A primer on optimization using solvers

Anwar Ali
www.theoptimizationexpert.com

Overview
 The slides are meant for those who are interested to
learn linear programming application but have no
time for proper graduate study
 The emphasis is on practical model building and
solving the models using solvers
 The slides are free preview of 3-day “Decision
Optimization” training offered by the author
 The author has 27 years semiconductor manufacturing
experience, of which 18 years are in Industrial
Engineering and Operations Research
A Primer on Optimization Using Solvers - Anwar Ali 2

Agenda
 Introduction to Analytics and Operations Research
 Optimization Modeling with Textbook Examples
 Formulating and Solving Mathematical Models
 Optimization Applications in Industry
 How to Get Started

Analytics Landscape
Descriptive
Prescriptive
Predictive
Degree of Complexity
CompetitiveAdvantage
Standard Reporting
Ad hoc reporting
Query/drill down
Alerts
Simulation
Forecasting
Predictive modeling
Optimization
What exactly is the problem?
What will happen next if ?
What if these trends continue?
What could happen…. ?
What actions are needed?
How many, how often, where?
What happened?
Stochastic Optimization
How can we achieve the best outcome?
How can we achieve the best outcome
including the effects of variability?
Source: IBM, Based on: Competing on Analytics,
Davenport and Harris, 2007

Analytics
 Descriptive analytics (what has occurred)
 The simplest class of analytics, condense big data into
smaller, more useful nuggets of information
 e.g. counts, likes, posts, views, sales, finance
 Predictive analytics (what will occur)
 Use available data to predict data we don’t have using
variety of statistical, modeling, data mining, and
machine learning techniques
 Prescriptive analytics (what should occur)
 Recommend one or more courses of action and showing
the likely outcome of each decision so that the business
decision-maker can take this information and act
Adapted from Information Week, definitions by Dr Michael Wu
http://www.informationweek.com/big-data/big-data-analytics/big-data-analytics-descriptive-vs-predictive-vs-prescriptive/d/d-id/1113279

Business Intelligence Framework
Back in Business, by Ronald K. Klimberg and Virginia Miori, OR/MS Today, Vol 37, No 5, October 2010,
[http://www.informs.org/ORMS-Today/Public-Articles/October-Volume-37-Number-5/Back-in-Business]
OR/MS =
Operations Research/
Management Science

What is Operations Research?
 O.R. is the discipline of applying advanced analytical
methods to help make better decisions
 Also called Management Science or Decision Science,
O.R. is the science of Decision-Making
 Employing techniques from mathematical sciences,
O.R. arrives at optimal or near-optimal solutions to
complex decision-making problems
 Determine the maximum (e.g. profit, performance, or
yield) or minimum (e.g. loss, risk, or cost)

Analytics Landscape
Descriptive
Prescriptive
Predictive
Degree of Complexity
CompetitiveAdvantage
Standard Reporting
Ad hoc reporting
Query/drill down
Alerts
Simulation
Forecasting
Predictive modeling
Optimization
What exactly is the problem?
What will happen next if ?
What if these trends continue?
What could happen…. ?
What actions are needed?
How many, how often, where?
What happened?
Stochastic Optimization
How can we achieve the best outcome?
How can we achieve the best outcome
including the effects of variability?
Adapted from: IBM, Based on: Competing on Analytics,
Davenport and Harris, 2007
Operations Research

O.R. Leading Edge Techniques
 Simulation
 Giving you the ability to try out approaches and test
ideas for improvement
 Optimization
 Narrowing your choices to the very best where there are
virtually innumerable feasible options and comparing
them is difficult
 Probability and statistics
 Helping you measure risk, mine data to find valuable
connections and insights, test conclusions, and make
reliable forecasts

 Simulation (predictive)
 Optimization (prescriptive)
 Narrowing your choices to the very best where there are
virtually innumerable feasible options and comparing
them is difficult
 Probability and statistics (predictive)
reliable forecasts

 Simulation
 Optimization – THIS TRAINING
 Narrowing your choices to the very best where there
are virtually innumerable feasible options and
comparing them is difficult
 Probability and statistics
reliable forecasts

Examples of O.R. Application
 Deciding where to invest capital in order to grow
 Figuring out the best way to run a call center
 Locating a warehouse or depot to deliver materials
over shorter distances at reduced cost
 Solving complex scheduling problems
 Deciding when to discount, and how much
 Getting more out of manufacturing equipment
 Optimizing a portfolio of investments

Terminology Evolved
Previous Current
 Operations Research
 Statistics
 Decision Support
 Data sets (structured)
 Data Analyst
 Business Analytics
 Data Science
 Business Intelligence / Analytics
 Big Data (unstructured)
 Data Scientist

What are the Benefits of O.R.?
 Operations Research is called “The Science of Better”,
i.e. using science to make:
 bold decisions and run everyday operations with less
risk and better outcomes (no more gut-feel)
 repeatable, quantitative decision analysis
Adapted from: The Guide to Operational Research, http://www.scienceofbetter.co.uk/

Signs O.R. Could Be Beneficial
 The management face complex decision making
 The management is not sure what the main problem is
 The management is uncertain about potential
outcomes
 The organization is having problems with decision
making processes
 Management is troubled by risk
 The organization is not making the most of its data
 The organization needs to beat stiff competition
The Guide to Operational Research, http://www.scienceofbetter.co.uk/

Agenda
 Introduction to Analytics and Operations Research √
 Optimization Modeling with Textbook Examples

Optimization Modeling
 Optimization models have
 Objective function
 Decision variables
 Constraints
 Formulated as mathematical equations
 Solved graphically (for problems with 2 decision
variables) or using ‘Solver’ (will be explained later)
 We will start with simple Linear Programming (LP)
model examples from textbooks

Example 1: Dorian Auto
 Operations Research:
Applications and Algorithms
 Wayne L. Winston
 Duxbury Press; 4th edition
(2003)

 Dorian Auto manufactures luxury cars and trucks
 The company believes that its most likely customers
are high-income women and men
 To reach these groups, Dorian Auto has embarked on
an ambitious TV advertising campaign and will
purchase 1-minute commercial spots on two type of
programs: comedy shows and football games

 Each comedy commercial is seen by 7 million high
income women and 2 million high-income men and
costs $50,000
 Each football game is seen by 2 million high-income
women and 12 million high-income men and costs
$100,000
 Dorian Auto would like for commercials to be seen by
at least 28 million high-income women and 24 million
high-income men
 We will use LP to determine how Dorian Auto can
meet its advertising requirements at minimum cost

Example 1: Solution
 Decision variables:
x = the number of 1-minute comedy ads
y = the number of 1-minute football ads
 The objective is to minimize advertising cost
 Minimize z = 50x + 100y
 Constraints:
 Ads must be seen by at least 28 million high-income
women; 7x + 2y ≥ 28
 Ads must be seen by at least 24 million high-income
men; 2x + 12y ≥ 24
 Sign restrictions; x ≥ 0 and y ≥ 0

Graphical Solution
x (comedy ads)
y (football ads)
4 8 12 16
4
12
16
8
2
6
10
14
2 6 10 14
Min z = 50x + 100y
subject to:
7x + 2y ≥ 28
2x + 12y ≥ 24
x, y ≥ 0

Graphical Solution
x (comedy ads)
y (football ads)
4 8 12 16
4
12
16
8
2
6
10
14
2 6 10 14
High-income women constraint; 7x + 2y ≥ 28
Min z = 50x + 100y
subject to:
7x + 2y ≥ 28
2x + 12y ≥ 24
x, y ≥ 0

Graphical Solution
x (comedy ads)
y (football ads)
4 8 12 16
4
12
16
8
2
6
10
14
2 6 10 14
High-income women constraint; 7x + 2y ≥ 28
High-income men constraint; 2x + 12y ≥ 24
Min z = 50x + 100y
subject to:
7x + 2y ≥ 28
2x + 12y ≥ 24
x, y ≥ 0

Unbounded
feasible region
Graphical Solution
x (comedy ads)
y (football ads)
4 8 12 16
4
12
16
8
2
6
10
14
2 6 10 14
High-income women constraint
High-income men constraint
Min z = 50x + 100y
subject to:
7x + 2y ≥ 28
2x + 12y ≥ 24
x, y ≥ 0

Unbounded
feasible region
Graphical Solution
x (comedy ads)
y (football ads)
4 8 12 16
4
12
16
8
2
6
10
14
2 6 10 14
Min z = 50x + 100y
subject to:
7x + 2y ≥ 28
2x + 12y ≥ 24
x, y ≥ 0

Unbounded
feasible region
Graphical Solution
x (comedy ads)
y (football ads)
4 8 12 16
4
12
16
8
2
6
10
14
2 6 10 14
x = 3.6
y = 1.4
Min z = 50x + 100y
subject to:
7x + 2y ≥ 28
2x + 12y ≥ 24
x, y ≥ 0

Optimal Answer
 To minimize advertising cost, purchase
 3.6 slots of comedy ads (x)
 1.4 slots of football ads (y)
 The total advertising cost (in thousands) is
z = 50x + 100 y
z = 50(3.6) + 100(1.4)
z = 320
 But in reality, it is not possible to purchase fractional
number of 1-minute ads. The decision variables x and
y must be integers

Integer Programming
 When an LP model has integer decision variable(s), it
is called integer linear programming (ILP). Why ILP?
 We cannot buy 3.6 slots of ads, must be either 3 or 4
 Yes/no decisions can be modeled as 0 or 1 variables
 When an LP model has mixture of continuous and
integer variables, it is called mixed integer linear
programming (MILP)
 ILP and MILP models are harder and take longer to
solve compared to LP models
 We will use the term “math programming” to refer to
LP, ILP, and MILP

Unbounded
feasible region
Graphical Integer Solution
x (comedy ads)
y (football ads)
4 8 12 16
4
12
16
8
2
6
10
14
2 6 10 14
High-income men constraint Feasible integer solutions
Min z = 50x + 100y
subject to:
7x + 2y ≥ 28
2x + 12y ≥ 24
x, y ≥ 0
x, y integers

Unbounded
feasible region
Graphical Integer Solution
x (comedy ads)
y (football ads)
4 8 12 16
4
12
16
8
2
6
10
14
2 6 10 14
Lowest z value
in feasible region
Optimal integer solutions
Feasible integer solutions
Min z = 50x + 100y
subject to:
7x + 2y ≥ 28
2x + 12y ≥ 24
x, y ≥ 0
x, y integers

Unbounded
feasible region
Graphical Integer Solutions
x (comedy ads)
y (football ads)
4 8 12 16
4
12
16
8
2
6
10
14
2 6 10 14
x = 6, y = 1
x = 4, y = 2
2 solutions with
z = 400
Min z = 50x + 100y
subject to:
7x + 2y ≥ 28
2x + 12y ≥ 24
x, y ≥ 0
x, y integers

Graphical Integer Solutions
 There are 2 solutions with z = 400
 4 slots of comedy ads (x) and 2 slots of football ads (y); z
= 50(4) + 100(2) = 400
 6 slots of comedy ads (x) and 1 slot of football ads (y); z
= 50(6) + 100(1) = 400
 For more complex problems which cannot be solve
graphically, branch-and-bound method is used

Example 2: Diet Problem
 Introduction to
Management Science
 Bernard W. Taylor III
 Prentice Hall, 7th edition
(2002); the diet problem is
from this edition
 Latest is 11th edition (2012)

Breakfast to include at least 420 calories, 5 milligrams of
iron, 400 milligrams of calcium, 20 grams of protein, 12
grams of fiber, and must have no more than 20 grams of
fat and 30 milligrams of cholesterol

 The objective is to minimize meal cost while meeting
the following nutritional requirement:
 Calories ≥ 420
 Iron ≥ 5
 Calcium ≥ 400
 Protein ≥ 20
 Fiber ≥ 12
 Fat ≤ 20
 Cholesterol ≤ 30

Example 2: Decision Variables
x1 = cups of bran cereal
x2 = cups of dry cereal
x3 = cups of oatmeal
x4 = cups of oat bran
x5 = eggs
x6 = slices of bacon
x7 = oranges
x8 = cups of milk
x9 = cups of orange juice
x10 = slices of wheat toast

Example 2: Problem Formulation
Minimize
0.18x1 + 0.22x2 + 0.10x3 + 0.12x4 + 0.10x5 + 0.09x6 + 0.40x7 + 0.16x8 + 0.50x9
+ 0.07x10
Subject to:
90x1 + 110x2 + 100x3 + 90x4 + 75x5 + 35x6 + 65x7 + 100x8 + 120x9 +
65x10 ≥ 420
6x1 + 4x2 + 2x3 + 3x4 + x5 + x7 + x10 ≥ 5
20x1 + 48x2 + 12x3 + 8x4 + 30x5 + 52x7 + 250x8 + 3x9 + 26x10 ≥ 400
3x1 + 4x2 + 5x3 + 64 + 7x5 + 2x6 + x7 + 9x8 + x9 + 3x10 ≥ 20
5x1 + 2x2 + 3x3 + 4x4 + x7 + 3x10 ≥ 12
2x2 + 2x3 + 2x4 + 5x5 + 3x6 + 4x8 + x10 ≤ 20
270x5 + 8x6 + 12x8 ≤ 30
x1, x2, x3, x4, x5, x6, x7, x8, x9, x10 ≥ 0

Example 2: Solution
 The diet problem cannot be solved graphically as it has
10 decision variables
 We will use ‘Solver’ to find solution for the problem

Solver
 Mathematical software, either stand-alone or library,
that 'solves' a mathematical programming problem
 Uses algorithms such as SIMPLEX and branch-and-
bound to solve the problem
 May include Integrated Development Environment
(IDE), e.g. GUI and editor
 Solvers used in this training:
 Excel Solver, free Excel ad-in with limited capability
 IBM ILOG CPLEX and LPSolve have complete IDE
 LPSolve is free (GNU lesser general public license) and
can be downloaded from sourceforge.net

Decision Modeling with Excel
 Knowledge on the following is required: named range
and SUMPRODUCT() function
 Named ranges can help make Excel spreadsheet
formulas more readable
 SUMPRODUCT() can simplify lengthy formulas if the
worksheet is designed properly
 We will explain how the Diet problem Excel Solver
model is developed using named ranges and
SUMPRODUCT() function
 We also need to enable Excel Solver Add-in
41A Primer on Optimization Using Solvers - Anwar Ali

Enabling Excel Solver Add-in
 Start Excel
 From File, Options,
highlight Add-Ins
 From Manage, select
Excel Add-ins drop down
menu and hit Go button
 Note: Excel Solver interface
screenshots are from Excel 2010.
Other screenshots are from
Excel 2007 & Excel 2010

Enabling Excel Solver Add-In
 Check the box for Solver
Add-in, then hit OK

Enter diet data into Excel

Name B3:B12 as serving
These cells are the decision variables x1 to x10

Name D3:D12 as cost

Name E3:E12 as calories

Naming Ranges: Continue with
 F3:F12 as fat
 G3:G12 as cholesterol
 H3:H12 as iron
 I3:I12 as calcium
 J3:J12 as protein
 K3:K12 as fiber

Add SUMPRODUCT() to col E:K

What is SUMPRODUCT()?
Given that serving is defined as B3:B12 and calories is defined as E3:E12,
SUMPRODUCT(serving,calories) equals to B3*E3 + B4*E4 + B5*E5 +
B6*E6 + B7*E7 + B8*E8 + B9*E9 + B10*E10 + B11*E11 + B12*E12

Enter SUMPRODUCT() @ cells
 E14 =SUMPRODUCT(serving,calories)
 F14 =SUMPRODUCT(serving,fat)
 G14 =SUMPRODUCT(serving,cholesterol)
 H14 =SUMPRODUCT(serving,iron)
 I14 =SUMPRODUCT(serving,calcium)
 J14 =SUMPRODUCT(serving,protein)
 K14 =SUMPRODUCT(serving,fiber)

Adding Constraints
 The ≥ and ≤ signs in row 15 are optional
 To make the spreadsheet more readable
 Add Nutritional Requirement values in row 16

Objective Function

Objective Function
 Name cell B18 as meal_cost
 Enter =SUMPRODUCT(serving,cost) into cell B18
 The objective function is to minimize cell B18,
meal_cost
 Now, we need to enter the objective function and the
constraints into Excel Solver

Excel Solver Interface
 From Excel menu, select Data. Solver should be visible
on the right. Select Solver

Excel Solver Interface
 Objective (Min)
 Enter meal_cost
 Select Min radio button
 By Changing Variable Cells
(Decision Variables)
 Enter serving
 Subject to the Constraints:
 Add the constraints one at
a time

Excel Solver Parameters

Solving….
 Hit Solve button and the
dialog box should
appear, with the answers
in cells serving
 Hit OK

Excel Solver Solution

Excel Solver
 Excel Solver has
determined these are the
optimal answers

Excel Solver
 But the solution requires
fractional cups and/or
slices of food
 How to make them as
round number?
 We will show how to
make the answer for
wheat toast slice as a
round number

Mixed-Integer Diet Problem
 Add another constraint
with type integer

Integer variable added

Solution with Integer Variable

Model in IBM ILOG CPLEX

IBM ILOG CPLEX Solution

CPLEX Model (Integer variable)

Model in LPSolve

LPSolve Solution

LPSolve Model (Integer Variable)

LPSolve Solution (Integer Variable)

Key Take Away
 In university, we were taught how to model and then
solve the problem by hand
 In practice, solvers like Excel Solver, ILOG CPLEX and
LPSolve can find the solution(s) very quickly
 SIMPLEX used for linear programming
 Brand-and-bound used for integer model
 It is important to understand the modeling concepts
and able to formulate the problems correctly
 But real-world models are a lot more complex than the
textbook examples
 May have multiple conflicting objectives
 Many (thousands) decision variables and constraints

Conflicting Objectives
CostProfit
Labor
Service
Time
Regulations
Policy Laws
Process
Quality Systems
Safety
Compliance

Choice of Solver
 The choice of solver depends on the problem size and
the ability to integrate with enterprise system
 Excel Solver is recommended for rapid prototyping
and quick-wins
 Demonstrate the concept to users and management
 Can be used if the problem is small
 When all data is local and no database interface is required
 Commercial solver is required for large problems and
data integration with enterprise system
 Scalable with powerful database interfaces

Agenda
 Optimization Modeling with Textbook Examples √

Problem Formulation
 Problem formulation is the most challenging part in
math programming
 Once the problem has been formulated correctly,
putting the problem into solvers is easy
 Need to use the correct approach in developing the
mathematical equations of a problem
 The more experience we have in problem formulation,
the easier it becomes

The formulation of a
problem is often more
essential than its
solution, which may be
merely a matter of
mathematical or
experimental skill
Albert Einstein
Picture from Wikipedia

Recommended Modeling Approach
 First, must understand the problem well
 e.g. business rules, objective(s), constraints, input data
and output/decisions required
 Talk to the experts how decisions are made without a
model
 Relate the problem to the relevant model types
 Look at examples of the relevant model types
 Many Excel Solver examples are downloadable from
Frontline Systems
 IBM ILOG CPLEX has examples of different complexity
 Develop and refine the model until it represents the
problem faithfully

Additional Reference – Williams
 Model Building in
Mathematical Programming
 H. Paul Williams
 John Wiley & Sons, Ltd.
5th edition (2013)

Bin packing / knapsack problem

Cut into different sizes and shapes and minimize the waste
Cutting stock problem

Start from a city, visit each city only once, and return to the original city
after all cities visited. Minimize the travel distance / cost
Traveling salesman problem (TSP)

Assign gates to planes considering plane type, schedule, domestic/international, airlines
Assignment problemA Primer on Optimization Using Solvers - Anwar Ali 84

Blending problem

Minimize breakfast cost and include at least 420
calories, 5 milligrams of iron, 400 milligrams of
calcium, 20 grams of protein, 12 grams of fiber, and
must have no more than 20 grams of fat and 30
milligrams of cholesterol
Diet problem

Summary of Problems
 Linear Programming
 Blending problem
 Diet problem
 Integer Programming
 Bin packing / knapsack problem
 Cutting stock problem
 Traveling salesman problem (TSP)
 Assignment problem
 We pick the interesting TSP problem and demonstrate
how it is formulated and solved

TSP Described
 You are given a set of n cities
 You are given the distances between the cities
 You start and terminate your tour at your home city
 You must visit each other city exactly once
 Your mission is to determine the shortest tour

TSP LP Formulation
 Set of cities 𝑁 = 1,2, … , 𝑛
 Decision variables, 𝑥𝑖𝑗
 𝑥𝑖𝑗 = 1 if we go from city i to city j
 𝑥𝑖𝑗 = 0 otherwise, 𝑖 ≠ 𝑗
 𝑑𝑖𝑗 = direct distance from between city i and city j
 Example: 𝑛 = 4, 𝑥 =
0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0
 represents the tour (1,2,3,4,1)

TSP Objective Function
𝑚𝑖𝑛 𝑧 =
𝑛
𝑖=1
𝑑𝑖𝑗 𝑥𝑖𝑗
𝑛
𝑗=1
𝑖≠𝑗

TSP Constraints
Each city must be “entered” exactly once and “exited” exactly once
𝑥𝑖𝑗
𝑛
𝑗=1
𝑗≠𝑖
= 1, ∀𝑖 ∈ 𝑁
𝑥𝑖𝑗
𝑛
𝑖=1
𝑖≠𝑗
= 1, ∀𝑗 ∈ 𝑁
Subject to:

But it may create sub-tours
 Example solution, 𝑥 =
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
 represents two sub-tours (1,2,1) and (3,4,3)
 This is not feasible for TSP
1
2
3
4

Sub-tour Elimination Constraint
 The various methods of adding sub-tour elimination
constraints were summarized by Orman, A. J. and
Williams, H. Paul
 http://eprints.lse.ac.uk/22747/
 For implementation in Excel, we select the Sequential
Formulation by Miller, Tucker and Zemlin as it has the
least number of decision variables and constraints

Sub-tour Elimination Constraint
 Add continuous variables,
 𝑢𝑖= sequence in which city i is visited (𝑖 ≠ 1)
 And add constraints
 𝑢𝑖 − 𝑢𝑗 + 𝑛𝑥𝑖𝑗 ≤ 𝑛 − 1, ∀𝑖, 𝑗 ∈ 𝑁 − 1 , 𝑖 ≠ 𝑗
 The simpler, layman version:
 𝑢𝑖 − 𝑢𝑗 + 𝑛𝑥𝑖𝑗 ≤ 𝑛 − 1, ∀𝑖, 𝑗 ∈ 2,3, . . , 𝑛 , 𝑖 ≠ 𝑗

Complete TSP Formulation
𝑚𝑖𝑛 𝑧 =
𝑛
𝑖=1
𝑑𝑖𝑗 𝑥𝑖𝑗
𝑛
𝑗=1
𝑖≠𝑗
𝑥𝑖𝑗
𝑛
𝑗=1
𝑗≠𝑖
= 1, ∀𝑖 ∈ 𝑁
𝑥𝑖𝑗
𝑛
𝑖=1
𝑖≠𝑗
= 1, ∀𝑗 ∈ 𝑁
Subject to:
𝑢𝑖 − 𝑢𝑗 + 𝑛𝑥𝑖𝑗 ≤ 𝑛 − 1, ∀𝑖, 𝑗 ∈ 𝑁 − 1 , 𝑖 ≠ 𝑗
𝑁 = 𝑐𝑖𝑡𝑖𝑒𝑠 {1,2, . . , 𝑛}

TSP Demo
 Due to limitations of free Excel Solver, we can only
solve a 10-city problem. Excel 2007 is used here
 Let’s travel to 10 capital cities of ASEAN
 Start and terminate the tour at a city
 Must visit each other city exactly once
 Objective: minimize the total air travel distance
Country Capital Airport BWN PNH CGK VTE KUL NYT MNL SIN BKK HAN
Brunei Bandar Seri Begawan BWN 1330 1534 1977 1487 2603 1255 1278 1833 2060
Cambodia Phnom Penh PNH 1330 1975 757 1038 1289 1782 1138 504 1081
Indonesia Jakarta CGK 1534 1975 2719 1129 3083 2788 882 2297 3042
Laos Vientiane VTE 1977 757 2719 1697 694 2007 1857 517 495
Malaysia Kuala Lumpur KUL 1487 1038 1129 1697 1970 2490 297 1221 2102
Myanmar Naypyidaw NYT 2603 1289 3083 694 1970 2696 2202 819 1016
Philippines Manila MNL 1255 1782 2788 2007 2490 2696 2375 2188 1773
Singapore Singapore SIN 1278 1138 882 1857 297 2202 2375 1417 2218
Thailand Bangkok BKK 1833 504 2297 517 1221 819 2188 1417 995
Vietnam Hanoi HAN 2060 1081 3042 495 2102 1016 1773 2218 995

Setup the Excel worksheet

Name D2:M11 as distance

Name D13:M22 as var_X

D23=SUM(D13:D22), copy to E23:M23

Name D23:M23 as constraint1

N13=SUM(D13:M13), copy to N14:N22

Name N13:N22 as constraint2

Name E25:M25 as var_U

Sub-tour elimination constraint
 We need to implement this into the Excel model
 While the equation looks simple, the indexing of
variable u makes it more challenging in Excel
 Use INDEX() function; row = index i, column = index j
 Name top left cell as ref_cell (where i, j = 1)
 Name the constraint range as subtour_elim with i, j ≥ 2
 Each cell in subtour_elim needs to calculate its i and j
indices by referencing to ref_cell
 We cannot exclude i ≠ j in the named ranges; the
model in Excel is not 100% correct but will still work
𝑢𝑖 − 𝑢𝑗 + 𝑛𝑥𝑖𝑗 ≤ 𝑛 − 1
∀𝑖, 𝑗 ∈ 𝑁 − {1 , 𝑖 ≠ 𝑗

Name D29 as ref_cell

Enter sub-tour elimination constraint

Copy to the range, subtour_elim

Name D40 as total_distance,
=SUMPRODUCT(distance,var_X)

Solver Parameters
Under Options, select the checkboxes
 Assume Linear Model
 Assume Non-Negative

Solution

Solution
BWN  CGK  SIN  KUL  PNH  BKK  NYT  VTE  HAN  MNL  BWN
Total distance = 9291 km

CPLEX OPL TSP Model
/*********************************************
* OPL 12.6.2.0 Model
* Author: Anwar Ali
* 10-city TSP model of ASEAN capital cities
*********************************************/
int n = 10;
range cities = 1..n;
range uvar = 2..n;
string city[cities] = ["BWN", "PNH", "CGK", "VTE", "KUL", "NYT", "MNL", "SIN", "BKK", "HAN"];
int dist[cities][cities] = [
[0, 1330, 1534, 1977, 1487, 2603, 1255, 1278, 1833, 2060],
[1330, 0, 1975, 757, 1038, 1289, 1782, 1138, 504, 1081],
[1534, 1975, 0, 2719, 1129, 3083, 2788, 882, 2297, 3042],
[1977, 757, 2719, 0, 1697, 694, 2007, 1857, 517, 495],
[1487, 1038, 1129, 1697, 0, 1970, 2490, 297, 1221, 2102],
[2603, 1289, 3083, 694, 1970, 0, 2696, 2202, 819, 1016],
[1255, 1782, 2788, 2007, 2490, 2696, 0, 2375, 2188, 1773],
[1278, 1138, 882, 1857, 297, 2202, 2375, 0, 1417, 2218],
[1833, 504, 2297, 517, 1221, 819, 2188, 1417, 0, 995],
[2060, 1081, 3042, 495, 2102, 1016, 1773, 2218, 995, 0]
];

CPLEX OPL TSP Model
dvar boolean x[cities][cities];
dvar float+ u[uvar];
minimize
sum(i in cities, j in cities: i!=j) x[i][j]*dist[i][j];
subject to {
forall(i in cities) sum(j in cities: j!=i) x[i][j] == 1;
forall(j in cities) sum(i in cities: i!=j) x[i][j] == 1;
forall(i in uvar, j in uvar: i!=j) u[i] - u[j] + n*x[i][j] <= n - 1;
}
execute {
var from;
var dest;
from = 1;
for (var cnt=1; cnt<=10; cnt++) {
for (dest=1; dest<=n; dest++)
if (x[from][dest]==1) {
write(city[from]," --> ");
from = dest;
break;
}
}
writeln(city[dest]);
}

CPLEX OPL Solution
 // solution (optimal) with objective 9291
 BWN --> CGK --> SIN --> KUL --> PNH --> BKK -->
NYT --> VTE --> HAN --> MNL --> BWN

CPLEX OPL Solution

TSP Key Learning
 Textbook TSP formulation can be directly used to solve
real-world problems
 As the formulation gets more complex, it becomes
harder to model in Excel, but still easy in OPL
 Excel named ranges are only for rectangles
 Some tricks required for indexing of variables

Agenda
 Introduction to Analytics & Operations Research √
 Formulating and Solving Mathematical Models √

Waste neither time nor
money, but make the
best use of both
Benjamin Franklin
Picture from Wikipedia

Supply Chain Optimization
 Plan
 Facilities
 Resources required
 Source
 Make vs Buy
 Materials
 Make
 Prod planning
 Deliver
 Warehouse
 Transportation
 Location (where to open)
 Min capital & headcount, max usage
 Make in-house or sub-con?
 How much to buy? Which supplier?
 Who does what? How much to make?
 Sequencing, batching decisions
 Location and inventory level
 Which air, sea & ground network

Supply Chain Optimization
 This can be very complex depending on company’s size
 For large companies, it is recommended to get
complete solutions from various solution providers
 It is not just system implementation, but business
processes changes as well
 It is better to start with small wins (phase-by-phase)
instead of big bang approach

Airline Industry Optimization
 Crew scheduling
 Aircraft scheduling
 Airport gates assignment
 Baggage handling
 Tickets pricing

TSP Application Examples
 Vehicle routing problem
 Modified (more complex) TSP. Find the optimal route
to deliver goods from central depot to customers who
placed orders for such goods
 Design and fabrication of integrated circuits (IC)
 Minimize time spent during lithography to reduce the
capital cost of semiconductor factory
 Design and manufacturing of printed circuit board
(PCB) and substrates
 Placement of components, routing, drilling

Vehicle Routing Problem

PCB Example

PCB Holes Drilling

PCB Holes Drilling
Without TSP
Less travel distance with TSP,
hence less drill time

Agenda
 Formulating and Solving Mathematical Models √
 Optimization Applications in Industry √

Getting Started with Optimization
 Get management sponsors
 Convince management the benefits of optimization
 Identify the challenges in decision making process
 Unable to predict the outcome?
 Complexity in decision making
 Drill down the decision making process
 Objectives, rules, and boundary conditions
 Input data required
 What kind of outcomes/decisions needed
 Build and demo quick-win optimization model(s)
 Refine it until it can replace the current process

Competencies Required
 Spreadsheet modeling
 Mathematical optimization
 Data integration
 Business acumen
 Hire consultant and/or upskill employees
 Read this:
 http://www.scienceofbetter.co.uk/

Training & Consulting
 Current training offered:
 3-day “Decision Optimization”
 1-day “Decision Optimization for Managers”
 If you need help with optimization modeling for your
business, I will be happy to offer my service. More
details here:
www.theoptimizationexpert.com
 My profile: https://www.linkedin.com/pub/anwar-ali-mohamed/54/2bb/57b

A primer on optimization using solvers

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