1. In the name of ALLAH The Most Beneficent The Most Merciful

2. Assist. Prof. Dr.-Ing. Mostafa Ranjbar
Multidisciplinary Engineering Design Optimization (MCE 540 Graduate Course – Mechanical Engineering Department)
Instructor:
Assist. Prof. Dr.-Ing. Mostafa Ranjbar
Ph.D. (Dr-Ing.), Multidisciplinary Engineering Design Optimization of Structures,Technische Universität Dresden, Germany, 2011
M.Sc., Vibration Monitoring and Fault Diagnosis of Structures, Tarbiat Modares University, Tehran, Iran, 2000
B.Sc., Mechanical Engineering, Shiraz university, Iran, 1998

3. MULTIDISCIPLINARY SYSTEM DESIGN Optimization
LECTURE # 3

4. INTRODUCTION
PHASE-I
Introduction to Multidisciplinary System Design Optimization
Terminology and Problem Statement
Introduction to Optimization
Classification of Optimization Problems
Numerical Optimization
MSDO Architectures
Practical Applications

5. OPTIMZATION
LECTURE # 3

6. INTRODUCTION
PHASE-I
Introduction to Multidisciplinary System Design Optimization
Terminology and Problem Statement
Introduction to Optimization
Classification of Optimization Problems
Numerical Optimization
MSDO Architectures
Practical Applications

7. INTRODUCTION to OPTIMIZATION

8. WHAT CAN BE ACHIEVED? Optimization techniques can be used for:
Getting a design/system to work
Reaching the optimal performance
Making a design/system reliable and robust
Also provide insight in
Design problem
Underlying physics
Model dynamics

9. OPTIMIZATION PROBLEM
General form of optimization problem:

10. SOLVING OPTIMIZATION PROBLEMS
Optimization problems are typically solved using an iterative algorithm:
Responses
Derivatives of responses
(design sensitivities)
Constants
Model
Design variables
Optimizer

11. CURSE OF DIMENSIONALITY
Looks complicated … why not just sample the design space, and take the best one?
Consider problem with n design variables
Sample each variable with m samples
Number of computations required: mn
                                                                    
Take 1 s per computation, 10 variables, 10 samples:
total time 317 years!

12. Parallel computing
Still, for large problems, optimization requires lots of computing power
Parallel computing

13. OPTIMIZATION IN THE DESIGN PROCESS
Conventional design process:
Collect data to describe the system
Estimate initial design
Analyze the system
Check performance criteria
Is design satisfactory?
Change design based on experience / heuristics / wild guesses
Done
Optimization-based design process:
Collect data to describe the system
Estimate initial design
Analyze the system
Check the constraints
Does the design satisfy convergence criteria?
Change the design using an optimization method
Done
Identify:
Design variables
Objective function
Constraints
Taken from J.S. Arora “Introduction to Optimum Design”, fig. 1-2.

14. OPTIMIZATION POPULARITY
Increasing availability of numerical modeling techniques
Increasing availability of cheap computer power
Increased competition, global markets
Better and more powerful optimization techniques
Increasingly expensive production processes (trial-and-error approach too expensive)
More engineers having optimization knowledge
Increasingly popular:

15. OPTIMIZATION PITFALLS!
Proper problem formulation critical!
Choosing the right algorithm for a given problem
Many algorithms contain lots of control parameters
Optimization tends to exploit weaknesses in models
Optimization can result in very sensitive designs
Some problems are simply too hard / large / expensive

16. OVERVIEW Traditional description of the design phases comprised of;
Problem Definition: Recognition of the original need is followed by a technical statement of the problem,
Synthesis: The creation of one or more physical configurations,
Analysis: The study of the configuration‘s performance using engineering science,
Optimization: The selection of "best" alternative.
The process concludes with testing of the prototype against the original need.

17. OVERVIEW Optimization deals with betterment and improvement.
“The concept of optimization is intrinsically tied to humanity’s desire to excel. Though we may not consciously recognize it, and though the optimization process takes different forms in a different field of endeavor, this drive to do better than before, whether we are athletes, artists, business-person, or engineers”
Optimization is defined as “the process of achieving the most favorable system condition on the basis of a metric or set of metrics [Merriam-Webster, 1998].
Optimization means the maximizing or minimizing of a given function possibly subject to some type of constraints and controlled by decision variables.

18. OVERVIEW Broadly, Narrowly,
The efforts and processes of making a decision, a design or a system as perfect, effective or functional as possible.
Narrowly,
The specific methodology, techniques and procedures used to decide on the one specific solution in a defined set of possible alternatives that will best satisfy a selected criterion.
“The main aim of OPTIMIZATION is to construct a model that can be easily understood and that provides good solutions in a reasonable amount of computing time”

19. OVERVIEW
Optimization, in engineering interpretation, is referred as the process of finding appropriate solutions with the intention of finding the best solutions to the system design problem.
Optimization is a computational design method which helps us select an optimal design among a number of (or an infinite set of ) possible options, such that a certain requirement or a set of requirements is best satisfied.

20. OVERVIEW
Optimization is applied in virtually all areas of human endeavor, including;
Engineering system design
Optical system design
Power systems
Water and land use
Transportation systems
Resource allocation
Personnel planning
Portfolio selection
Mining operations
Structural design
Control systems

21. OVERVIEW
The first (key) step in modern optimization is to obtain a mathematical description of the process or system to be optimized.
System models used in optimization is classified in various ways,
Linear versus nonlinear
Static versus dynamic
Deterministic versus stochastic
Time-invariant versus time-varying.

22. OVERVIEW
In forming a model for use with optimization, all of the important aspects of the problem should be included, so that they will be taken into account in the solution.
In some cases, the constraints and objective values or goals can be exchanged.
The model of a system must account for constraints that are imposed on the system. Some of the constraints restrict the values that can be assumed by variables of a system.
The types of constraints involved in any given problem are determined by the physical nature of the problem and by the level of complexity used in forming the mathematical model.

23. OVERVIEW
When constraints have been established, it is necessary to determine if there are any solutions to the problem that simultaneously satisfy all of the constraints. Any such solution is called a feasible solution.
A key step in the formulation of any optimization problem is the assignment of performance measures that are to be optimized.
“The success of any optimization result is critically dependent on the selection of meaningful performance measures”

24. OPTIMIZATION

25. OPTIMIZATION------WHY?????
Today’s mantra of “Faster, Better and Cheaper” has caused problem solvers to rethink how to reach at possible solution of a given problem.
Maybe the only way forward in the current daily technological advances.
The continuing push for reducing design costs and cycle time using computer-based models makes the use of optimization tools inevitable.
Suitable for generating more than a single solution, and this added information gives more flexibility to the user to choose a few solutions for further investigation.
“Need of modern era is not only to design a system which fits a customer’s needs, but it is also required to deliver an optimized system”

26. OPTIMIZATION------WHY?????
DESIGN
ANALYSIS
EVALUATION
FINE
YES
HOW TO MODIFY THE DESIGN????
MODIFY THE DESIGN NO
WITH EXPERIENCE….TRIAL and ERROR
WITH OPTIMIZATION METHODS

27. OPTIMIZATION------WHY?????
Is there one aircraft which is the fastest, most efficient, quietest, most inexpensive, most light weight ??????

28. OPTIMIZATION------WHY?????

29. OVERVIEW
Informally, but rigorously, we can say that design optimization involves:
1. The selection of a set of variables to describe the design alternatives.
2. The selection of an objective (criterion), expressed in terms of the design variables, which we seek to minimize or maximize.
3. The determination of a set of constraints, expressed in terms of the design variables, which must be satisfied by any acceptable design.
4. The determination of a set of values for the design variables, which minimize (or maximize) the objective, while satisfying all the constraints.
This definition of optimization suggests a philosophical and tactical approach during the design process.

30. OVERVIEW
It is not a phase in the process but rather a pervasive viewpoint.
Philosophically, optimization formalizes what humans (and designers) have always done.
Operationally, it can be used in design, in any situation where analysis is used, and is therefore subjected to the same limitations.

31. OPTIMIZATION MODEL

32. Definition of OPTIMIZATION Problem---OPTIMIZATION MODEL
The first step in modern optimization is to obtain a mathematical description of the process or the system to be optimized.
Design Space: The space of working (Hill in this case)
Objective: Find the Highest Point.
Design Variables: Longitude and latitude.
Constraints: Stay inside the fences.

33. OPTIMIZATION MODEL

34. OPTIMIZATION MODEL

35. OPTIMIZATION MODEL Constraint surfaces in a hypothetical two-dimensional design space

36. OPTIMIZATION MODEL Constraint surfaces in a hypothetical two-dimensional design space
Each point in the n-dimensional design space is called a design point and represents either a possible or an impossible solution to the design problem
In many practical problems, the design variables cannot be chosen arbitrarily; rather, they have to satisfy certain specified functional and other requirements.
The restrictions that must be satisfied to produce an acceptable design are collectively called design constraints.
Constraints that represent limitations on the behavior or performance of the system are termed behavior or functional constraints.
Constraints that represent physical limitations on design variables, such as availability, fabricability, and transportability, are known as geometric or side constraints.

37. OPTIMIZATION MODEL Constraint surfaces in a hypothetical two-dimensional design space

38. OPTIMIZATION MODEL Constraint surfaces in a hypothetical two-dimensional design space
Figure shows a hypothetical two-dimensional design space where the infeasible region is indicated by hatched lines.
A design point that lies on one or more than one constraint surface is called a bound point , and the associated constraint is called an active constraint .
Design points that do not lie on any constraint surface are known as free points.

39. OPTIMIZATION MODEL Constraint surfaces in a hypothetical two-dimensional design space
Depending on whether a particular design point belongs to the acceptable or unacceptable region, it can be identified as one of the following four types
1. Free and acceptable point
2. Free and unacceptable point
3. Bound and acceptable point
4. Bound and unacceptable point

40. OPTIMIZATION MODEL Constraint surfaces in a hypothetical two-dimensional design space
Depending on whether a particular design point belongs to the acceptable or unacceptable region, it can be identified as one of the following four types
1. Free and acceptable point
2. Free and unacceptable point
3. Bound and acceptable point
4. Bound and unacceptable point

41. OPTIMIZATION MODEL _________________ DESIGN SPACE

42. OPTIMIZATION MODEL
DESIGN SPACE

43. OPTIMIZATION MODEL
DESIGN SPACE
Design Space
Design Space

44. OPTIMIZATION MODEL _________________ DESIGN OBJECTIVE

45. OPTIMIZATION MODEL DESIGN OBJECTIVE
Design objectives usually represent the desires of the decision makers (designers).
A design objective can be considered as a criterion about whether or not a given design is better than others.
A design objective is determined by the objective function f(d)= f(d1,d2,…,dn).

46. OPTIMIZATION MODEL _________________ DESIGN VARIABLES

47. OPTIMIZATION MODEL DESIGN VARIABLES
A design variable is a decision variable or a control variable that can be changed by designers during a design process.
A design variable have an impact on the performances of a design.
Different combinations of design variables represent different designs.
Design variables should be as independent of each other as possible.
Optimization is the process of choosing the design variables that yield an optimum design

48. OPTIMIZATION MODEL DESIGN VARIABLES
Design variables are also known as design parameters and will be represented by the vector x. They are the variables in the problem that we allow to vary in the design process.
During design space exploration or optimization we change the entries of x in some rational fashion to achieve a desired effect.

49. OPTIMIZATION MODEL ___________________ DESIGN CONSTRAINTS

50. OPTIMIZATION MODEL DESIGN CONSTRAINTS
Few practical engineering optimizations problems are unconstrained.
We can not optimize our objectives infinitely because we have limited resources.
Constraints on the design variables are called bounds and are easy to enforce.
Constraints are restrictions or requirements imposed to a design.
A constraint function is expressed in a mathematical form in terms of design variables.
gi(d)<0, i=1,2,…,n
hj(d)=0, j=1,2,…,n
mk(d)>0, k=1,2,…,n

51. OPTIMIZATION MODEL DESIGN CONSTRAINTS
Constraints that must be satisfied are called rigid constraints or absolute goals.
Physical variables often are restricted to be nonnegative; for example, the amount of a given material used in a system is required to be greater than or equal to zero.
Soft constraints are those constraints that are negotiable to some degree.
These constraints can be viewed as goals that are associated with target values.
Equality constraints (e.g. 2nd stage diameter should be equal to 1st stage)
InEquality constraints (e.g. Max stress does not exceed the strength of material)

52. OPTIMIZATION MODEL Constraints vs. Objectives
It can be difficult to choose whether a condition is a constraint or an objective.
For example: should we try to minimize cost, or should we set a constraint stating that cost should not exceed a given level.
The two approaches can lead to different designs.
Sometimes, the initial formulation will need to be revised in order to fully understand the design space.
In some formulations, all constraints are treated as objectives (physical programming).

53. OPTIMIZATION MODEL ___________________ OBJECTIVE FUNCTION

54. OPTIMIZATION MODEL OBJECTIVE FUNCTION
In general, an optimization problem consists of an objective function which needs to be minimized or maximized, have some constraints and controlled by design variables.
Objective function is a “measure of goodness” that enables us to compare two designs quantitatively.
Objective function may be linear or nonlinear and may or not be given explicitly. We will represent it by the scalar f.

55. OPTIMIZATION MODEL OBJECTIVE FUNCTION
The objective function is the goal of the optimization.
For example, to minimize the mass of some structure or to maximize the speed of an object can be an objective function.
“If we select the wrong goal, it doesn’t matter how good the analysis is, or how efficient the optimization method is”

56. OPTIMIZATION MODEL OBJECTIVE FUNCTION FORMULATION
Optimization techniques are used to find a set of design parameters,
x = [x1, x2, .., xn], that can in some way be defined as optimal.
In a simple case this might be the minimization or maximization of some system characteristic that is dependent on x.
In a more advanced formulation the objective function, f(x), to be minimized or maximized, might be subject to constraints in the form of equality constraints

57. OPTIMIZATION MODEL _____________________ PROBLEM FORMULATION

58. OPTIMIZATION MODEL PROBLEM FORMULATION Objective Function Constraints
A general problem description is stated as
Minimize J (x)
gi (x) = 0, i=1,……,me
gi (x) ≤ 0 , i= me + 1,…..m
x is the vector of length n design parameters
J(x) is the objective function, which returns a scalar value
The vector function g(x) returns a vector of length m containing the values of the equality and inequality constraints evaluated at x.
Objective Function
Constraints
Design Variables

59. OPTIMIZATION MODEL PROBLEM FORMULATION
f : objective function, output (e.g. structural weight).
x : vector of design variables, inputs (e.g. aerodynamic shape); bounds can be
set on these variables.
h : vector of equality constraints (e.g. lift); in general these are nonlinear functions of the design variables.
g : vector of inequality constraints (e.g. structural stresses), may also be nonlinear and implicit.

60. OPTIMIZATION MODEL PROBLEM FORMULATION
An efficient and accurate solution to the problem depends not only on the size of the problem in terms of the number of constraints and design variables but also on characteristics of the objective function and constraints.
In the formulations discussed above, the objective is to minimize some function. If instead the objective was to maximize the objective function, one would minimize the negative of the objective function in order to be consistent in the manner of posing the problem.

61. OPTIMIZATION MODEL _________ OPTIMIZER

62. OPTIMIZATION OPTIMIZER
As part of the procedure, the optimizer performs the following functions:
Select specific values for some of the variables
Assign variables that are functions of time or other independent variables
Satisfy constraints that are imposed on the variables
satisfy certain goals and account for uncertainties or random aspects of the system

63. OPTIMIZATION MODEL

64. OPTIMIZATION MODEL OPTIMIZER
Objectives are what we are trying to achieve
Constraints are what we cannot violate
Design variables what we can play with
Optimizer is the tool to do all this
Constraints
Design Variables
Objective Function
OPTIMIZER

65. OPTIMIZATION MODEL _____________ EXAMPLE

66. Aerodynamic Optimization of Wing
Design Variables
Wing shape parameters
Airfoil at selected span stations

67. Aerodynamic Optimization of Wing
Design Constraints
Geometric Constraints
Airfoil maximum thickness at selected span stations
Limitations of planform geometric parameters
Performance Constraints
CL > specific value
Cmo > specific value
CLmax > specific value
CLalpha > specific value

68. Aerodynamic Optimization of Wing
Design Objective
Minimum drag (CD) at the design point
Maximum CL/CD at the design point
Desired pressure distribution at selected span stations

69. Aerodynamic Optimization of Wing
Sample design variables for configuration optimization

70. Placement of Big Telescope Problem
Put a big telescope on a high mountain peak of Central Pacific Ocean. Find a peak.
Design Space
Central Pacific Ocean
Feasible Region
Anything nod underwater
Considerations
Discontinuous feasible regions within the design space

71. OPTIMIZATION MODEL _______________ FEASIBLE REGION

72. FEASIBLE REGION
When solving an optimization problem, it is not only necessary to search for the design variables that give the best objective function but it is also necessary to ensure that the solution is feasible.
When constraints have been established, it is important to determine if there are any solutions to the problem that simultaneously satisfy all of the constraints.
A feasible solution is one in which all constraints are satisfied, whereas an infeasible solution is one in which one or more constraints are unsatisfied.

73. FEASIBLE REGION

74. FEASIBLE REGION

75. LOCAL AND GLOBAL OPTIMA
LOCAL OPTIMA
maxima
Local maxima
Local minima
minima
GLOBAL MINIMA

76. OPTIMIZATION _________________ POINTS TO PONDER

77. OPTIMIZATION----points to ponder
Optimization models are representations of a decision-making process that is restricted to what can be modeled mathematically.
A proper identification of the system and its elements that represent the design is crucial for successful formulation and solution of mathematical optimization models.
An underlying requirement of any optimization study is a good quantitative understanding of the behavior of the artifact or process to be optimized.

78. OPTIMIZATION----points to ponder
Optimization became an engineer‘s partner in design.
It excels at handling the quantitative side of design.
It‘s applications range from component to systems.
It‘s utility is dramatically increasing with the advent of massively concurrent computing.
Current trend: extend optimization to entire life cycle with emphasis on economics, include uncertainties.
Engineer remains the principal creator, data interpreter, and design decision maker.

79. OPTIMIZATION MODEL _________________ SEARCH DIRECTION

80. SEARCH DIRECTION
Find a search direction that will improve the objective while staying inside the fences.
Search in this direction until no more improvement is made.

81. OPTIMIZATION MODEL _________________ CONVERGENCE CRITERIA

82. CONVERGENCE CRITERIA Algorithm has converged when….
No change in the objective function is obtained OR
The maximum number of iterations is reached.
Once the “optimal” solution has been obtained, the KKT conditions should be checked.

83. OPTIMIZATION ______________ CLASSIFICATION

84. OPTIMIZATION----CLASSIFICATIONS
The goal of design optimization is to find the optimum (from the Latin word optimus, meaning best) solution to the design problem.
Throughout history numerous techniques have been developed in order to carry out design optimization.
Modern computers, with their incredibly fast computational power, have turned optimization theory into a rapidly growing branch of applied mathematics.

85. OPTIMIZATION----CLASSIFICATIONS
Constraints: Constrained vs. Unconstrained
Design Variables: Single-variable vs. Multivariable; continuous vs. discrete.
Linearity: Linear objective function and linear constraints (Linear Programming) vs. Non-linear objective or constraints are (Non-linear Programming).
Time: dynamic vs. static optimization.
Data: Deterministic vs. stochastic optimization.

86. OPTIMIZATION CLASSIFICATION
OPERATIONS RESEARCH METHODS

87. OPTIMIZATION CLASSIFICATION
OPERATIONS RESEARCH METHODS

88. OPTIMIZATION CLASSIFICATION

89. OPTIMIZATION CLASSIFICATION

90. OPTIMIZATION CLASSIFICATION

91. OPTIMIZATION CLASSIFICATION

92. LINEAR PROGRAMMING
LECTURE #3

93. OPTIMIZATION Optimization
Optimization is defined as the process of achieving the most favorable system condition on the basis of a metric or set of metrics [Merriam-Webster, 1998].
Optimization means the maximizing or minimizing of a given function possibly subject to some type of constraints.
The objective function is the goal of the optimization

94. OPTIMIZATION Constrained optimization
Finding the optimal solution to a problem given that certain constraints must be satisfied by the solution.
A form of decision making that involves situations in which the set of acceptable solutions is somehow restricted.
Recognizes scarcity—the limitations on the availability of physical and human resources.
Seeks solutions that are both efficient and feasible in the allocation of resources.

95. OPTIMIZATION LINEAR PROGRAMMING
A family of mathematical techniques (algorithms) that can be used for constrained optimization problems with linear relationships.
Graphical method
Simplex method
Karmakar’s method
The problems must involve a single objective, a linear objective function, and linear constraints and have known and constant numerical values

96. LINEAR PROGRAMMING CHARACTERISTICS of LP MODELS
LECTURE #3

97. LINEAR PROGRAMMING
CHARACTERISTICS

98. Sample of LP Decision variables Objective function Constraints
Let xi be denoted as xi product to be produced, and i = 1, 2 or Let x1 be numbers of product x1 to be produced and x2 be numbers of product 21 to be produced Maximize Z=$40x1 + 50x2 subject to 1x1 + 2x2  40 hours of labor 4x2 + 3x2  120 pounds of clay x1, x2  0
Decision variables
Objective function
Constraints

99. General LP format subject to xij ≥ 0, for i=1,…,m, j=1,…,n
Max/Min Z : Σ cixi
subject to
Σ aij xij (=, ≤, ≥) bj , j = 1,…., n
xij ≥ 0, for i=1,…,m, j=1,…,n
General steps for LP formulation
It means there are total of m decision variables
n resource constraints

100. General LP format

101. LINEAR PROGRAMMING STEPS
LECTURE #3

102. Steps for LP formulation
Step 1: define decision variables
Step 2: define the objective function
Step 3: state all the resource constraints
Step 4: define non-negativity constraints

103. LINEAR PROGRAMMING MAXIMIZATION PROBLEM
LECTURE #3

104. Example 1: Max Problem A Maximization Model
Example The Beaver Creek Pottery Company produces bowls and mugs. The two primary resources used are special pottery clay and skilled labour. The two products have the following resource requirements for production and profit per item produced (that is, the model parameters).
Resource available: 40 hours of labour per day and 120 pounds of clay per day. How many bowls and mugs should be produced to maximizing profits give these labour resources?
LP formulation

106. Max LP problem Step 1: define decision variables
Let x1=number of bowls to produce/day
x2= number of mugs to produce/day
Step 2: define the objective function
maximize Z = $40x1 + 50x where Z= profit per day
Step 3: state all the resource constraints
1x1 + 2x2  40 hours of labor ( resource constraint 1)
4x1 + 3x2  120 pounds of clay (resource constraint 2)
Step 4: define non-negativity constraints
x10; x2  0

107. Max LP problem Complete Linear Programming Model:
maximize Z=$40x1 + 50x2
subject to
1x1 + 2x2  40
4x2 + 3x2  120
x1, x2  0

108. A Maximization Example
LP Model Formulation
A Maximization Example
Product mix problem - Beaver Creek Pottery Company
How many bowls and mugs should be produced to maximize profits given labor and materials constraints?
Product resource requirements and unit profit:

109. A Maximization Example
LP Model Formulation
A Maximization Example
Resource hrs of labor per day
Availability: 120 lbs of clay
Decision x1 = number of bowls to produce per day
Variables: x2 = number of mugs to produce per day
Objective Maximize Z = $40x1 + $50x2
Function: Where Z = profit per day
Resource x1 + 2x2  40 hours of labor
Constraints: 4x1 + 3x2  120 pounds of clay
Non-Negativity x1  0; x2  0
Constraints:

110. A Maximization Example
LP Model Formulation
A Maximization Example
Complete Linear Programming Model:
Maximize Z = $40x1 + $50x2
subject to: x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0

111. A FEASIBLE SOLUTION
A feasible solution does not violate any of the constraints:
Example
x1 = 5 bowls
x2 = 10 mugs
Z = $40x1 + $50x2 = $700
Labor constraint check:
1(5) + 2(10) = 25 < 40 hours, within constraint
Clay constraint check:
4(5) + 3(10) = 70 < 120 pounds, within constraint

112. An INFEASIBLE SOLUTION
An infeasible solution violates at least one of the constraints:
Example
x1 = 10 bowls
x2 = 20 mugs
Z = $1400
Labor constraint check:
1(10) + 2(20) = 50 > 40 hours, violates constraint

113. Don’t send an elephant to kill an ant

114. LINEAR PROGRAMMING GRAPHICAL SOLUTIONS
LECTURE #3

115. Graphical Solution of LP Models
Graphical solution is limited to linear programming models containing only two decision variables (can be used with three variables but only with great difficulty).
Graphical methods provide visualization of how a solution for a linear programming problem is obtained.

116. Graphing the Model The graphical approach:
Plot each of the constraints.
Determine the region or area that contains all of the points that satisfy the entire set of constraints.
Determine the optimal solution.

117. Feasible solution space Corner point Redundant constraint Slack
Key Terms in Graphing
Optimal solution
Feasible solution space
Corner point
Redundant constraint
Slack
Surplus

118. Graphical Solution of Maximization Model (1 of 12)
Coordinate Axes
Graphical Solution of Maximization Model (1 of 12)
Maximize Z = $40x1 + $50x2
subject to:
1x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0
Figure 2.2 Coordinates for Graphical Analysis

119. Graphical Solution of Maximization Model (2 of 12)
Labor Constraint
Graphical Solution of Maximization Model (2 of 12)
Maximize Z = $40x1 + $50x2
subject to:
1x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0
Figure 2.3 Graph of Labor Constraint

120. Graphical Solution of Maximization Model (3 of 12)
Labor Constraint Area
Graphical Solution of Maximization Model (3 of 12)
Maximize Z = $40x1 + $50x2
subject to:
1x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0
Figure 2.4 Labor Constraint Area

121. Graphical Solution of Maximization Model (4 of 12)
Clay Constraint Area
Graphical Solution of Maximization Model (4 of 12)
Maximize Z = $40x1 + $50x2
subject to:
1x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0
Figure 2.5 Clay Constraint Area

122. Graphical Solution of Maximization Model (5 of 12)
Both Constraints
Graphical Solution of Maximization Model (5 of 12)
Maximize Z = $40x1 + $50x2
subject to:
1x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0
Figure 2.6 Graph of Both Model Constraints

123. Feasible Solution Area
Graphical Solution of Maximization Model (6 of 12)
Maximize Z = $40x1 + $50x2
subject to:
1x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0
Figure Feasible Solution Area

124. Objective Function Solution = $800
Graphical Solution of Maximization Model (7 of 12)
Maximize Z = $40x1 + $50x2
subject to:
1x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0
Figure 2.8 Objection Function Line for Z = $800

125. Alternative Objective Function Solution Lines
Graphical Solution of Maximization Model (8 of 12)
Maximize Z = $40x1 + $50x2
subject to:
1x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0
Figure 2.9 Alternative Objective Function Lines

126. Graphical Solution of Maximization Model (9 of 12)
Optimal Solution
Graphical Solution of Maximization Model (9 of 12)
Maximize Z = $40x1 + $50x2
subject to:
1x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0
Figure Identification of Optimal Solution

127. Optimal Solution Coordinates
Graphical Solution of Maximization Model (10 of 12)
Maximize Z = $40x1 + $50x2
subject to:
1x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0
Figure 2.11 Optimal Solution Coordinates

128. Extreme (Corner) Point Solutions
Graphical Solution of Maximization Model (11 of 12)
Maximize Z = $40x1 + $50x2
subject to:
1x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0
Figure Solutions at All Corner Points

129. Optimal Solution for New Objective Function
Graphical Solution of Maximization Model (12 of 12)
Maximize Z = $70x1 + $20x2
subject to:
1x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0
Figure 2.13 Optimal Solution with Z = 70x1 + 20x2

130. Slack Variables
Standard form requires that all constraints be in the form of equations (equalities).
A slack variable is added to a  constraint (weak inequality) to convert it to an equation (=).
A slack variable typically represents an unused resource.
A slack variable contributes nothing to the objective function value.

131. Linear Programming Model: Standard Form
Max Z = 40x1 + 50x2 + s1 + s2
subject to:1x1 + 2x2 + s1 = 40
4x1 + 3x2 + s2 = 120
x1, x2, s1, s2  0
Where:
x1 = number of bowls
x2 = number of mugs
s1, s2 are slack variables
Figure 2.14 Solution Points A, B, and C with Slack

132. LINEAR PROGRAMMING MINIMIZATION PROBLEM
LECTURE #3

133. Example 2: Min Z
A farmer is preparing to plant a crop in the spring. There are two brands of fertilizer to choose from, Supper-gro and Crop-quick. Each brand yields a specific amount of nitrogen and phosphate, as follows:
The farmer’s field requires at least 16 pounds of nitrogen and 24 pounds of phosphate. Super-gro costs $6 per bag and Crop-quick costs $3 per bag. The farmer wants to know how many bags of each brand to purchase in order to minimize the total cost of fertilizing.
LP formulation

135. Min Z
Step 1: define their decision variables x1  number of bags of Super-gro, x2  number of bags of Crop-quick. Step 2: define the objective function Minimise Z  6x1  3x2 Step 3: state all the resource constraints 2x1  4x2  16, (resource 1) 4x1  3x2  24 (resource 2) Step 4: define the non-negativity constraints x1  0, x2  0 Overall LP: Minimise Z  6x1  3x2 subject to 2x1  4x2  16, 4x1  3x2  24,

136. A Minimization Example (1 of 7)
LP Model Formulation
A Minimization Example (1 of 7)
Two brands of fertilizer available - Super-Gro, Crop-Quick.
Field requires at least 16 pounds of nitrogen and 24 pounds of phosphate.
Super-Gro costs $6 per bag, Crop-Quick $3 per bag.
Problem: How much of each brand to purchase to minimize total cost of fertilizer given following data ?

137. A Minimization Example (2 of 7)
LP Model Formulation
A Minimization Example (2 of 7)
Decision Variables: x1 = bags of Super-Gro
x2 = bags of Crop-Quick
Objective Function:
Minimize Z = $6x1 + 3x2
Where:
$6x1 = cost of bags of Super-Gro
$3x2 = cost of bags of Crop-Quick
Model Constraints:
2x1 + 4x2  16 lb (nitrogen constraint)
4x1 + 3x2  24 lb (phosphate constraint)
x1, x2  0 (non-negativity constraint)

138. LP Model Formulation and Constraint Graph
A Minimization Example (3 of 7)
Minimize Z = $6x1 + $3x2
subject to: 2x1 + 4x2  16
4x1 + 3x2  24
x1, x2  0
Figure Graph of Both Model Constraints

139. Feasible Solution Area A Minimization Example (4 of 7)
Minimize Z = $6x1 + $3x2
subject to: 2x1 + 4x2  16
4x1 + 3x2  24
x1, x2  0
Figure 2.17 Feasible Solution Area

140. Optimal Solution Point A Minimization Example (5 of 7)
Minimize Z = $6x1 + $3x2
subject to: 2x1 + 4x2  16
4x1 + 3x2  24
x1, x2  0
Figure Optimum Solution Point

141. A Minimization Example (6 of 7)
Surplus Variables
A Minimization Example (6 of 7)
A surplus variable is subtracted from a  constraint to convert it to an equation (=).
A surplus variable represents an excess above a constraint requirement level.
Surplus variables contribute nothing to the calculated value of the objective function.
Subtracting slack variables in the farmer problem constraints:
2x1 + 4x2 - s1 = 16 (nitrogen)
4x1 + 3x2 - s2 = 24 (phosphate)

142. A Minimization Example (7 of 7)
Graphical Solutions
A Minimization Example (7 of 7)
Minimize Z = $6x1 + $3x2 + 0s1 + 0s2
subject to: 2x1 + 4x2 – s1 = 16
4x1 + 3x2 – s2 = 24
x1, x2, s1, s2  0
Figure 2.19 Graph of Fertilizer Example

143. Irregular Types of Linear Programming Problems
For some linear programming models, the general rules do not apply.
Special types of problems include those with:
Multiple optimal solutions
Infeasible solutions
Unbounded solutions

144. Multiple Optimal Solutions Beaver Creek Pottery Example
Objective function is parallel to to a constraint line.
Maximize Z=$40x1 + 30x2
subject to:
1x1 + 2x2  40
4x1 + 3x2  120
x1, x2  0
Where:
x1 = number of bowls
x2 = number of mugs
Figure Example with Multiple Optimal Solutions

145. An Infeasible Problem Figure 2.21 Graph of an Infeasible Problem
Every possible solution violates at least one constraint:
Maximize Z = 5x1 + 3x2
subject to: 4x1 + 2x2  8
x1  4
x2  6
x1, x2  0
Figure Graph of an Infeasible Problem

146. An Unbounded Problem Figure 2.22 Graph of an Unbounded Problem
Value of objective function increases indefinitely:
Maximize Z = 4x1 + 2x2
subject to:
x1  4
x2  2
x1, x2  0
Figure Graph of an Unbounded Problem

147. Characteristics of Linear Programming Problems
A linear programming problem requires a decision - a choice amongst alternative courses of action.
The decision is represented in the model by decision variables.
The problem encompasses a goal, expressed as an objective function, that the decision maker wants to achieve.
Constraints exist that limit the extent of achievement of the objective.
The objective and constraints must be definable by linear mathematical functional relationships.

148. Properties of Linear Programming Models
Proportionality - The rate of change (slope) of the objective function and constraint equations is constant.
Additivity - Terms in the objective function and constraint equations must be additive.
Divisibility -Decision variables can take on any fractional value and are therefore continuous as opposed to integer in nature.
Certainty - Values of all the model parameters are assumed to be known with certainty (non-probabilistic).

149. Example Problem No. 1 (1 of 3)
Problem Statement
Example Problem No. 1 (1 of 3)
Hot dog mixture in 1000-pound batches.
Two ingredients, chicken ($3/lb) and beef ($5/lb).
Recipe requirements:
at least 500 pounds of chicken
at least 200 pounds of beef
Ratio of chicken to beef must be at least 2 to 1.
Determine optimal mixture of ingredients that will minimize costs.

150. Example Problem No. 1 (2 of 3)
Solution
Example Problem No. 1 (2 of 3)
Step 1:
Identify decision variables.
x1 = lb of chicken in mixture (1000 lb.)
x2 = lb of beef in mixture (1000 lb.)
Step 2:
Formulate the objective function.
Minimize Z = $3x1 + $5x2
where Z = cost per 1,000-lb batch
$3x1 = cost of chicken
$5x2 = cost of beef

151. Example Problem No. 1 (3 of 3)
Solution
Example Problem No. 1 (3 of 3)
Step 3:
Establish Model Constraints
x1 + x2 = 1,000 lb
x1  500 lb of chicken
x2  200 lb of beef
x1/x2  2/1 or x1 - 2x2  0
x1, x2  0
The Model: Minimize Z = $3x1 + 5x2
subject to: x1 + x2 = 1,000 lb
x1  50
x2  200
x1 - 2x2  0
x1,x2  0

152. Example Problem No. 2 (1 of 3)
Solve the following
model graphically:
Maximize Z = 4x1 + 5x2
subject to: x1 + 2x2  10
6x1 + 6x2  36
x1  4
x1, x2  0
Step 1: Plot the constraints as equations
Figure 2.23 Constraint Equations

153. Example Problem No. 2 (2 of 3)
Maximize Z = 4x1 + 5x2
subject to: x1 + 2x2  10
6x1 + 6x2  36
x1  4
x1, x2  0
Step 2: Determine the feasible solution space
Figure 2.24 Feasible Solution Space and Extreme Points

154. Example Problem No. 2 (3 of 3)
Maximize Z = 4x1 + 5x2
subject to: x1 + 2x2  10
6x1 + 6x2  36
x1  4
x1, x2  0
Step 3 and 4: Determine the solution points and optimal solution
Figure Optimal Solution Point

155. LINEAR PROGRAMMING SUCCESSFUL APPLICATIONS
LECTURE #4

156. LP – SUCCESSFUL APPLICATIONS

157. Marketing Applications
Applications of linear programming in marketing
Media selection—how to allocate an advertising budget among different media of advertising.
Determination of the optimal assignment of salespeople that work for the company among the sales territories.
Marketing research which the objective is to determine the best number of interviews, mailings, or phone calls, given a number of client-specified constraints and the cost considerations.

158. LINEAR PROGRAMMING MODELING EXAMPLES
LECTURE # 3

159. MODELING EXAMPLES Product Mix Example Diet Example Investment Example
Marketing Example
Transportation Example
Blend Example
Multi-Period Scheduling Example
Data Envelopment Analysis Example

160. QUESTIONS

161. THANK YOU FOR YOUR INTEREST