Anwar Ali, D.Eng anwar.ali.mohamed@gmail€¦ · A Primer on Optimization Using Solvers - Anwar Ali 5 Business Intelligence Framework Back in Business, by Ronald K. Klimberg and Virginia

Anwar Ali, D.Eng

[email protected]

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 a 3-day training “Practical Business Optimization Using Solvers” 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

Com

petitive A

dvanta

ge

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 A Primer on Optimization Using Solvers - Anwar Ali 4

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


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


http://www.informs.org/ORMS-Today/Public-Articles/October-Volume-37-Number-5/Back-in-Business



















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

Com

petitive A

dvanta

ge

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


O.R. Leading Edge Techniques Simulation (predictive)


Optimization (prescriptive)


Probability and statistics (predictive)



O.R. Leading Edge Techniques Simulation


Optimization – THIS TRAINING


Probability and statistics



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/


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



How to Get Started


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)


Example 1: Dorian Auto 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


Example 1: Dorian Auto 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 A Primer on Optimization Using Solvers - Anwar Ali 22

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



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



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




Graphical Solution

x (comedy ads)

y (football ads)

4 8 12 16

4

12

16

8

2

6

10

14

2 6 10 14






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



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



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


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




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



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)


Example 2: Diet Problem

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


Example 2: Diet Problem 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

41 A 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

Cost Profit 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



Optimization Modeling with Textbook Examples √



How to Get Started


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) A Primer on Optimization Using Solvers - Anwar Ali 83

Assign gates to planes considering plane type, schedule, domestic/international, airlines Assignment problem


Blending problem A Primer on Optimization Using Solvers - Anwar Ali 85

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 A Primer on Optimization Using Solvers - Anwar Ali 86

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 00 0 1 00 0 0 11 0 0 0

represents the tour (1,2,3,4,1)


TSP Objective Function

𝑚𝑖𝑛 𝑧 =

𝑛

𝑖=1

𝑑𝑖𝑗𝑥𝑖𝑗

𝑛

𝑗=1𝑖≠𝑗


TSP Constraints

Each city must be “entered” exactly once and “exited” exactly once

𝑥𝑖𝑗

𝑛

𝑗=1𝑗≠𝑖

= 1, ∀𝑖 ∈ 𝑁

𝑥𝑖𝑗

𝑛

𝑖=1𝑖≠𝑗

= 1, ∀𝑗 ∈ 𝑁

Subject to:


But it may create sub-tours

Example solution, 𝑥 =

0 1 0 01 0 0 00 0 0 10 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


http://eprints.lse.ac.uk/22747/

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

𝑑𝑖𝑗𝑥𝑖𝑗

𝑛

𝑗=1𝑖≠𝑗

𝑥𝑖𝑗

𝑛

𝑗=1𝑗≠𝑖

= 1, ∀𝑖 ∈ 𝑁

𝑥𝑖𝑗

𝑛

𝑖=1𝑖≠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 √


How to Get Started


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




Formulating and Solving Mathematical Models √

Optimization Applications in Industry √

How to Get Started


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:





Training & Consulting Current training:

3-day “Practical Business Optimization Using Solvers”

If you need help with optimization modeling for your business needs, I will be happy to offer my service

My email: [email protected]

My profile: https://www.linkedin.com/pub/anwar-ali-mohamed/54/2bb/57b

Location: Penang & Kuala Lumpur, Malaysia


mailto:[email protected]

https://www.linkedin.com/pub/anwar-ali-mohamed/54/2bb/57b







Documents

Anwar Ali, D.Eng anwar.ali.mohamed@gmail€¦ · A Primer on Optimization Using Solvers - Anwar Ali 5 Business Intelligence Framework Back in Business, by Ronald K. Klimberg and Virginia