
45,000
October
35,000
November
50,000
December
45,000
By Louisiana law, a reactor employee can actually work no more than 130 hours per
month. (Slightly over one hour per day is used for check-in and check-out record keeping
and for daily radiation health scans.) Policy at South Central Utilities also dictates that
layoffs are not acceptable in those months when the nuclear plant is overstaffed. So, if more
trained employees are available than are needed in any month, each worker is still fully paid,
even though he or she is not required to work the 130 hours. Training new employees is
an important and costly procedure. It takes one month of one-on-one classroom instruction
before a new technician is permitted to work alone in the reactor facility. Therefore, South
Central must hire trainees one month before they are actually needed. Each trainee teams
up with a skilled nuclear technician and requires 90 hours of that employee’s time, meaning
that 90 hours less of the technician’s time are available that month for actual reactor work.
Personnel department records indicate a turnover rate of trained technicians at 5 percent
per month. In other words, about 5 percent of the skilled employees at the start of any
month resign by the end of that month.
A trained technician earns an average monthly salary of $ 2,000 (regardless of the number
of hours worked, as noted earlier). Trainees are paid $ 900 during their one month of
instruction.
Formulate a linear program to minimize total salaries paid and meet all the constraints.
3.
Easy Ryder
Easy Ryder Transport ships avocados, mangos, and limes once a week from South Florida
to New York. This morning they have 39 tons of avocados, 37 tons of mangos and 35 tons
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DMOP: Sample problems – Deterministic Optimization
of limes on hand. The fruits all go to the Hunts Point Market in New York and therefore
can be mixed on the company’s four trucks.
The capacities of trucks 1 to 4 are 27, 28, 30, and 20 tons respectively. Easy Ryder
paid $160 per ton for avocados, $160 per ton for mangos, and $100 per ton for limes. Fruit
is packed in 20 pound cartons. (There are 2000 pounds in a ton.) Ryder sells unspoiled
avocados in New York for $2 per carton, mangos for $3 per carton and limes for$1.75 per
carton. There is no salvage value for fruits left in Florida. The traveling costs of the trucks
are negligible and independent of what fruits to ship; all four trucks will be sent.
Spoilage occurs during the transportation of fruit from Florida to New York. Because of
differences in their refrigeration systems, the fruit losses differ by truck as shown in Table 2.
The numbers in the table give the percentage losses for each truck-fruit combination.
Truck no.
Avocado
Mango
Lime
1
4.0
10.0
2.9
2
4.4
12.4
2.4
3
3.6
11.3
2.6
4
4.8
11.2
1.2
Table 2: Percentage losses in transit.
Formulate a linear program that will produce an optimal shipping plan for Easy Ryder.
4.
ELP Team Selection
You have been hired to be an ELP-consultant to guide one of the ELP teams. Prior to being
hired, students ranked their desire (Table 3 )to be on your team – 1 being most preferred,
3 being least preferred. Your task is to create a ELP team from the following 10 students.
You wish to select your team based on the rank they have given.
The Administration has given you a number of guidelines to follow in creating the team:
1. The team must consist of at least 5 students; but no more than 6.
2. There must be at least one female in the group.
3. It is not possible to assign all three international students to the team.
4. Students 2 and 4 insist on being on the same learning team (either they both are
assigned to your team, or neither of them is assigned).
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DMOP: Sample problems – Deterministic Optimization
Student Gender International Rank
1
Male
Yes
1
2
Male
No
2
3
Female
No
3
4
Male
No
1
5
Female
Yes
2
6
Male
Yes
1
7
Male
No
1
8
Female
No
3
9
Male
No
1
10
Male
No
2
Table 3: Student desire to be on your team.
Formulate an integer program that will select the team for which the sum of all the ranks
is minimized.
5.
Dorian auto: either-or constraints
Dorian Auto is considering manufacturing three types of autos: compact, midsize, and large.
The resources required for, and the profits yielded by, each type of car are shown in Table
below. Currently, 6000 tons of steel and 60,000 hours of labor are available. For production
of a type of car to be economically feasible, at least 1000 cars of that type must be produced.
Formulate an IP to maximize Dorian’s profit.
Resources and profits for the three types of cars are shown in Table 4.
Car Type
Resource
Compact
Midsize
Large
Steel Required
1.5 tons
3 tons
5 tons
Labor required
30 hours
25 hours
40 hours
Profit yielded ($)
2000
3000
4000
Table 4: Resources and profits for Dorian Auto.