Wiederholung
Operations Research
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Operations Research
Operations Research (OR) is the field
of how to form mathematical models
of complex management decision
problems and how to analyze the
models to gain insight about possible
solutions.
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OR Process
Assessment
Real world problem
Abstraction
Model
Real world solution
Interpretation
Analysis
Model solution
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Operations Research
Operations Research deals with
decision problems by formulating and
analyzing mathematical models –
mathematical representations of
pertinent problem features.
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Operations Research
The model-based OR approach to
problem solving works best on
problems important enough to
warrant the time and resources for a
careful study.
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Mathematical Programming
Optimization Models
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OR models
The three fundamental concerns of forming
operations research models are
• decisions open to decision makers,
• the constraints limiting decision choices, and
• the objectives making some decisions
preferred to others.
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Mortimer Middleman

Min 3.50 r  55 
q
2

2000
q / 55
s.t. q  100
r  55
r *  55
q  250.7
*
c( r * , q* )  $45630
8
Mathematical Programming
Deterministic Optimization
• Maximise/Minimise
– a single real function
– of real or integer variables
• subject to constraints on the variables
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Variables
• Variables in optimization models
represent the decisions to be taken.
• Variable-type constraints specify the
domain of definition for decision
variables: the set of values for which the
variables have meaning.
10
Main constraints
• Main constraints of optimization
models specify the restriction and
interactions, other than variable type,
that limit decision variables.
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Objective Functions
• Objective functions in optimization
models quantify the decision
consequences to be maximized or
minimized.
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Mortimer Middleman
•
•
•
•
•
•
d ... weekly demand
f ... fixed cost of replenishment
h ... cost per carat per week holding
s ... cost per carat lost sales
l ... lead time
m ... minimum order size
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Mortimer Middleman
Min r ,q

h r  l d  
q
2

f
q/d
s.t. q  m
r  ld
14
Parameters – Output Variables
• Parameters – quantities taken as given
– Weekly demand, fixed cost of
replenishment, cost for holding inventory,
cost per carat lost sales, lead time,
minimum order size.
• Parameters and decision variables
determine results measured as output
variables
– c(r,q ; d,f,h,s,l,m)
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Canonical Form of a (NonLinear) Optimization Problem
•
Maximize f(x)
subject to g(x) <= 0
x >= 0
• Key Components of Optimization Pb.
– Objective Function
– Decision Variables
– Constraints
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Two Crude Petroleum Case
Min : 50x1  60x2
s.t. 0.3x1  0.4 x2  2.0
gasoline
0.4 x1  0.2 x2  1.5
jet fuel
0.2 x1  0.3x2  0.5
lubricant
x1
9
x2  6
NNC : x1  0; x2  0;
Saudi
Venezuelan
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Fence Excercise
Max : l w
s.t. 2l  2 w  80
NNC : l  0; w  0;
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Howie’s Hot Tub Problem
•
Blue Ridge Hot Tubs manufactures and sells two models of hot tubs:
the Acqua-Spa and the Hydro-Lux. Howie Jones, the owner and
manager of the company, needs to decide how many of each type of
hot tub to produce during his next production cycle. Howie buys
prefabricated fiberglass hot tub shells from a local supplier and adds
the pump and tubing to the shells to create his hot tubs. (This supplier
has the capacity to deliver as many hot tub shells as Howie needs.)
Howie installs the same type of pump into both hot tubs. He will have
only 200 pumps available during his next production cycle. From a
manufacturing standpoint, the main difference between the two models
of hot tubs is the amount of tubing and labor required. Each AcquaSpa requires 9 hours of labor and 12 feet of tubing. Each Hydro-Lux
requires 6 hours of labor and 16 feet of tubing. Howie expects to have
1,566 production labor hours and 2,880 feet of tubing available during
the next production cycle. Howie earns a profit of $350 on each AquaSpa he sells and $300 on each Hydro-Luc he sells. He is confident that
he can sell all the hot tubs he produces. The question is, how many
Acqua-Spas and Hydro-Luxes should Howie produce if he wants to
maximize
his profitsBook
during the next production cycle?
Taken
from Ragsdale’s
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Howie’s Decision Problem
• Let
– X1 = # of Aqua-spas produced
– X2 = # of Hydro-Luxs produced
• Maximize Z = 350 X1 + 300 X2
s.t.
X1 + X2 <= 200
(pumps)
9 X1 + 6 X2 <= 1,566
(labor hours)
12 X1 + 16 X2 <= 2880
(feet of tubing)
X1, X2 >= 0
(non-negativity)
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Feasible
• The feasible set (or region) of an
optimization model is the collection of
choices for decision variables satisfying
all model constraints.
• The feasible set for an optimization model
is plotted by introducing constraints one
by one, keeping track of the region
satisfying all at the same time.
21
Optimal Solution
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.
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Graphing Objective Functions
Objective functions are normally
plotted in the same coordinate
system as the feasible set of
optimization model by introducing
contours – lines or curves through
points having equal objective
function values.
23
Optimal Solution
Optimal solutions show graphically as
points lying on the best objective
function contour that intersects the
feasible region.
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Graphical Solution
(Only practical for 2D Pbs.)
• Plot the constraints
• Identify the feasible region
• Draw contours (level curves; iso-value
lines) of objective function
• Most desirable level curve will intersect
feasible region
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Mathematical Programming
Graphical Solution
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Howie’s hot tube problem
Excel Workbook
Lawrence W. Robinson
Johnson Grad. School of Mgmt, Cornell University
27
Optimal Value
• The optimal value in an optimization
model is the objective function value
of any optimal solution.
• An optimization model can have only
one optimal value.
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Use Graphical Solution to
Develop Some Intuition
• Alternate optimal solutions
– If obj. fn. is parallel to a binding constraint
• Redundant constraints
– Plays no role in determining feasible region
• Unbounded solution
– Can occur if feasible region is unbounded
• Infeasible problem
– There is no feasible region; constraints are
inconsistent
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Fence Excercise
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Mathematical Programming
Large Scale Optimisation
31
Pi Hybrids Example
Mingjian Zuo, Way Kuo, and Keith L.
McRoberts (1991),
„Application of Mathematical programming
to a Large-Scale Agricultural Production
and Distribution System“,
Journal of Operational Research Society,
42, 639-648
32
Pi Hybrids Example
• l = 20 facilities
• m = 25 hybrid corn
• n = 30 sales region
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Pi Hybrids Example
• The producing cost($/bag)
• The corn processing capacity (bushels)
• The corn needed to produce a bag
(bushels/bag)
• Hybrid corn demanded (bag)
• The cost per bag shipping ($/bag)
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Indexing
The first step in formulating a large
optimization model is to choose
appropriate indexes for the different
dimensions of the problem.
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Pi Hybrids Example
Indexes:
• f = 1...l (facilities)
• h = 1...m (hybrid variety)
• r = 1...n (sales region)
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Indexing parameters
To describe large-scale optimization
models compactly it is usually
necessary to assign indexed
symbolic names to variables and to
most input parameters, even though
they are being treated as constant.
Summation Notation
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Pi Hybrids Example
Variables:
• xf,h
f = 1,...,l; h = 1,...,m
– bags h at facility f
• yf,h,r
f = 1,...,l; h = 1,...m, r = 1,...,n
– bags h from facility f to region r
38
Pi Hybrids Example
Parameters:
• pf,h
f = 1,...,l; h = 1,...,m
– production cost ($/bag)
• sf,h,r
f = 1,...,l; h = 1,...m, r = 1,...,n
– shipping cost ($/bag)
39
Pi Hybrids Example
Parameters (continued):
• uf
f = 1,...,l;
– capacity (bushel)
• ah
h = 1,...,m;
– (bushel/bag)
• dh,r
h = 1,...,m; r = 1,...,n
– demand (bag)
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Indexed 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.
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Pi Hybrids Example
42
Large-scale
Optimization models become large
mainly by a relatively small number
of objective function and constraint
elements being repeated many times
for different periods, locations,
products, and so on.
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Mathematical Programming
Linear or Nonlinear
44
LP Model
An optimization model is a linear
program (or LP) if it has continuous
variables, a single linear objective
function, and all constraints are
linear equalities or inequalities.
45
Linear functions
• A function is linear if it is a constantweighted sum of decision variables.
Otherwise, it is nonlinear.
• Linear functions implicitly assume
that each unit increase in a decision
variable has the same effect as the
preceding increase: equal returns to
scales.
46
Linearity
• Proportional
– regular hourly wage rates
– machine output per hour
–…
• Non-Proportional
– wage rates for over time
– freight rates
– quantity purchasing discout
47
f(x) is linear if it is a sum of constants
times the components of x
• Linear
– y = f(x) = a x + b
– f(x) = c0 + c1 x1 + c2 x2 + c3 x3 + ...
• Not linear
– f(x) = sin(x)
– f(x1, x2) = x1/x2
– f(x) = ex
48
Linear Programming:
A Special Kind of NLP
• Suppose
– Objective function is linear
– Constraints are linear
– Decision variables are continuous
• Max cT x
(i.e., c1 x1 + c2 x2 + ...)
st A x <= b (a1,1 x1 + a1,2 x2 + ... <= b1
a2,1 x1 + a2,2 x2 + ... <= b2)
x >= 0
(i.e., x1 >= 0, x2 >= 0, ...)
49
E-Mart
P. Doyle and J. Saunders (1990),
„Multiproduct Advertising Budgeting“,
Marketing Science, 9, 97-113
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E-Mart
51
Mathematical Programming
Discrete (Integer) vs. Continuous
52
Discrete decision var.
• A variable is discrete if it is limited to
a fixed countable set of values.
Often, the choices are integer or
only binary (0 and 1).
• A variable is continuous if it can
take on any value in a specified
interval.
53
Integer Program
An optimization model is an integer
program (IP) if any one if its
decision variables is discrete. If all
variables are discrete, the model is
a (pure) integer program; otherwise,
it is a mixed-integer program (MIP).
54
Bethlehem Ingot Mold
F.J. Vasko, F.E. Wolf, K.S. Stott, J.W.
Scheirer (1989),
„Selecting Optimal Ingot Sizes for
Bethlehem Steel“,
Interfaces, 19:1, 68-84
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Bethlehem Ingot Mold
56
Integer Program
• A discrete or integer programming
model is an integer linear program (ILP)
if its (single) objective function and all
main constraints are linear.
• A discrete or integer programming
model is an integer nonlinear program
(INLP) if its (single) objective function or
any of its main constraints is nonlinear.
57
Exam Scheduling
C.J. Horan and W.D. Coates (1990)
„Using More Than ESP to Schedule Final
Exams: Purdue‘s Examination
Scheduling Procedure II (ESP II)“
College and University Computer Users
Conference Proceedings, 35, 133-142
58
Exam Scheduling
59
LP Models are preferred
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.
60
Mathematical Programming
Multiobjective Optimization Models
61
DuPage Land Use
Deepak Bammi and Dalip Bammi (1979)
„Development of a Comprehensive Land
Use Plans by means of a Multiple
Objective Mathematical Progamming
Model,“
Interfaces, 9:2, part 2, 50-63
62
DuPage Land Use
1.
2.
3.
4.
5.
6.
7.
Single-family residential
Multiple-family residential
Commerical
Offices
Manufacturing
Schools and other institutions
Open space
63
DuPage Land Use
64
LP Model
• Linear programming requires a single
objective function
• If not:
– including objectives as constraints in the model +
Sensitivity Analysis
– Goal Programming
– MCDM
65
Single objectives are preferred
When there is an option, singleobjective optimization models are
preferred to multiobjective ones
because conflicts among objectives
usually make multiobjective models
less tractable.
66
Beispiel 1
Production Allocation: The Acme Axle Company produces
both car and track axles for national and international
markets. Each axle must complete two manufacturing
processes: molding and finishing. Each car axle requires
16 units of molding and 10 units of finishing, whereas a
truck axle requires 24 units of molding and 20 units of
finishing. Weekly, 480 units of molding and 360 units of
finishing are available. The demand for Acme‘s axles is
such that the firm may sell all it produces. Acme achieves
a profit of $50 per car axle and $60 per truck axle. Acme
also has an agreement with the Spitz Motor Company to
supply 12 car axles and 8 truck axles weekly. Given the
above constraints and requirements, Acme desires to
know what amounts of car and truck axles to produce
weekly in order to maximize profit. Formulate an LP
Model to gain insights on the optimal production mix.
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Beispiel 2
Large scale: Suppose that the decision variables of a mathematical
programming model are
xi l t … amount of product i produced on manufacturing line l during
week t
where i=1,…,17; l=1,…,5; t=1,…,7. Use summation and indexed
notation to write expressions for each of the following systems
of constraints in terms of these decision variables, and
determine how many constraints belong to each system:
•
•
•
Total production on any line in any week should not exceed 200.
The total 7-week production of product i=5 should not exceed
4000.
At least 100 units of each product should be produced each
week.
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Bsp - Evaluierung
Evaluation: Five car salespeople had the following sales for the
past two months:
Salesperson
Fred
Mary
John
Jane
Chris
Luxury Cars
3
7
1
2
5
SUV’s
6
4
4
3
5
Mid-sized
12
15
18
24
16
The general manager believes that total dollar sales doesn’t
adequately capture performance and would like to use a
weighted average of luxury car, SUV, and mid-sized sales
instead. The manager asks each salesperson to come up
with a (positive) weight for each car category, but stipulates
that weights cannot allow anyone’s total weighted score to
exceed 100. For example, defining w1 = luxury weight, w2 =
SUV weight, and w3 = mid-sized weight, Fred’s weighted
score would be: 3 w1 + 6 w2 + 12 w3. Develop an LP model
that will find a set of weights that will make John’s weighted
score as large as possible.
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Break
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