Sample Problems

DMOP: Sample problems – Deterministic Optimization

June 24, 2008

1

DMOP: Sample problems – Deterministic Optimization 1

1. Linear Programming

Here are some sample problems on Linear Programming formulations, sensitivity analysis,

and duality.

1.1 FreeFlight Tours

FreeFlight Tours is planning a charter trip to a major resort in Hawaii. The eight-day-seven-

night 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 10% of the packages must be of the deluxe type.

2. At least 35% but no more than 70% of the packages must be of the standard type.

3. At least 30% 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.


1.2 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

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 employees time, meaning

that 90 hours less of the technicians 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.


1.3 The Astros and Cosmos Mix

ASCO manufactures two television models, Astros and Cosmos. Each Astro set sells for

$300 and each Cosmo sells for $250 a set. ASCO purchases components for and Astro set

for $260 and components for a Cosmo set cost $190.

Production of each model involve circuit board fabrication, picture tube construction

and chassis assembly. There are two separate and completely automated lines for circuit

board fabrication, one for Astro and one for Cosmo. However, the picture tube and chassis

assembly departments are shared in the production of both sets.

The capacity of the Astro circuit board fabrication line is 70 sets per day, while that of

the Cosmo line is 50 sets per day. The picture tube department has 20 workstations, while

the chassis department has 16 workstations. Each workstation can process one TV set at a

time and is operated 6 hours a day. Each Astro set takes 1 hour for chassis assembly and 1

hour for tube production. Each Cosmo set requires 2 hours for picture tube production and

1 hour for chassis assembly.

Workers in the picture tube and chassis assembly departments are paid $10 an hour.

Heating, Lighting, and other overhead charges amount to $1000 per day.

1. How many Astros and Cosmos should ASCO produce each day? what will be the

maximum profit?

2. How should they allocate the available resources among the two models? Where are

the bottlenecks?

3. Suppose that due to raw material shortage ASCO could make only 30 circuit boards

for Cosmos each day. What will be the effect on their operation?

4. Suppose workers in the picture tube department are willing to work overtime for a

premium of $ 21 an hour. How many hours of overtime, if any, should they employ?

How will they use it ?

5. If a workstation in the picture tube department breaks down, how will it affect the

ASCO’s production and profit ?

6. If a chassis assembly workstation breaks down, how will it affect ASCO’s production

plan and profit?

7. How much would you be willing to pay to increase Astro’s circuit board capacity?


8. Suppose ASCO has developed a new model that uses same circuit boards as a Cosmo

and requires 3 hours of the picture tube time. if its profit margin is expected to be

high $42, should they produce it?

9. If the profit margin on Astro goes up to $ 40 a set, how would it affect the firm’s

production plan and the daily profit? What if it goes down by $10 a set?

10. How much must Cosmo’s price increase before you will consider producing more Cos-

mos?

2. Networks

This section contains problems on minimum-cost network flow problems (shortest paths,

assignment, and transportation problems.)

2.1 Producing umbrellas

An umbrella company has estimated its monthly demand for the next year, with di being

the estimated demand for each month i. Each month of the year, the company can produce

umbrellas. The cost per umbrella produced in month i is pi. (Because of the varying seasonal

costs of raw materials, pi might not be the same from month to month.)

The company must always meet its monthly demand. However, it can also overproduce;

that is, it can produce more than is necessary to meet demand, and then hold the extra

umbrellas in inventory. Holding inventory costs hi dollars per umbrella in month i.

At the beginning of the year, heading into the first month, the company has 10 umbrellas

already in inventory. They would like to end the year with the same number of umbrellas in

inventory.

Suppose there was a limit of ui on the amount that could be produced by regular-time

workers in month i, but that they could exceed that limit by giving workers overtime pay.

The first ui umbrellas could be produced at the standard cost pi each, and an additional vi

(or fewer) umbrellas could be produced at the overtime rate of oi per umbrella, with oi > pi.

1. Draw this scenario as a network problem. If it is a special type of network problem,

specify what type it is.

2. Specify the corresponding linear program.


2.2 The knapsack problem

An buyer must decide which items to pick from p items so as to maximize the total utility of

all the picked items. An item i is priced at wi (always integral) and has ui (integral) utility

for the buyer. The buyer’s budget is W . This is called as the knapsack problem (see Happy

Hiker’s dilemma.) The buyer either purchases an item or does not. It is possible to represent

this problem on a network. Try to represent the following instance of the knapsack problem,

where W = 6, as a network problem. Show all the nodes, arcs, and costs.

Item (i) 1 2 3 4

ui 40 15 20 10

wi 4 2 3 1

In this particular instance what kind of a network problem must be solved to obtain a

solution to the buyer’s maximization problem?

2.3 The hopping plane problem

1 2 3 n-1 n. . .

Figure 1: The hopping plane problem.

A small commuter airline uses a plane, with a capacity to carry at most p passengers,

on a “hopping flight,” as shown in Figure 1. The hopping plane visits the cities 1, 2, 3, . . . ,

n, in a fixed sequence. The plane can pick up passengers at any node and drop them off at

any other node. Let bij denote the number of passengers available at any station (node) i

wanting to travel to station (node) j and let fij denote the fare per passenger from station

i to station j.

The airline would like to determine the number of passengers that the plane should

carry between the various stations in order to maximize the total fare per trip, while never

exceeding the capacity of the plane.

1. How do we represent this as a network optimization problem? Draw the network.

2. What kind of a network problem is it?


2.4 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

of limes on hand. The fruits all go to the Hunts Point Market in New York and therefore

can be mixed on the companys 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.

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: Losses in transit.

Formulate a linear program that will produce an optimal shipping plan for Easy Ryder.

3. Integer Programming

Here are some sample problems illustrating Integer Programming formulations and logical

constraints.

3.1 Happy Hiker’s Dilemma

Happy is going on an overnight hike. There are five major items she is considering taking

along. The weight of each item (in Pounds) and its utility (a measure of its usefulness on

the trip) are given below.


Item Weight Utility

Blanket 5 16

Tent 7 22

Food Can 4 12

Stove 3 8

Kitchen sink 100 5

If happy would like to carry only up to 14 pounds of stuff in the knapsack, which item

should she take along?

3.2 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.

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.

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).

5. A request has been made by student 7 that if student 8 is assigned to the team, then

he cannot (i.e. does not want to) be assigned to the team.

Formulate an integer program that will select the team for which the sum of all the ranks

is minimized.

3.3 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.

3.4 Euling gas: piece-wise linear functions

Euling Gas produces two types of gasoline (gas 1 and gas 2) from two types of oil (oil 1 and

oil 2). Each gallon of gas 1 must contain at least 50 percent oil 1, and each gallon of gas 2

must contain at least 60 percent oil 1. Each gallon of gas 1 can be sold for $12, and each

gallon of gas 2 can be sold for $14. Currently, 500 gallons of oil 1 and 1000 gallons of oil

2 are available. As many as 1500 more gallons of oil 1 can be purchased at the following

prices: first 500 gallons, $25 per gallon; next 500 gallons, $20 per gallon; next 500 gallons, $

per gallon. Formulate an IP that will maximize Euling’s profit(revenues - purchasing costs.)

Sample Problems

Documents

Transcript of Sample Problems