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Chapter 2

Deterministic Optimization Models

in Operations Research

Introduction

• Deterministic Models in OR: models where it is

reasonable to assume all problem data to be known

with certainty.

• Advantages of Deterministic Models:

– Often produce valid enough results to be useful;

– They are always easier to analyze than stochastic

models.

• Deterministic Models are also called mathematical

programs.

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EXAMPLE 2.1:

Two Crude Petroleum

Two Crude Petroleum runs a small refinery on the

Texas coast. The refinery distills crude petroleum from

two sources, Saudi Arabia and Venezuela, into the three

main products: gasoline, jet fuel and lubricants.

The two crudes differ in chemical composition and

yield different product mixes. Each barrel of Saudi crude

yields 0.3 barrel of gasoline, 0.4 barrel of jet fuel, and 0.2

barrel of lubricants. Each barrel of Venezuelan crude

yields 0.4 barrel of gasoline, 0.2 barrel of jet fuel and 0.3

barrel of lubricants. The remaining 10% is lost to refining.

EXAMPLE 2.1:

Two Crude Petroleum

The crudes differ in cost and availability. Two Crude

can purchase up to 9000 barrels per day from Saudi

Arabia at $20 per barrel. Up to 6000 barrels per day of

Venezuelan petroleum are available at the lower cost of

$15 per barrel.

Two contracts require it to produce 2000 barrels per

day of gasoline,1500 barrels per day of jet fuel and 500

barrels per day of lubricants. How can these requirements

be fulfilled most efficiently?

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EXAMPLE 2.1:

Two Crude Petroleum

Saudi Arabia Venezuela Requirements

(barrels / day)

Yields /barrel gasoline 0.3 barrel 0.4 barrel 2000

jet fuel 0.4 barrel 0.2 barrel 1500

lubricant 0.2 barrel 0.3 barrel 500

lost to refining 0.1 barrel 0.1 barrel

Availability barrels / day 9000 6000

Purchase cost per barrel $20 $15

2.1 Decision Variables, Constraints,

and Objective Functions

• Decision Variables: Variables in optimization models

represent the decisions to be taken. [2.1]

• Input parameters: fixed information

– Yields, Cost, Availability, Requirements

• Decision Variables:

𝑥1 ≜ barrels of Saudi crude refined /day (in 1000s)

𝑥2 ≜ barrels of Venezuelan crude refined /day (in

1000s) (2.1)

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Constraints

• Variable-type Constraints specify the domain of

definition for decision variables: the set of values for

which the variables have meaning. [2.2]

Nonnegativity: 𝑥1, 𝑥2 0 (2.2)

Constraints

• Main Constraints of optimization models specify the

restrictions and interactions, other than variable-type,

that limit decision variable values. [2.3]

0.3 𝑥1 + 0.4 𝑥2 2.0 (gasoline)

0.4 𝑥1 + 0.2 𝑥2 1.5 (jet fuel)

0.2 𝑥1 + 0.3 𝑥2 0.5 (lubricants)

𝑥1 9 (Saudi)

𝑥2 6 (Venezuelan)

(2.3)

(2.4)

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Objective Functions

• Objective Functions in optimization models quantity the

decision consequences to be maximized or minimized.

[2.4]

min20 𝑥1 + 15 𝑥2 (2.5)

Standard Model

The standard statement of an optimization model has the

form

max or min (objective function(s))s.t. (main constraints)

(variable-type constraints)

• min 20 𝑥1 + 15 𝑥2 (total cost)s.t.

[2.5]

0.3 𝑥1 + 0.4 𝑥2 2.0 (gasoline)

0.4 𝑥1 + 0.2 𝑥2 1.5 (jet fuel)

0.2 𝑥1 + 0.3 𝑥2 0.5 (lubricants)

𝑥1 9 (Saudi)

𝑥2 6 (Venezuelan)

𝑥1 , 𝑥2 0 (nonnegativity)

(2.6)

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Exercise 2.1: Formulating Formal

Optimization Models

Suppose that we wish to enclose a rectangular equipment yard

by at most 80 meters of fencing. Formulate an optimization

model to find the design of maximum area.

2.2 Graphic Solution and

Optimization Outcomes

• Graphic solution solves 2 and 3-variable optimization

models by plotting elements of the model in a

coordinate system corresponding to the decision

variables.

• Feasible set (or region) of an optimization model is the

collection of choices for decision variables satisfying all

model constraints. [2.6]

• Graphic solution begins with a plot of the choices for

the decision variables that satisfy variable-type

constraints. [2.7]

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Graphing Feasible Set (Region)

Variable-type Constraints

1 2 3 4 5 6 7 8 9 10

x2

x1

1

2

3

4

5

6

7

8

Feasible Set (Region)

Main Constraints

• The set of points satisfying an equality constraint plots

as a line or curve. [2.8]

• The set of points satisfying an inequality constraint plots

as a boundary line or curve, where the constraint holds

with equality, together with all points on whichever side

of the boundary satisfy the constraint as an inequality.

[2.9]

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Graphing Feasible Set (Region)

Main Constraints

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10

x2

x1

7

8

Feasible Set (Region)

Main Constraints

• The feasible set (or region) for an optimization model is

plotted by introducing constraints one by one, keeping

track of the region satisfying all at the same time. [2.10]

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Graphing Feasible Set (Region)

Main Constraints

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10

x2

x1

7

8

𝑥1 9

𝑥2 6

Exercise 2.2: Graphing Constraints

and Feasible Sets

Graph the feasible sets corresponding to each of the following

systems of constraints.

a) 𝑥1 + 𝑥2 ≤ 23 𝑥1 + 𝑥2 3𝑥1 , 𝑥2 0

b) 𝑥1 + 𝑥2 ≤ 23 𝑥1 + 𝑥2 = 3𝑥1 , 𝑥2 0

c) (𝑥1)2 +(𝑥2)

2 ≤ 4| 𝑥1 | − 𝑥2 ≤ 0

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Objective Functions

𝑐 𝑥1, 𝑥2 ≜ 20𝑥1 + 15𝑥2• Objective functions are normally plotted in the same

coordinate system as the feasible set of an optimization

model by introducing contours – lines or curves through

points having equal objective function value. [2.11]

(2.8)

Graphing Objective Functions

(3D View)

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Graphing Objective Functions

(2D with Contours)

1 2 3 4 5 6 7 8 9 10

𝑥2

𝑥1

1

2

3

4

5

6

7

8

60

90

120

20𝑥1 + 15𝑥2

Exercise 2.3: Plotting Objective

Function Contours

Show contours of each of the following objective functions over

the feasible region defined by 𝑦1 + 𝑦2 ≤ 2, 𝑦1 , 𝑦2 0a) Min 3𝑦1 + 𝑦2

b) Max 3𝑦1 + 𝑦2

c) Max 2(𝑦1)2 + 2(𝑦2)

2

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Optimal Solutions

• An optimal solution is a feasible choice for decision

variables with objective function value at least equal to

that of any other solution satisfying all constraints.

[2.12]

• Optimal solutions show graphically as points lying on

the best objective function contour that intersects the

feasible region. [2.13]

Optimal Solutions

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10

x2

x1

7

8

𝑥1 9

𝑥2 6

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Exercise 2.4: Formulating Formal

Optimization Models

Suppose that we wish to

enclose a rectangular

equipment yard by at most

80 meters of fencing. Solve

the optimization model to

find the design of maximum

area.

Optimal Values

• An optimal value in an optimization model is the

objective function value of any optimal solutions. [2.14]

• An optimization model can have only one optimal value.

[2.15]

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Unique versus Alternative

Optimal Solutions

• An optimization model may have a unique optimal

solution or several alternative optimal solutions. [2.16]

• Unique optimal solutions show graphically by the optimal-

value contour intersecting the feasible set at exactly one

point. If the optimal-value contour intersects at more than

one point, the model has alternative optimal solutions. [2.17]

Alternative Optimal Solutions

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10

𝑥2

𝑥1

7

8

𝑥1 9

𝑥2 6

20𝑥1 + 10𝑥2

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Exercise 2.5: Identifying Unique

and Alternative Optimal Solutions

Determine graphically which of the following optimization

models has a unique optimal solution and which has

alternative optima.

a) max 3𝑤1 + 3𝑤2

s.t. 𝑤1 +𝑤2 ≤ 2𝑤1 , 𝑤2 0

b) max 3𝑤1 + 3𝑤2

s.t. 𝑤1 +𝑤2 ≤ 𝟐𝑤1 , 𝑤2 0

Infeasible Models

• An optimization model is infeasible if no choice of

decision variables satisfies all constraints. [2.18]

• An infeasible model shows graphically by no point falling

within the feasible region for all constraints. [2.19]

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Infeasible Models

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10

x2

x1

7

8

x1 2

x2 2

Exercise 2.6: Identifying Infeasible

Models Graphically

Determine graphically which of the following model is feasible

and which is infeasible. a) max 3𝑤1 + 𝑤2

s.t. 𝑤1 +𝑤2 ≤ 2𝑤1 +𝑤2 ≥ 1𝑤1 , 𝑤2 0

b) max 3𝑤1 + 𝑤2

s.t. 𝑤1 +𝑤2 ≤ 2𝑤1 +𝑤2 ≥ 3𝑤1 , 𝑤2 0

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Unbounded Models

• An optimization model is unbounded when feasible

choices of the decision variables can produce arbitrarily

good objective function values. [2.20]

• Unbounded models show graphically by there being points

in the feasible set lying on ever-better objective function

contours. [2.21]

Unbounded Models

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10

x2

x1

7

8

x2 6

-2x1+15x2

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Exercise 2.7: Identifying

Unbounded Models Graphically

Determine graphically which of the following optimization

model has an optimal solution and which is unbounded. a) max −3𝑤1 + 𝑤2

s.t. −𝑤1 + 𝑤2 ≤ 1𝑤1 , 𝑤2 0

b) max 3𝑤1 + 𝑤2

s.t. −𝑤1 + 𝑤2 ≤ 1𝑤1 , 𝑤2 0

2.3 Large-scale Optimization

Models and Indexing

• Indexing or subscripts permit representing collections of

similar quantities with a single symbol.

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EXAMPLE 2.2:

Pi Hybrids

Pi Hybrid, a large manufacturer of corn seed, operates

l=20 facilities producing seeds of m=25 hybrid corn

varieties and distributes them to customers in n=30 sales

regions. They want to know how to carry out these

production and distribution operations at minimum cost.

Parameters:

• Cost per bag of producing each hybrid at each facility

• Corn processing capacity of each facility in bushels

• Number of bushels of corn must be processed

• Demand (bags) of each hybrid in each region

• Cost of shipping (per bag) from facility to region

Indexing

• The first step in formulating a large optimization model

is to choose appropriate indexes for the different

dimensions of the problem. [2.22]

f ≜ production facility number (f = 1, …, l)

h ≜ hybrid variety number (h = 1,…, m)

r ≜ sales region number (r = 1, …, n)

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Indexing Decision Variables

• It is usually appropriate to use separate indexes for

each problem dimension over which a decision variable

or input parameter is defined. [2.23]

xf,h ≜ number of bags of hybrid h produced at facility f (f

= 1, …, l; h = 1,…, m)

yf,h,r ≜ number of bags of hybrid h shipped from facility f

to sales region r (f=1, …, l; h=1,…, m; r=1, …, n)

Exercise 2.8: Identifying

Unbounded Models Graphically

Suppose that an optimization model employs decision

variables 𝑤𝑖,𝑗,𝑘,𝑙, where 𝑖 and 𝑘 range over 1,…,100, while 𝑗

and 𝑙 index through 1,…,50. Compute the total number of

decision variables.

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Indexing Input Parameters

• To describe large-scale optimization models, it is

usually necessary to assign indexed symbolic names to

most input parameters, even though they are being

treated as constant. [2.24]

pf,h ≜ cost per bag of producing hybrid h at facility f

uf ≜ corn processing capacity (in bushels) of facility f

ah ≜ number of bushels of corn must ne processed for

a bag of hybrid h

dh,r ≜ demand of hybrid h in sales region r

sf,h,r ≜ cost per bag of of shipping hybrid h from facility f

to sales region r

Exercise 2.9:

Using Summation Notation

a) Write the following sum more compactly with summation

notation:

2𝑤1,5 + 2𝑤2,5 + 2𝑤3,5 + 2𝑤4,5 + 2𝑤5,5

b) Write out terms separately of the sum

𝑖=1

4

𝑖𝑤𝑖

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Objective Function

• Total cost = total production cost + total shipping cost

𝑚𝑖𝑛

𝑓=1

𝑙

ℎ=1

𝑚

𝑝𝑓,ℎ 𝑥𝑓,ℎ +

𝑓=1

𝑙

ℎ=1

𝑚

𝑟=1

𝑛

𝑠𝑓,ℎ,𝑟 𝑦𝑓,ℎ,𝑟

Indexing Families of Constraints

• Families of similar constraints distinguished by indexes

may be expressed in a single-line format

(constraint for fixed indexes) (ranges of indexes)

which implies one constraint for each combination of

indexes in the ranges specified. [2.25]

ℎ=1

𝑚

𝑎ℎ 𝑥𝑓,ℎ ≤ 𝑢𝑓 𝑓 = 1,… , 𝑙 ∀𝑓 (𝑓𝑜𝑟 𝑎𝑙𝑙 𝑓)

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Exercise 2.10:

Using Indexed Families of Constraints

An optimization model must decide how to allocate available

supplies 𝑠𝑖 at sources 𝑖 = 1,… , 𝑝 to meet requirements 𝑟𝑗 at customer

𝑗 = 1,… , 𝑞. Using decision variables

𝑤𝑖,𝑗 ≜amount allocated from source 𝑖 to customer 𝑗

Formulate each of the following requirements in a single line.

a) The amount allocated from source 32 cannot exceed the supply

available at 32.

b) The amount allocated from each source 𝑖 cannot exceed the

supply available at 𝑖.c) The amount allocated to customer 𝑛 should equal the

requirement at 𝑛.

d) The amount allocated to each customer 𝑗 should equal the

requirement at 𝑗.

Exercise 2.11:

Counting Indexed Constraints

Determine the number of constraints in the following systems.

a) 𝑖=122 𝑧𝑖,3 ≥ 𝑏3

b) 𝑖=122 𝑧𝑖,𝑝 ≥ 𝑏𝑝 , 𝑝 = 1,… , 45

c) 𝑘=110 𝑧𝑖,𝑗,𝑘 ≤ 𝑔𝑗, 𝑖 = 1,… , 14; 𝑗 = 1,… , 30

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Pi Hybrids Example Model

𝑚𝑖𝑛

𝑓=1

𝑙

ℎ=1

𝑚

𝑝𝑓,ℎ 𝑥𝑓,ℎ +

𝑓=1

𝑙

ℎ=1

𝑚

𝑟=1

𝑛

𝑠𝑓,ℎ,𝑟 𝑦𝑓,ℎ,𝑟

s.t. ℎ=1𝑚 𝑎ℎ 𝑥𝑓,ℎ ≤ 𝑢𝑓 𝑓 = 1, … , 𝑙

𝑓=1𝑙 𝑦𝑓,ℎ,𝑟 = 𝑑ℎ,𝑟 ℎ = 1, … ,𝑚; 𝑟 = 1,… , 𝑛

𝑟=1𝑛 𝑦𝑓,ℎ,𝑟 = 𝑥𝑓,ℎ 𝑓 = 1,… , 𝑙; ℎ = 1,… ,𝑚

𝑥𝑓,ℎ ≥ 0 𝑓 = 1,… , 𝑙; ℎ = 1,… ,𝑚

𝑦𝑓,ℎ,𝑟 ≥ 0 𝑓 = 1,… , 𝑙; ℎ = 1,… ,𝑚; 𝑟 = 1, … , 𝑛

(2.10)

How Models Become Large

• Optimization models become large mainly by relatively

small number of objective function and constraint

elements being repeated many times for different

periods, locations, products, and so on. [2.26]

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2.4 Linear and Nonlinear

Programs

The general form of a mathematical program or (single

objective) optimization model is

min or max f(x1, …, xn)

subject to:

𝑔𝑖(𝑥1,…, 𝑥𝑛)≤=≥

𝑏𝑖 𝑖 = 1,… ,𝑚

Where f, g1,…,gm are given functions of decision variables

x1,…,xn, and b1, …, bm are specified constant parameters.

[2.27]

Two Crude Petroleum

min 20 x1 + 15 x2

s.t.

0.3 x1 + 0.4 x2 2.0

0.4 x1 + 0.2 x2 1.5

0.2 x1 + 0.3 x2 0.5

x1 9

x2 6

x1 , x2 0

f(x1, x2) 20 x1 + 15 x2

g1(x1, x2) 0.3 x1 + 0.4 x2

g2(x1, x2) 0.4 x1 + 0.2 x2

g3(x1, x2) 0.2 x1 + 0.3 x2

g4(x1, x2) x1

g5(x1, x2) x2

g6(x1, x2) x1

g7(x1, x2) x2

RHSs:

b1 = 2.0, b2 = 1.5, b3 = 0.5,

b4 = 9, b5 = 6, b6 = 0,

b7 = 0

(2.11)

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Exercise 2.12: Expressing Models

in Functional Form

Assuming that the decision variables are 𝑤1, 𝑤1, 𝑤1, express

the following optimization model in general functional format

[2.27] and identify all required functions and right-hand sides:

Max (𝑤1)2+8𝑤2 + (𝑤3)

2

s.t. 𝑤1 + 6𝑤2 ≤ 10 + 𝑤2

(𝑤3)2= 7

𝑤1 ≥ 𝑤3

𝑤1, 𝑤2 ≥ 0

Linear Functions

A function is linear if it is a constant-weighted sum of

decision variables. Otherwise, it is nonlinear. [2.28]

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Exercise 2.13:

Recognizing Linear Functions

Assuming that 𝑥’s are decision variables and all other symbols are

constant, determine whether each of the following functions is linear

or nonlinear.

a) 𝑓 𝑥1, 𝑥2, 𝑥3 ≜ 9𝑥1 − 17𝑥3

b) 𝑓 𝑥1, 𝑥2, 𝑥3 ≜ 𝑗=13 𝑐𝑗 𝑥𝑗

c) 𝑓 𝑥1, 𝑥2, 𝑥3 ≜5

𝑥1+ 3𝑥2 − 6𝑥3

d) 𝑓 𝑥1, 𝑥2, 𝑥3 ≜ 𝑥1𝑥2 + 𝑥23 − ln(𝑥3)

e) 𝑓 𝑥1, 𝑥2, 𝑥3 ≜ 𝑒𝛼𝑥1 + ln(β) 𝑥3

f) 𝑓 𝑥1, 𝑥2, 𝑥3 ≜𝑥1+𝑥2

𝑥2−𝑥3

Linear and Nonlinear Programs

Defined

• An optimization model in functional form [2.27] is a linear

program (LP) if the (single) objective function f and all

constraint functions g1, …, gm are linear in the decision

variables. Also, decision variables should be able to

take on whole-number or fractional values. [2.29]

• An optimization model in functional form [2.27] is a

nonlinear program (NLP) if the (single) objective function

f or any of the constraint functions g1, …, gm is nonlinear

in the decision variables. Also, decision variables should

be able to take on whole-number or fractional values.

[2.30]

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Exercise 2.14: Recognizing Linear

and Nonlinear Programs

Assuming that 𝑦’s are decision variables and all other symbols

are constant, determine whether each of the following

mathematical programs is linear program or a nonlinear

program.

a) min 𝛼(3𝑦1 + 11𝑦4 )

s.t. 𝑗=15 𝑑𝑗𝑦𝑗 ≤ 𝛽

𝑦𝑗 1 𝑗 = 1,… , 9

b) min 𝛼(3𝑦1 + 11𝑦4 )2

s.t. 𝑗=15 𝑑𝑗𝑦𝑗 ≤ 𝛽

𝑦𝑗 1 𝑗 = 1,… , 9

c) max 𝑗=19 𝑦𝑗

s.t. 𝑦1𝑦2 ≤ 100𝑦𝑗 1 𝑗 = 1,… , 9

Example 2.3: E-mart

E-mart, a large European variety store, sells products in m=12

major merchandise groups, such as children’s wear, candy,

music, toys, and electric. Advertising is organized into n=15

campaign formats promoting specific merchandise groups

through a particular medium (catalog, press, or television). For

example, one variety of campaign advertises children’s wear in

catalogs, another promotes the same product line in

newspapers and magazines, while a third sells toys with

television. The profit margin (fraction) for each merchandise

group is known, and E-mart wishes to maximize the profit

gained from allocating its limited advertising budget across the

campaign alternatives.

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Indexing, Parameters, and

Decision Variables for E-mart

• Indexing

g ≜ merchandise group number (g = 1, …, m)

c ≜ campaign type number (c = 1, …, n)

• Input parameters

pg ≜ profit, as a fraction of sales, realized from

merchandise group g

b ≜ available advertising budget

• Decision variables

xc ≜ amount spent on campaign type c

Nonlinear Response

• When there is an option, linear constraint and objective

functions are preferred to nonlinear ones in optimization

models because each nonlinearity of an optimization model

usually reduces its tractability as compared to linear forms.

[2.31]

• Linear functions implicitly assume that each unit increase in

a decision variable has the same effect as the preceding

increase: equal returns to scale. [2.32]

(sales increase in group g due to campaign c) =

sg,clog (xc +1)

where sg,c≜ parameter relating advertising expenditure in

campaign c to sales growth in merchandise group g

(2.12)

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E-mart Model

𝑚𝑎𝑥

𝑔=1

𝑚

𝑝𝑔

𝑐=1

𝑛

𝑠𝑔,𝑐log(𝑥𝑐 + 1)

s.t. 𝑐=1𝑛 𝑥𝑐 ≤ 𝑏

𝑥𝑐 ≥ 0 𝑐 = 1, … , 𝑛

(2.13)

2.5 Discrete or Integer Programs

• Discrete optimization models include decisions of a

logical character qualitatively different from those of

linear or nonlinear programs.

• Discrete optimization models are also called integer

programs, mixed-integer programs, and combinatorial

optimization problems.

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Example 2.4:

Bethlehem Ingot MoldBethlehem Steel Corporation needs to choose ingot sizes and molds. In

their process for making steel products, molten output from main furnaces is

poured into large molds to produce rectangular blocks called ingots. After

the molds have been removed, the ingots are reheated and rolled into

product shapes such as l-beams and flat sheets.

Bethlehem’s mills using this process make approximately n = 130

different products. The dimensions of ingots directly affect efficiency. For

example. ingots of one dimension may be easiest to roll into l-beams, but

another produces sheet steel with less waste. Some ingot sizes cannot be

used at all in making certain products.

A careful examination of the best mold dimensions for different products

yielded m = 600 candidate designs. However, it is impractical to use more

than a few because of the cost of handling and storage. We wish to select at

most p = 6 and to minimize the waste associated with using them to produce

all n products.

Indexing and Parameters of the

Bethlehem Example

• Indexing

i ≜ mold design number (i = 1, …, m)

j ≜ product number (j = 1, …, n)

• Input parameters

ci,j ≜ amount of waste caused by using mold i on

product j

Ij ≜ collection of indexes i corresponding to molds

that could be used for product j . If i ∈ Ij , mold

i is feasible for product j

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Discrete versus Continuous

Decision Variables

• A variable is discrete if it is limited to a fixed or

countable set of values. Often, the choices are only 0

and 1. [2.33]

• Decision variables

yi ≜ 10

if ingot mold i is selectedif not

xi,j ≜ 10

if ingot mold i is used for product jif not

Discrete versus Continuous

Decision Variables

• A variable is continuous if it can take any value in a

specified interval. [2.34]

• When there is an option, such as when optimal variable

magnitudes are likely to be large enough that fractions

have no practical importance, modeling with continuous

variables is preferred to discrete because optimizations

over continuous variables are generally more tractable

than are ones over discrete variables. [2.35]

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Exercise 2.15: Choosing Discrete

versus Continuous Variables

Decide whether a discrete or a continuous variable would be

best employed to model each of the following quantities.

a) The operating temperature of a chemical process

b) The warehouse slot assigned a particular product

c) Whether a capital project is selected for investment

d) The amount of money converted from yen to dollars

e) The number of aircraft produced on a defense contract

Constraints with

Discrete Variables

𝑖=1

𝑚

𝑦𝑖 ≤ 𝑝

𝑖∈𝐼𝑗

𝑥𝑖,𝑗 = 1 𝑗 = 1, … , 𝑛

𝑥𝑖,𝑗 ≤ 𝑦𝑖 𝑖 ∈ 𝐼𝑗; 𝑗 = 1,… , 𝑛

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Exercise 2.16: Expressing

Constraints in 0-1 Variables

In choosing among a collection of 16 investment projects,

variables

𝑤𝑗 ≜ 1 𝑖𝑓 𝑝𝑟𝑜𝑗𝑒𝑐𝑡 𝑗 𝑖𝑠 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Express each of the following constraints in terms of these

variables.

a) At least one of the first eight projects must be selected.

b) At most three of the last eight projects can be selected

c) Either project 4 or 9 must be selected, but not both.

d) Project 11 can be selected only if project 2 is also.

Bethlehem Ingot Mold Example

Model

𝑚𝑖𝑛

𝑗=1

𝑛

𝑖∈𝐼

𝑐𝑖,𝑗𝑥𝑖,𝑗

s.t. 𝑖=1𝑚 𝑦𝑖 ≤ 𝑝

𝑖∈𝐼 𝑥𝑖,𝑗 = 1 j = 1,… , 𝑛

𝑥𝑖,𝑗 ≤ 𝑦𝑖 𝑖 ∈ 𝐼𝑗; 𝑗 = 1, … , 𝑛

𝑦𝑖 = 0 𝑜𝑟 1 𝑖 = 1, … ,𝑚𝑥𝑖,𝑗 = 0 𝑜𝑟 1 𝑖 ∈ 𝐼𝑗; 𝑗 = 1,… , 𝑛

(2.14)

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Integer and Mixed Integer

Programs

• A mathematical program is a discrete optimization

model if it includes any discrete variable at all.

Otherwise, it is a continuous optimization model.

• An optimization model is an integer program (IP) if any

one of its decision variables is discrete. If all variables

are discrete, the model is a pure integer program;

otherwise, it is a mixed-integer program. [2.36]

Exercise 2.17: Recognizing Integer

Programs

Determine whether an optimization model over each of the

following systems of variables is an integer program, and if so,

state whether it is pure or mixed.

a) 𝑤𝑗 ≥ 0, 𝑗 = 1,… , 𝑞

b) 𝑤𝑗 = 0 𝑜𝑟 1, 𝑗 = 1,… , 𝑝

𝑤𝑝+1 ≥ 0 and integer

c) 𝑤𝑗 ≥ 0, 𝑗 = 1,… , 𝑝

𝑤𝑝+1 ≥ 0 and integer

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Integer Linear versus

Integer Nonlinear Programs

• A discrete or integer programming model is an integer

linear program (ILP) if its (single) objective function and

all main constraints are linear. [2.37]

• A discrete or integer programming model is an integer

nonlinear program (INLP) if its (single) objective

function or of its main constraints is linear. [2.38]

Exercise 2.18: Recognizing ILPs

and INLPs

Assuming that 𝑤’s are decision variables, determine whether each of

the following mathematical programs is best described as a linear

program (LP), a nonlinear program (NLP), an integer linear program

(ILP), an integer nonlinear program (INLP), a) max 3𝑤1 + 14𝑤2 −𝑤3

s.t. 𝑤1 ≤ 𝑤2

𝑤1 +𝑤2 +𝑤3 = 10𝑤𝑗 = 0 𝑜𝑟 1, 𝑗 = 1, … , 3

b) min 3𝑤1 + 14𝑤2 −𝑤3

s.t. 𝑤1𝑤2 ≤ 1𝑤1 +𝑤2 +𝑤3 = 10𝑤𝑗 ≥ 0, 𝑗 = 1,… , 3

𝑤1 integerc) min 3𝑤1 + 9

ln(𝑤2)

𝑤3

s.t. 𝑤1 ≤ 𝑤2

𝑤1 +𝑤2 +𝑤3 = 10𝑤2, 𝑤3 ≥ 1,𝑤1 ≥ 0

d) max 19𝑤1

s.t. 𝑤1 ≤ 𝑤2

𝑤1 +𝑤2 +𝑤3 = 10𝑤2, 𝑤3 ≥ 1; 𝑤1 ≥ 0

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Example 2.5:

Purdue Final Exam SchedulingIn a typical term Purdue University picks one of n = 30 final exam

time periods for each of over m = 2000 class units on its main

campus. Most exams involve just one class section, but there are a

substantial number of "unit exams" held at a single time for multiple

sections.

The main issue in this exam scheduling is "conflicts," instances

where a student has more than one exam scheduled during the same

time period. Conflicts burden both students and instructors because a

makeup exam will be required in at least one of the conflicting

courses. Purdue’s exam scheduling procedure begins by processing

enrollment records to determine how many students are jointly

enrolled in each pair of course units. Then an optimization scheme

seeks to minimize total conflicts as it selects time periods for all class

units.

Indexing, Parameters, and Decision

Variables for Purdue Finals Example

• Indexing

i ≜ class unit number (i = 1, …, m)

t ≜ exam time period number (t = 1, …, n)

• Decision variables

xi,t ≜ 10

if class i is assigned to time period totherwise

• Input parameters

ei,i’ ≜ number of students taking an exam in both

class i and class i’

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Nonlinear Objective Function

• Conflicts

xi,t xi’,t ≜ 10

if i and i′ are both sched at time period tif not

• Objective function

𝑚𝑖𝑛

𝑖=1

𝑚−1

𝑖′=𝑖+1

𝑚

𝑒𝑖,𝑖′

𝑡=1

𝑛

𝑥𝑖,𝑡𝑥𝑖′,𝑡

Purdue Final Exam Scheduling

Example Model

𝑚𝑖𝑛

𝑖=1

𝑚−1

𝑖′=𝑖+1

𝑚

𝑒𝑖,𝑖′

𝑡=1

𝑛

𝑥𝑖,𝑡𝑥𝑖′,𝑡

s.t. 𝑡=1𝑛 𝑥𝑖,𝑡 = 1 𝑖 = 1, … ,𝑚

𝑥𝑖,𝑡 = 0 𝑜𝑟 1 𝑖 = 1,… ,𝑚; 𝑡 = 1, … , 𝑛

(2.16)

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2.6 Multi-objective Optimization

Models

• A multi-objective optimization model is required to

capture all the perspectives – one that maximizes or

minimizes more than one objective function at the time.

Example 2.6: DuPage Land Use

Planning

Perhaps no public-sector problem involves more conflict

between different interests and perspectives than land use

planning. That is why a multi-objective approach was adopted

when government officials in DuPage County, Illinois, which is a

rapidly growing suburban area near Chicago, sought to

construct a plan controlling use of its undeveloped land.

Table 2.1 shows a simplified classification with m = 7 land

use types. The problem was to decide how to allocate among

these uses the undeveloped land in the county’s n = 147

planning regions.

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Example 2.6: DuPage Land Use

Planning

TABLE 2.1 Land Use Types in DuPage Example

i Land Use Type

1 Single-family residential

2 Multiple-family residential

3 Commercial

4 Offices

5 Manufacturing

6 Schools and other institutions

7 Open space

Example 2.6: DuPage Land Use

Planning: Multiple Objectives

1. Compatibility: an index of the compatibility between each possible

use in a region and the existing uses in and around the region.

2. Transportation: the time incurred in making trips generated by the

land use to/from major transit and auto links.

3. Tax load: the ratio of added annual operating cost for government

services associated with the use versus increase in the property

tax assessment base.

4. Environmental impact: the relative degradation of the environment

resulting from the land use.

5. Facilities: the capital costs of schools and other community

facilities to support the land use.

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Indexing, Parameters, and Decision

Variables for DuPage Land Use Planning

• Indexing

i ≜ land use type (i = 1, …, m)

j ≜ planning region (j = 1, …, n)

• Decision variables

xi,j ≜ number of undeveloped acres assigned to

land use i in planning region j

Indexing, Parameters, and Decision

Variables for DuPage Land Use Planning

• Input parameters

ci,j ≜ compatibility index per acre of land use i in

planning region j

ti,j ≜ transportation trip time generated per acre of

land use i in planning region j

ri,j ≜ property tax load ratio per acre of land use i in

planning region j

ei,j ≜ relative environmental degradation per acre of

land use i in planning region j

fi,j ≜ capital costs for community facilities per acre

of land use i in planning region j

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Multiple Objectives

𝑚𝑎𝑥

𝑖=1

𝑚

𝑗=1

𝑛

𝑐𝑖,𝑗 𝑥𝑖,𝑗

𝑚𝑖𝑛

𝑖=1

𝑚

𝑗=1

𝑛

𝑡𝑖,𝑗 𝑥𝑖,𝑗

𝑚𝑖𝑛

𝑖=1

𝑚

𝑗=1

𝑛

𝑟𝑖,𝑗 𝑥𝑖,𝑗

𝑚𝑖𝑛

𝑖=1

𝑚

𝑗=1

𝑛

𝑒𝑖,𝑗 𝑥𝑖,𝑗

𝑚𝑖𝑛

𝑖=1

𝑚

𝑗=1

𝑛

𝑓𝑖,𝑗 𝑥𝑖,𝑗

Constraints of the DuPage Land Use

Planning Example

• Constraints

bj ≜ number of undeveloped acres in planning region j

li ≜ county-wide minimum number of acres allocated to

land use type i

ui ≜ county-wide maximum number of acres allocated

to land use type i

oj ≜ number of acres in planning region j consisting of

undevelopable floodplains, rocky areas, etc.

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Constraints of the DuPage Land

Use Planning Example

s.t. 𝑖=1𝑚 𝑥𝑖,𝑗 = 𝑏𝑗 𝑗 = 1,… , 𝑛

𝑗=1𝑛 𝑥𝑖,𝑗 ≥ 𝑙𝑖 𝑖 = 1, … ,𝑚

𝑗=1

𝑛

𝑥𝑖,𝑗 ≤ 𝑢𝑖 𝑖 = 1,… ,𝑚

Additional Constraints of the DuPage

Land Use Planning Example

𝑥7,𝑗 ≥ 𝑜𝑗 𝑗 = 1, … , 𝑛 (all undevelopable land is

assigned to parks and other open space)si ≜ new acres of land use i implied by allocation of an

acre of undeveloped land to single-family residential

di ≜ new acres of land use i implied by allocation of an

acre of undeveloped land to multiple-family residential

𝑥𝑖,𝑗 ≥ 𝑠𝑖 𝑥1,𝑗 + 𝑑𝑖 𝑥2,𝑗 𝑖 = 3,6,7; 𝑗 = 1, … , 𝑛

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Conflict among Objectives

• When there is an option, single-objective optimization

models are preferred to multi-objective ones because

conflicts among objectives usually make multi-objective

models less tractable. [2.39]

Exercise 2.19: Understanding

Multiple-Objective Conflict

Consider the mathematical program

Max 3𝑧1 + 𝑧2Min 3𝑧1 + 𝑧2

s.t. 𝑧1 + 𝑧2 ≤ 3𝑧1, 𝑧2 ≥ 0

Graph the feasible

region and show that

the best solutions for

the two objective

functions conflict.

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2.7 Classification Summary