© 2013 OptiRisk India (P) Ltd, All rights reserved
Bala. Padmakumar
Director & CEO
OptiRisk India
Ph: +91 98406 18472/ +91...
1. Operations Research
2. Optimization
3. Linear Programming
4. Case Study (Linear Programming )
5. Summary
2© 2010-13 Opt...
3© 2010-12 OptiRisk India (P) Ltd, All rights reserved
 Analysing Decision Problems,
 Formulating Mathematical Models
 ...
1. Operations Research
2. Optimization
3. Linear Programming
4. Case Study (Linear Programming )
5. Summary
4© 2010-13 Opt...
Optimization is a
decision making process
and a set of related tools that
employ mathematics,
Algorithms, and computer sof...
6© 2010-13 OptiRisk India (P) Ltd, All rights reserved
Route
#
Possible
Routes
Distance(KM)
1 A – B – C 245
2 A – C – B 31...
Wherever decisions are made !Wherever decisions are made !
Where, Decisions require resources for implementation !!Where, ...
FacilityLayoutPlanning
OR Key
Areas
Production
Planning
Allocation
Job,
Machines
Inventory
Optimization
Transportation
(Ro...
SC Network optimization
Locate facilities
Match Supply & Demand
Managing seasonality, and
Reduce carbon footprint.
Benefit...
10© 2010-13 OptiRisk India (P) Ltd, All rights reserved
What-If Analysis
Collaboration
Courtesy: IBM Limited
11© 2010-13 OptiRisk India (P) Ltd, All rights reserved
2 Chilean Forestry firms* Timber Harvesting $20M/yr + 30% fewer trucks
UPS* Air Network Design $40M/yr + 10% fewer planes
...
13© 2013 OptiRisk India (P) Ltd, All rights reserved
14© 2013 OptiRisk India (P) Ltd, All rights reserved
1. Operations Research
2. Optimization
3. Linear Programming
4. Case Study (Linear Programming )
5. Summary
15© 2010-13 Op...
 A Carpenter can produce 2 products:
Chairs and Tables.
 The cost of production of a chair is Rs
50 and it sells for Rs ...
 Chair needs 1 log of wood and 3
steel rods
 Table needs 2 logs of wood and 2
steel rods
 And there are only 6 logs of ...
Brute force
Graphical Method
Mathematical Programming
• Using Excel
• Using ILOG CPLEX
Brute force
Graphical Method
Mathem...
19© 2013 OptiRisk India (P) Ltd, All rights reserved
Chair Table
Number to be
produced
x y
Revenue 60 75
Cost 50 60
Profit...
20© 2013 OptiRisk India (P) Ltd, All rights reserved
Subject to:
Maximize z =
Objective function
Maximize total contributi...
21© 2013 OptiRisk India (P) Ltd, All rights reserved
Chair Table
Costing
Revenue 60 75
Cost 50 60
Resource Requirement
Woo...
3
Steel Constraints
Infeasible
Wood Constraints
62x ≤+ y
142x3 ≤+ y
Y
X
22© 2010-12 OptiRisk India (P) Ltd, All rights res...
23© 2013 OptiRisk India (P) Ltd, All rights reserved
Excel Demo
24© 2013 OptiRisk India (P) Ltd, All rights reserved
Model Code
ResultsDebug
Project Explorer
25© 2013 OptiRisk India (P) Ltd, All rights reserved
ILOG CPLEX Demo
26© 2010-12 OptiRisk India (P) Ltd, All rights reserved
modeling language/systemmodeling language/system
min f (x, y)
g(x,...
1. Operations Research
2. Optimization
3. Linear Programming
4. Case Study (Linear Programming )
5. Summary
27© 2010-13 Op...
28© 2013 OptiRisk India (P) Ltd, All rights reserved
29© 2013 OptiRisk India (P) Ltd, All rights reserved
XYZ Chemical Ltd manufactures three chemicals, namely CH-A, CH-B, and...
30© 2013 OptiRisk India (P) Ltd, All rights reserved
Decision Variables:
= Number of Kgs of chemical p produced each day. ...
31© 2013 OptiRisk India (P) Ltd, All rights reserved
∑=
P
p
pp xv
1
∑=
≤
P
p
spps Cxa
1
,
0≥px for p = 1, …, P=Max num of ...
32© 2013 OptiRisk India (P) Ltd, All rights reserved
Subject to:
Maximize z =
Objective function
Maximize total contributi...
Objective function
Maximize Contribution
33© 2013 OptiRisk India (P) Ltd, All rights reserved
/***************************...
Post Processing
Print Results
Constraint
Capacity Constraint
// Constraints
// 1. Capacity Constraints
subject to {
forall...
Input Data
Parameters
Index
/*********************************************
* OPL 12.5 Data
* Author: OptiRisk
* Creation D...
36© 2013 OptiRisk India (P) Ltd, All rights reserved
// solution (optimal) with objective 10077.5
Maximum Contribution Pos...
1. Operations Research
2. Optimization
3. Linear Programming
4. Case Study (Linear Programming )
5. Summary
37© 2010-13 Op...
38© 2010-13 OptiRisk India (P) Ltd, All rights reserved
__________________________________________________________________________________________________________________________...
40© 2010-13 OptiRisk India (P) Ltd, All rights reserved
Asia Pacific, Africa, Australia &
Middle East:
Europe & America:
No 12, 25th Cross Street
Thiruvalluvar Nagar ,
Thiruvanmi...
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Mathematical Programming Introduction

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  • In simple terms, OR deals with Optimization <<Another slide>> This subject is highly interesting to a few, and extremely boring to many. It is boring on the surface, until you delve deep.
  • << Two clicks>> With respect to planning, what does optimization mean? Optimization is a decision making process. It employs mathematics and algorithms. It can generate better plans, which requires less resources for a given business objective. And, It can come up with plans faster than humans.
  • In a Manufacturing company, manager undertake various planning exercises, Some of which are listed here. [1] – Supply Chain Network Optimization This is a planning exercise, undertaken yearly or once in every few years, to find out the best places to locate plants and warehouses, considering Various cost, supply and demand patterns, so that the Overall production and distribution cost is minimized. [2] – Sourcing selection Given a demand pattern and production target, and various suppliers location, cost, transportation, Taxes, and service level, whom to pick and how much To order, and in what lot sizes. [3] – Production Planning Given a production target and resources, Come up with a production plan and schedule that leads to overall least production cost. [4] – Revenue / Yield management Based on flucation in demand, competition, and The inventory build-up, identify best prices, discounts, and Commissions to increase the sales and profit. [5] – Transportation scheduling
  • Add this  Conference on “ Warehousing & Cold Chain Infrastructure” 28 June 2013, Hotel Hilton, Chennai
  • Mathematical Programming Introduction

    1. 1. © 2013 OptiRisk India (P) Ltd, All rights reserved Bala. Padmakumar Director & CEO OptiRisk India Ph: +91 98406 18472/ +91 44 4501 8472 Web: http://www.optiriskindia.com Email : optimize@optiriskindia.com
    2. 2. 1. Operations Research 2. Optimization 3. Linear Programming 4. Case Study (Linear Programming ) 5. Summary 2© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    3. 3. 3© 2010-12 OptiRisk India (P) Ltd, All rights reserved  Analysing Decision Problems,  Formulating Mathematical Models  Coming-up the best possible (optimal) solutions  Testing the solution for sensitivity, “what-if”, etc OR deals with Optimization
    4. 4. 1. Operations Research 2. Optimization 3. Linear Programming 4. Case Study (Linear Programming ) 5. Summary 4© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    5. 5. Optimization is a decision making process and a set of related tools that employ mathematics, Algorithms, and computer software not only to sort and organize data, but to use that data to make Recommendations faster and better than humans can. - The Optimization Edge, Steve Sashihara 5© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    6. 6. 6© 2010-13 OptiRisk India (P) Ltd, All rights reserved Route # Possible Routes Distance(KM) 1 A – B – C 245 2 A – C – B 315 3 B – A – C 370 4 B – C – A 315 5 C – A – B 370 6 C – B – A 245 Example : Find the shortest path for delivery Deliveries Possible Routes Time to process manually 3 3! = 6 10 10! = 3,628,800 13 13! = 6,227,020,800 3 centuries 20 20! = 11 billion centuries 10 seconds to calculate each route manually A B C Warehouse 80 35 115 60 70 105 Best Routes
    7. 7. Wherever decisions are made !Wherever decisions are made ! Where, Decisions require resources for implementation !!Where, Decisions require resources for implementation !! Capital AllocateCapital Allocate People Acquire, schedule, assign, TrainPeople Acquire, schedule, assign, Train Equipment Acquire, schedule, LocateEquipment Acquire, schedule, Locate Facilities Locate, scheduleFacilities Locate, schedule Vehicles Acquire, route, scheduleVehicles Acquire, route, schedule Raw Material Acquire, assignRaw Material Acquire, assign Time Allocate, ScheduleTime Allocate, Schedule 7© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    8. 8. FacilityLayoutPlanning OR Key Areas Production Planning Allocation Job, Machines Inventory Optimization Transportation (Routing, Scheduling) Network Optimization 8© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    9. 9. SC Network optimization Locate facilities Match Supply & Demand Managing seasonality, and Reduce carbon footprint. Benefits: 5-15% reduction in SC costs Better service level Sourcing Selection Select Suppliers Determine Order Quantities Benefits : Enhanced S&OP capability, 2-5% reduction in mfg costs Transportation Scheduling Fleet Selection Route Plan / Schedule Benefits : 10-30% reduction in Transport cost Increased on-time deliveries Production planning Optimal production schedule Meet plant floor constraints. Benefits : Improved throughput Reduced costs, Reduced inventory Leaner plants Operational Inventory Management Order quantity Order time Supplier identification Benefits : 10-30% reduction in inventory Improved service level Demand Planning Choose best price Indentify Optimal Commission Benefits : Higher sales volume Better Margins 9© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    10. 10. 10© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    11. 11. What-If Analysis Collaboration Courtesy: IBM Limited 11© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    12. 12. 2 Chilean Forestry firms* Timber Harvesting $20M/yr + 30% fewer trucks UPS* Air Network Design $40M/yr + 10% fewer planes South African Defense* Force/Equip Planning $1.1B/yr Motorola* Procurement Mgmt $100M-150M/yr Samsung Electronics* Semiconductor Mfg 50% reduction in cycle times SNCF (French RR)* Scheduling & Pricing $16M/yr rev + 2% lower op ex Continental Airlines* Crew Re-scheduling $40M/yr AT&T* Network Recovery 35% reduction spare capacity Grant Mayo van Otterloo* Portfolio Optimization $4M/yr Indeval* Securities Trade Settlement $150M/yr financing costs Midwest ISO* Energy Grid Management $2.1B to $3B savings over 4 years *Franz Edelman Competition Finalists, Science of Better, http://www.scienceofbetter.org , Published Case Studies 12© 2010-13 OptiRisk India (P) Ltd, All rights reserved 12© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    13. 13. 13© 2013 OptiRisk India (P) Ltd, All rights reserved
    14. 14. 14© 2013 OptiRisk India (P) Ltd, All rights reserved
    15. 15. 1. Operations Research 2. Optimization 3. Linear Programming 4. Case Study (Linear Programming ) 5. Summary 15© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    16. 16.  A Carpenter can produce 2 products: Chairs and Tables.  The cost of production of a chair is Rs 50 and it sells for Rs 60  The cost of production of a table is Rs 60 and it sells for Rs 75  The Carpenter want to know how much he has to produced so that he can maximize his profit. A classic decision DilemmaA classic decision Dilemma 16© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    17. 17.  Chair needs 1 log of wood and 3 steel rods  Table needs 2 logs of wood and 2 steel rods  And there are only 6 logs of wood and 14 steel rods available ... Cont 17© 2013 OptiRisk India (P) Ltd, All rights reserved
    18. 18. Brute force Graphical Method Mathematical Programming • Using Excel • Using ILOG CPLEX Brute force Graphical Method Mathematical Programming • Using Excel • Using ILOG CPLEX 18© 2013 OptiRisk India (P) Ltd, All rights reserved
    19. 19. 19© 2013 OptiRisk India (P) Ltd, All rights reserved Chair Table Number to be produced x y Revenue 60 75 Cost 50 60 Profit 10 15 Chair Table Max Available Number to be produced x y Wood Required 1 2 6 Steel Required 3 2 14 Total Profit = yx 1510 + 62 ≤+ yx 1423 ≤+ yx Objective FunctionObjective Function ConstraintsConstraints Decision VariablesDecision Variables Number of chairs x Number of Tables y
    20. 20. 20© 2013 OptiRisk India (P) Ltd, All rights reserved Subject to: Maximize z = Objective function Maximize total contribution(profit) Constraint Resource Availability Constraints Non negativity Constraint Production quantity must be non negative 62 ≤+ yx 1423 ≤+ yx 0,0 ≥≥ yx yx 1510 +
    21. 21. 21© 2013 OptiRisk India (P) Ltd, All rights reserved Chair Table Costing Revenue 60 75 Cost 50 60 Resource Requirement Wood Req 1 2 Steel Reg 3 2 S. No # Chairs # Tables Wood Req Steel Reg Wood Avail Steel Avail Revenue Cost Profit Solution Optimal 1 0 0 0 0 6 14 0 0 0 Feasible 2 0 1 2 2 6 14 75 60 15 Feasible 3 0 2 4 4 6 14 150 120 30 Feasible 4 0 3 6 6 6 14 225 180 45 Feasible 5 0 4 8 8 6 14 300 240 60 Infeasible 6 1 0 1 3 6 14 60 50 10 Feasible 7 1 1 3 5 6 14 135 110 25 Feasible 8 1 2 5 7 6 14 210 170 40 Feasible 9 1 3 7 9 6 14 285 230 55 Infeasible 10 2 0 2 6 6 14 120 100 20 Feasible 11 2 1 4 8 6 14 195 160 35 Feasible 12 2 2 6 10 6 14 270 220 50 Feasible 13 2 3 8 12 6 14 345 280 65 Infeasible 14 3 0 3 9 6 14 180 150 30 Feasible 15 3 1 5 11 6 14 255 210 45 Feasible 16 3 2 7 13 6 14 330 270 60 Infeasible 17 4 0 4 12 6 14 240 200 40 Feasible 18 4 1 6 14 6 14 315 260 55 Feasible YES 19 4 2 8 16 6 14 390 320 70 Infeasible 20 5 0 5 15 6 14 300 250 50 Infeasible
    22. 22. 3 Steel Constraints Infeasible Wood Constraints 62x ≤+ y 142x3 ≤+ y Y X 22© 2010-12 OptiRisk India (P) Ltd, All rights reserved 7 4.6 0 6 y15x10max + Objective Function Feasible X – No of Chairs Y – No of Tables X – No of Chairs Y – No of Tables Optimal Point
    23. 23. 23© 2013 OptiRisk India (P) Ltd, All rights reserved Excel Demo
    24. 24. 24© 2013 OptiRisk India (P) Ltd, All rights reserved Model Code ResultsDebug Project Explorer
    25. 25. 25© 2013 OptiRisk India (P) Ltd, All rights reserved ILOG CPLEX Demo
    26. 26. 26© 2010-12 OptiRisk India (P) Ltd, All rights reserved modeling language/systemmodeling language/system min f (x, y) g(x, y) = 0 ...model building... Data Know How Experience Variables Constraints Objective Function Solver: CPLEX, FortMP, Gurobi, etc Feasibility Optimality High Quality Low Costs h(x, y) 0>_ Model type: LP, IP, MILP, Modeling Language: OPL, AMPL, GAMS
    27. 27. 1. Operations Research 2. Optimization 3. Linear Programming 4. Case Study (Linear Programming ) 5. Summary 27© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    28. 28. 28© 2013 OptiRisk India (P) Ltd, All rights reserved
    29. 29. 29© 2013 OptiRisk India (P) Ltd, All rights reserved XYZ Chemical Ltd manufactures three chemicals, namely CH-A, CH-B, and CH-C. All three chemicals have to go through three processes namely MIXING, CONVERSION, and BAGGING. Each process has finite daily capacity, which are given in the below table. Each chemical requires different processing time, which are also given in the below table. Process Production minutes per Kg of product Capacity of production in minutes per dayCH-A CH-B CH-C MIXING CONVERSION BAGGING 2 3 1 2 1 4 1 1 0 400 450 390 Contribution /Kg Rs. 30 Rs. 55 Rs. 23 The company can sell whatever it produces (there is no constraint on demand). The company’s management wants to find out how much of each chemical it should produce to maximize the contribution (profit).
    30. 30. 30© 2013 OptiRisk India (P) Ltd, All rights reserved Decision Variables: = Number of Kgs of chemical p produced each day. p = 1, …, P = Number of minutes per day available at process s. ConstraintsConstraints px Constraints: sC = Number of minutes required by product p at production stage s. s = 1, …, S   = Profit per unit of product p Parameters: psa , pv Objective Function: z = Maximize total daily profit
    31. 31. 31© 2013 OptiRisk India (P) Ltd, All rights reserved ∑= P p pp xv 1 ∑= ≤ P p spps Cxa 1 , 0≥px for p = 1, …, P=Max num of products for s = 1, …, S=Max num of processes Subject to: Maximize z = Objective function Maximize total contribution (profit) Constraint Process Capacity Constraints Non negativity Constraint Production quantity must be non negative
    32. 32. 32© 2013 OptiRisk India (P) Ltd, All rights reserved Subject to: Maximize z = Objective function Maximize total contribution(profit) Constraint Process Capacity Constraints Non negativity Constraint Production quantity must be non negative 321 22 xxx ++ 3213 xxx ++ 21 4xx + 0,0,0 321 ≥≥≥ xxx 321 235530 xxx ++ ≤ 400 ≤ 450 ≤ 390
    33. 33. Objective function Maximize Contribution 33© 2013 OptiRisk India (P) Ltd, All rights reserved /********************************************* * OPL 12.5 Model * Author: OptiRisk * Creation Date: 27-Aug-2013 at 10:02:32 AM *********************************************/ // Input - indices {string} Chemicals = ...; {string} Processes = ...; // Input parameters float Capacity[Processes] = ...; float ProcessTime[Chemicals][Processes] = ...; float Contribution[Chemicals] = ...; // Decision Variable // Production Quantity of each Chemical dvar float+ ChemicalQty[Chemicals]; // Objective Function // Maximize total contribution maximize sum( p in Chemicals ) Contribution[p] * ChemicalQty[p]; Input Parameters Decision Variables
    34. 34. Post Processing Print Results Constraint Capacity Constraint // Constraints // 1. Capacity Constraints subject to { forall( s in Processes ) ctCapacity: sum( p in Chemicals ) ProcessTime[p][s] * ChemicalQty[p] <= Capacity[s]; } // Post processing to print results. execute { writeln("Maximum Contribution Possible = Rs. ", cplex.getObjValue()); writeln("n----- RESULT: Optimal Product Mix---------"); // Print the optimal product quantities. for( var p in Chemicals){ writeln(p, " ---> ", ChemicalQty[p], " Kgs"); } writeln("n---- END: Optimal Product Mix-------------"); } 34© 2013 OptiRisk India (P) Ltd, All rights reserved
    35. 35. Input Data Parameters Index /********************************************* * OPL 12.5 Data * Author: OptiRisk * Creation Date: 27-Aug-2013 at 10:02:32 AM *********************************************/ Chemicals = {"CH-A","CH-B","CH-C"}; Processes = {"MIXING", "CONVERSION", "BAGGING"}; // Capacity in minutes of each process Capacity = [400, 450, 390]; // Processing time for each chemical, for every process ProcessTime = [[2,3,1],[2,1,4],[1,1,0]]; // Contribution (profit) for each chemical Contribution = [30, 55, 23]; 35© 2013 OptiRisk India (P) Ltd, All rights reserved
    36. 36. 36© 2013 OptiRisk India (P) Ltd, All rights reserved // solution (optimal) with objective 10077.5 Maximum Contribution Possible = Rs. 10077.5 ----------- RESULT: Optimal Product Mix--------- CH-A ---> 0 Kgs CH-B ---> 97.5 Kgs CH-C ---> 205 Kgs ----------- END: Optimal Product Mix------------- // solution (optimal) with objective 10077.5 Maximum Contribution Possible = Rs. 10077.5 ----------- RESULT: Optimal Product Mix--------- CH-A ---> 0 Kgs CH-B ---> 97.5 Kgs CH-C ---> 205 Kgs ----------- END: Optimal Product Mix-------------
    37. 37. 1. Operations Research 2. Optimization 3. Linear Programming 4. Case Study (Linear Programming ) 5. Summary 37© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    38. 38. 38© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    39. 39. ___________________________________________________________________________________________________________________________ 39© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    40. 40. 40© 2010-13 OptiRisk India (P) Ltd, All rights reserved
    41. 41. Asia Pacific, Africa, Australia & Middle East: Europe & America: No 12, 25th Cross Street Thiruvalluvar Nagar , Thiruvanmiyur, Chennai –600041, India OptiRisk R&D House, One Oxford, Uxbridge, Middlesex, UB9 4DA, United Kingdom Bala. Padmakumar Ph: +91 98406 18472 / +91 44 4501 8472 Email: optimize@optiriskindia.com Web: http://www.optiriskindia.com/ 41© 2013 OptiRisk India (P) Ltd, All rights reserved Development Center: Corporate Head Quarters: No 1/4 , Justice Dwarkanath Road, Kolkata - 700020, India.

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