Chapter 1 Making Economic Decisionsweb.eng.fiu.edu/leet/OR_1/OR1_chap2_2014.pdf · lubricant 0.2...
Transcript of Chapter 1 Making Economic Decisionsweb.eng.fiu.edu/leet/OR_1/OR1_chap2_2014.pdf · lubricant 0.2...
8/8/2014
1
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.
8/8/2014
2
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?
8/8/2014
3
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)
8/8/2014
4
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)
8/8/2014
5
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)
8/8/2014
6
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]
8/8/2014
7
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]
8/8/2014
8
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]
8/8/2014
9
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
8/8/2014
10
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)
8/8/2014
11
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
8/8/2014
12
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
8/8/2014
13
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]
8/8/2014
14
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
8/8/2014
15
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]
8/8/2014
16
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
8/8/2014
17
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
8/8/2014
18
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.
8/8/2014
19
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)
8/8/2014
20
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.
8/8/2014
21
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
8/8/2014
22
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)
8/8/2014
23
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
8/8/2014
24
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]
8/8/2014
25
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)
8/8/2014
26
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]
8/8/2014
27
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]
8/8/2014
28
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.
8/8/2014
29
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)
8/8/2014
30
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.
8/8/2014
31
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
8/8/2014
32
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]
8/8/2014
33
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
8/8/2014
34
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)
8/8/2014
35
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
8/8/2014
36
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
8/8/2014
37
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’
8/8/2014
38
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)
8/8/2014
39
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.
8/8/2014
40
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.
8/8/2014
41
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
8/8/2014
42
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.
8/8/2014
43
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
8/8/2014
44
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.
8/8/2014
45
2.7 Classification Summary