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IE 3340 – Operations Research Lecture 3: Formulation of Linear Programming Problems

Eduardo Perez, Ph.D.

Ingram School of Engineering

Texas State University

601 University Dr., San Marcos, TX 78666

[email protected]

http://uweb.txstate.edu/~e_p86

E. Perez IE TXSTATE 1

Ch.2 Modeling with Linear Programming

mailto:[email protected]

http://uweb.txstate.edu/~e_p86

Learning Objectives


Learn how to formulate linear programming models of real-life

situations

Today’s Agenda

Guidelines for Model Formulation

Diet Problem

Production Planning and Inventory Control

Multi-period Production Planning and Inventory Control

Manpower Planning

Bus Scheduling Model

Post Office

Urban Development Planning

Project Selection

Blending and Refining


Formulation is like a Chess Game


I can teach you the rules, review “standard moves”, and discuss

examples.

But every game (problem) is different and requires thought.

Apply what you know: does this new problem fit a “standard

structure”?

Build a library of experience with a variety of problem types.

The Formulation Process


Identify ___________________: what must be decided?

Identify & formulate the __________ of the problem (e.g., max profit or

min cost)

Identify __________: what resources are in limited supply?

Identify how decision variables must be ________ in constraints &

formulate constraints

Constraints must define the feasible values of all decision variables

Make dimensional analyses to make sure that the _________ and

each ___________ are consistent

Formulation is an iterative process

Propose a reasonable definition of variables and begin model formulation

If you cannot incorporate all issues, you will have learned how to redefine

variables to do so

decision variables

objective

constraints

related

objective

constraint

Diet Problem


A Diet Problem

Many LP models arise from situations in which a decision maker

wants to minimize the cost of meeting certain requirements

Example: Bob wants to plan a nutritious diet, but he is on a limited

budget, so he wants to spend as little money as possible. His

nutritional requirements are as follows:


1. 2000 kcal

2. 55 g protein

3. 800 mg calcium

A Diet Problem

Nutritional values

Bob is considering the following foods:


Food Serving Size Energy (kcal) Protein (g) Calcium (mg) Price per serving

Oatmeal 28 g 110 4 2 $0.30

Chicken 100 g 205 32 12 $2.40

Eggs 2 large 160 13 54 $1.30

Whole milk 237 cc 160 8 285 $0.90

Cherry pie 170 g 420 4 22 $0.20

Pork and beans 260 g 260 14 80 $1.90

A Diet Problem

Question to be answered

Indices

Decision Variables

We can represent the number of servings of each type

of food in the diet by the variables:


How many servings of each type of food?

i = type of food

xi = number of servings food type i

x1 servings of oatmeal

x2 servings of chicken

x3 servings of eggs

x4 servings of milk

x5 servings of cherry pie

x6 servings of pork and beans

A Diet Problem

Objective Function


Minimize Cost


Oatmeal 28 g 110 4 2 $0.30

Chicken 100 g 205 32 12 $2.40

Eggs 2 large 160 13 54 $1.30

Whole milk 237 cc 160 8 285 $0.90

Cherry pie 170 g 420 4 22 $2.00

Pork and beans 260 g 260 14 80 $1.90

0.3x1 + 2.40x2 + 1.30x3 + 0.90x4 + 2.0x5 + 1.9x6

A Diet Problem

Constraints



Oatmeal 28 g 110 4 2 $0.30

Chicken 100 g 205 32 12 $2.40

Eggs 2 large 160 13 54 $1.30

Whole milk 237 cc 160 8 285 $0.90

Cherry pie 170 g 420 4 22 $2.00

Pork and beans 260 g 260 14 80 $1.90

x1

x2

x3

x4

x5

x6

Nutritional requirements 1. 2000 kcal

2. 55 g protein

3. 800 mg calcium

KCAL constraint:

110x1 + 205x2 + 160x3 + 160x4 + 420x5 + 260x6 2000

A Diet Problem

LP Formulation


110x1 + 205x2 + 160x3 + 160x4 + 420x5 + 260x6 2000

4x1 + 32x2 + 13x3 + 8x4 + 4x5 + 14x6 55

2x1 + 12x2 + 54x3 + 285x4 + 22x5 + 80x6 800

Minimize

subject to:

0,,,,, 654321 xxxxxx

Cost

Nutritional requirements

Bounds

0.3x1 + 2.40x2 + 1.30x3 + 0.90x4 + 2.0x5 + 1.9x6

Single Period Production ModelExample:

In preparation for the winter season, a clothing company is manufacturing

parka and goose overcoats, insulated pants, and gloves.

All products are manufactured in four different departments: cutting, insulating,

sewing, and packaging.

The company has received firm orders for its products.

The contract stipulates a penalty for undelivered items.

Devise an optimal production plan for the company based on the following data:



Indices

Decision Variables


Single Period Production Model

How many units of each type of product to produce?

i = type of product

xi = number of units of product type i

Objective

Function


Maximize the net profit, defined as: Total profit - Total penalty

The total penalty requires the use of a new variables s representing the shortage in

demand for product j


The total profit: 30x1 + 40x2 + 20x3 + 10x4

The total penalty: 15s1 + 20s2 + 10s3 + 8s4

The objective function:

Max z = 30x1 + 40x2 + 20x3 + 10x4 – (15s1 + 20s2 + 10s3 + 8s4 )

Constraints



LP Formulation


Maximize

subject to:

Profit

Bounds


Multiperiod Production Planning and

Inventory Control


Multiperiod Production Planning and Inventory Control

Example:

Acme Manufacturing Company has a contract to deliver 100, 250, 190, 140,

220, and 110 home windows over the next 6 months.

Production cost (labor, material, and utilities) per window varies by period and

is estimated to be $50, $45, $55, $48, $52, and $50 over the next 6 months.

To take advantage of the fluctuations in manufacturing cost, Acme can

produce more windows than needed in a given month and hold the extra units

for delivery in later months.

This will incur a storage cost at the rate of $8 per window per month, assessed

on end-of-month inventory.

Develop a linear program to determine the optimum production schedule.



Indices

Decision Variables



How many units to produce per month?

How many units left in inventory per month?

i = month of the year

Variables

The variables of the problem include the monthly production

amount and the end-of-month inventory. For i = 1, 2, …, 6

The relationship between these variables and the monthly

demand over the 6-month horizon is represented schematically

in the Figure. The system starts empty



Objective Function


Minimize the total cost of production and end-of-month inventory:


Total production cost: 50x1 + 45x2 + 55x3 + 48x4 + 52x5 + 50x6

Total inventory (storage) cost: 8 (I1 + I2 + I3 + I4 + I5 + I6)


Min z = 50x1 + 45x2 + 55x3 + 48x4 + 52x5 + 50x6 + 8 ( I1 + I2 + I3 + I4 + I5 + I6)

Constraints



The constraints of the problem can be determined directly from the figure

using the following balance equation:

Beginning inventory + Production amount - Ending inventory = Demand

LP Formulation


Minimize

subject to:

Cost

Bounds


Manpower Planning


Bus Scheduling ProblemExample:

Progress City is studying the feasibility of introducing a mass-transit bus

system to reduce in-city driving. The study seeks the minimum number of

buses that can handle the transportation needs.

The minimum number of buses needed fluctuated with time of the day

The required number of buses could be approximated by constant values over

successive 4-hour intervals.

To carry out the required daily maintenance, each bus can operate 8

successive hours a day only

Develop a linear program to determine the optimum number of buses in operation.



Indices

Decision Variables


Bus Scheduling Problem

How many buses are needed per shift?

i = shift number

xi = number of buses needed in shift i

Variables

The variables of the model are the number of buses needed in

each shift



Objective Function


Minimize the total number of buses in operation:


The objective function: Min z = x1 + x2 + x3 + x4 + x5 + x6

Constraints


Because of the overlapping of the shifts, the number of buses for the

successive 4-hour periods can be computed as


LP Formulation


Minimize

subject to:

Number of buses

Bounds


Post Office ProblemProblem:

A post office requires different numbers of full-time employees on different

days of the week.

Union rules state that each fulltime employee must five consecutive days and

then two days off.

The post office wants to meet its daily requirements using only full-time

employees.

Formulate an LP that the post office can use to minimize the number of full time

employees who must be hired.



Indices

Decision Variables


Post Office Problem

How many employees begin to work on each day ?

i = day of the week

xi = number of employees beginning work on day i

Objective Function


Minimize the total number of employees:

Post Office Problem

Number of full time employees = # of employees who start to work on Monday + #

of employees who start to work on Tuesday + … + # of employees who start to

work on Sunday

The objective function: Min z = x1 + x2 + x3 + x4 + x5 + x6 + x7

Constraints


The post office must ensure that enough employees are working every day

Who is working on Monday?

Everybody except the employees who begin to work on Tuesday or on

Wednesday

They get, respectively, Sunday and Monday, and Monday and Tuesday off

To ensure that at least 17 employees are working on Monday we require

that the constraint

Post Office Problem

x1 + x4 + x5 + x6 + x7 ≥ 17

Constraints


Adding similar constraints for the other six days of the week:

Post Office Problem

x1 + x4 + x5 + x6 + x7 ≥ 17 (Monday constraint)

x1 + x2 + x5 + x6 + x7 ≥ 13 (Tuesday constraint)

x1 + x2 + x3 + x6 + x7 ≥ 15 (Wednesday constraint)

x1 + x2 + x3 + x4 + x7 ≥ 19 (Thursday constraint)

x1 + x2 + x3 + x4 + x5 ≥ 14 (Friday constraint)

x2 + x3 + x4 + x5 + x6 ≥ 16 (Saturday constraint)

x3 + x4 + x5 + x6 + x7 ≥ 11 (Sunday constraint)

x1, x2, x3, x4, x5, x6, x7, ≥ 0

LP Formulation


Minimize

subject to:

Total number of employees

Bounds

Post Office Problem

Min z = x1 + x2 + x3 + x4 + x5 + x6 + x7

x1 + x4 + x5 + x6 + x7 ≥ 17 (Monday constraint)

x1 + x2 + x5 + x6 + x7 ≥ 13 (Tuesday constraint)

x1 + x2 + x3 + x6 + x7 ≥ 15 (Wednesday constraint)

x1 + x2 + x3 + x4 + x7 ≥ 19 (Thursday constraint)

x1 + x2 + x3 + x4 + x5 ≥ 14 (Friday constraint)

x2 + x3 + x4 + x5 + x6 ≥ 16 (Saturday constraint)

x3 + x4 + x5 + x6 + x7 ≥ 11 (Sunday constraint)

x1, x2, x3, x4, x5, x6, x7, ≥ 0

Urban Renewal ModelProblem:

The city of Erstville is faced with a severe budget shortage.

The city council votes to improve the tax base by condemning an

inner-city housing area and replacing it with a modern development.

The project involves two phases:

1. Demolishing substandard houses to provide land for the new

development and

2. Building the new development.


Urban Renewal ModelThe following is a summary of the situation:

1. As many as 300 substandard houses can be demolished. Each house

occupies a .25-acre lot. The cost of demolishing a condemned house is $2000.

2. Lot sizes for new single-, double-, triple-, and quadruple-family homes (units)

are .18, .28, .4, and .5 acre, respectively. Streets, open space, and utility

easements account for 15% of available acreage.

3. In the new development, the triple and quadruple units account for at least

25% of the total. Single units must be at least 20% of all units and double units

at least 10%.

4. The tax levied per unit for single, double, triple, and quadruple units is $1000,

$1900, $2700, and $3400, respectively.

5. The construction cost per unit for single-, double-, triple-, and quadruple-family

homes is $50,000, $70,000, $130,000, and $160,000, respectively.

6. Financing through a local bank is limited to $15 million.

How many units of each type should be constructed to maximize tax collection?



Indices

Decision Variables


Urban Renewal Model

How many units of each type should be constructed/demolish to

maximize tax collection?

i = type of home

xi = number of units of type of home i

Variables

The variables of the model are:

the number of units to be constructed of each type of housing,

the number houses must be demolished to make room for the new

development.


Urban Renewal Model

Objective Function


Maximize total tax collection from all four types of homes

Urban Renewal Model

The objective function: Max z = 1000 x1 + 1900 x2 + 2700 x3 + 3400 x4

Constraints


The first constraint of the problem deals with land availability

The acreage needed for new homes =

To determine the available acreage, each demolished home occupies a .25-

acre lot, thus netting .25x5 acres.

Allowing for 15% open space, streets, and easements, the net acreage

available is .85 (.25x5 ) = .2125 x5 The resulting constraint is :

Urban Renewal Model

Constraints


The number of demolished homes cannot exceed 300, which translates to

The constraints limiting the number of units of each home type

Urban Renewal Model

Constraints


Keep the demolition/construction cost within the allowable budget

Expressing all the costs in thousands of dollars, we get

Urban Renewal Model

LP Formulation


Maximize

subject to:

Total tax collection

Bounds

Urban Renewal Model

Project Selection


Project Selection ProblemProblem: Star Oil Company is considering five different investment opportunities.

The cash out-flows and net present values (in millions) are given in the Table.

Star Oil has $40million available for investment now (time 0); it estimates that

one year from now (time1) $20 million will be available for investment.

Star Oil may purchase any fraction of each investment. In this case, the cash

outflows and NPV are adjusted accordingly. For example, if Star Oil purchases one-fifth of investment 3, then a cash outflow of (1/5)(5) =$1

million would be required at time 1. The one-fifth share of investment 3 would yield an NPV

of $3.2million.

Star Oil wants to maximize the NPV that can be obtained by investing

Formulate an LP that will help achieve this goal. Assume that any funds leftover at

time 0 cannot be used at time 1.



Indices

Decision Variables


Project Selection Problem

What fraction of each investment to purchase?

i = type of investment

xi = fraction of investment i to purchase by Star Oil

Objective Function


Maximize the NPV earned from investments:


The objective function: Max z = 13x1 + 16x2 + 16x3 + 14x4 + 39x5

Constraints


Star Oil’s constraints may be expressed as follows:

Constraint 1 Star cannot invest more than $40 million at time 0.


Constraint 3 Star cannot purchase more than 100% of investment


Constraints




Constraint 3 Star cannot purchase more than 100% of investment


LP Formulation


Maximize

subject to:

the NPV earned from investments

Bounds


Blending and Refining Problem:

Shale Oil has a capacity of 1,500,000 bbl of crude oil per day.

The final products from the refinery include three types of unleaded

gasoline with different octane numbers (ON):

1. regular with ON = 87,

2. premium with ON = 89,and

3. super with ON = 92

The refining process encompasses three stages:

1. A distillation tower that produces feedstock (ON=82) at the rate of .2 bbl

per bbl of crude oil,

2. A cracker unit that produces gasoline stock (ON=98) by using a portion of

the feedstock produced from the distillation tower at the rate of .5 bbl per

bbl of feedstock, and

3. A blender unit that blends the gasoline stock from the cracker unit and the

feedstock from the distillation tower.


Blending and Refining Problem:

The company estimates the net profit per barrel of the three types of

gasoline to be $6.70, $7.20, and $8.10, respectively.

The input capacity of the cracker unit is 200,000 barrels of feedstock a

day.

The demand limits for regular, premium, and super gasoline are

50,000, 30,000, and 40,000 barrels, respectively, per day.

Develop a model for determining the optimum production schedule for the

refinery.



Indices

Decision Variables



How many barrels/day to produce certain type of final product?

i = bbl/day of input stream i

j = blend of final product j

Variables

The variables can be defined in terms of two input streams to

the blender (feedstock and cracker gasoline) and the three final

products.



Objective Function


Maximize total profit



Max z = 6.70 (x11 + x21) + 7.20 (x12 + x22) + 8.10 (x13 + x23)

Constraints



Constraints



Daily crude oil supply does not exceed 1,500,000 bbl/day

Cracker unit input capacity does not exceed 200,000 bbl/day

Daily demand for regular does not exceed 50,000 bbl

Constraints



Daily demand for premium does not exceed 30,000:

Daily demand for super does not exceed 40,000 bbl::

Constraints



The octane number of a gasoline product is the weighted average of the

octane numbers of the input streams used in the blending process and can

be computed as:

Octane number (ON) for premium is at least 87:

Constraints





LP Formulation


Maximize

subject to:

Revenue

Bounds


Questions?



Diet Problem


Multi-period Production Planning and Inventory Control

Manpower Planning

Bus Scheduling Model

Post Office


Project Selection


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