Prepared by Mr. Prabhu Assistant Professor, Mechanical Department

1. Operation and Planning Control U7MEA37 Preparedby Mr. Prabhu Assistant Professor, Mechanical Department VelTech Dr.RR & Dr.SR Technical University

2. Unit I Linear programming

3. Introduction to Operations Research • Operations research/management science • Winston: “a scientific approach to decision making, which seeks to determine how best to design and operate a system, usually under conditions requiring the allocation of scarce resources.” • Kimball & Morse: “a scientific method of providing executive departments with a quantitative basis for decisions regarding the operations under their control.”

4. Introduction to Operations Research • Provides rational basis for decision making • Solves the type of complex problems that turn up in the modern business environment • Builds mathematical and computer models of organizational systems composed of people, machines, and procedures • Uses analytical and numerical techniques to make predictions and decisions based on these models

5. Introduction to Operations Research • Draws upon • engineering, management, mathematics • Closely related to the "decision sciences" • applied mathematics, computer science, economics, industrial engineering and systems engineering

6. Identify the Problem or Opportunity Understand the System Formulate a Mathematical Model Verify the Model Select the Best Alternative Present the Results of the Analysis Implement and Evaluate Methodology of Operations Research*The Seven Steps to a Good OR Analysis

7. Identify the Problem or Opportunity Understand the System Formulate a Mathematical Model Verify the Model Select the Best Alternative Present the Results of the Analysis Implement and Evaluate Methodology of Operations Research*The Seven Steps to a Good OR Analysis • What are the objectives? • Is the proposed problem too narrow? • Is it too broad?

8. Identify the Problem or Opportunity Understand the System Formulate a Mathematical Model Verify the Model Select the Best Alternative Present the Results of the Analysis Implement and Evaluate Methodology of Operations Research*The Seven Steps to a Good OR Analysis • What data should be collected? • How will data be collected? • How do different components of the system interact with each other?

9. Identify the Problem or Opportunity Understand the System Formulate a Mathematical Model Verify the Model Select the Best Alternative Present the Results of the Analysis Implement and Evaluate Methodology of Operations Research*The Seven Steps to a Good OR Analysis • What kind of model should be used? • Is the model accurate? • Is the model too complex?

10. Identify the Problem or Opportunity Understand the System Formulate a Mathematical Model Verify the Model Select the Best Alternative Present the Results of the Analysis Implement and Evaluate Methodology of Operations Research*The Seven Steps to a Good OR Analysis • Do outputs match current observations for current inputs? • Are outputs reasonable? • Could the model be erroneous?

11. Identify the Problem or Opportunity Understand the System Formulate a Mathematical Model Verify the Model Select the Best Alternative Present the Results of the Analysis Implement and Evaluate Methodology of Operations Research*The Seven Steps to a Good OR Analysis • What if there are conflicting objectives? • Inherently the most difficult step. • This is where software tools will help us!

12. Identify the Problem or Opportunity Understand the System Formulate a Mathematical Model Verify the Model Select the Best Alternative Present the Results of the Analysis Implement and Evaluate Methodology of Operations Research*The Seven Steps to a Good OR Analysis • Must communicate results in layman’s terms. • System must be user friendly!

13. Identify the Problem or Opportunity Understand the System Formulate a Mathematical Model Verify the Model Select the Best Alternative Present the Results of the Analysis Implement and Evaluate Methodology of Operations Research*The Seven Steps to a Good OR Analysis • Users must be trained on the new system. • System must be observed over time to ensure it works properly.

14. Linear Programming

15. Objectives • Requirements for a linear programming model. • Graphical representation of linear models. • Linear programming results: • Unique optimal solution • Alternate optimal solutions • Unbounded models • Infeasible models • Extreme point principle.

16. Objectives - continued • Sensitivity analysis concepts: • Reduced costs • Range of optimality--LIGHTLY • Shadow prices • Range of feasibility--LIGHTLY • Complementary slackness • Added constraints / variables • Computer solution of linear programming models • WINQSB • EXCEL • LINDO

17. 3.1 Introduction to Linear Programming • A Linear Programming model seeks to maximize or minimize a linear function, subject to a set of linear constraints. • The linear model consists of the following components: • A set of decision variables. • An objective function. • A set of constraints. • SHOW FORMAT

18. The Importance of Linear Programming • Many real static problems lend themselves to linear programming formulations. • Many real problems can be approximated by linear models. • The output generated by linear programs provides useful “what’s best” and “what-if” information.

19. Assumptions of Linear Programming • The decision variables are continuous or divisible, meaning that 3.333 eggs or 4.266 airplanes is an acceptable solution • The parameters are known with certainty • The objective function and constraints exhibit constant returns to scale (i.e., linearity) • There are no interactions between decision variables

20. Methodology of Linear Programming Determine and define the decision variables Formulate an objective function verbal characterization Mathematical characterization Formulate each constraint

21. MODEL FORMULATION • Decisions variables: • X1 = Production level of Space Rays (in dozens per week). • X2 = Production level of Zappers (in dozens per week). • Objective Function: • Weekly profit, to be maximized

22. The Objective Function Each dozen Space Rays realizes $8 in profit. Total profit from Space Rays is 8X1. Each dozen Zappers realizes $5 in profit. Total profit from Zappers is 5X2. The total profit contributions of both is 8X1 + 5X2 (The profit contributions are additive because of the linearity assumption)

23. The Linear Programming Model Max 8X1 + 5X2 (Weekly profit) subject to 2X1 + 1X2 < = 1200 (Plastic) 3X1 + 4X2 < = 2400 (Production Time) X1 + X2 < = 800 (Total production) X1 - X2 < = 450 (Mix) Xj> = 0, j = 1,2 (Nonnegativity)

24. 3.4 The Set of Feasible Solutions for Linear Programs The set of all points that satisfy all the constraints of the model is called a FEASIBLE REGION

25. Using a graphical presentation we can represent all the constraints, the objective function, and the three types of feasible points.

26. The plastic constraint: 2X1+X2<=1200 The Plastic constraint Productionmix constraint: X1-X2<=450 X2 1200 Total production constraint: X1+X2<=800 Infeasible 600 Feasible Production Time 3X1+4X2<=2400 X1 600 800 • Interior points. • There are three types of feasible points • Boundary points. Extreme points.

27. Linear Programming- Simplex method “...finding the maximum or minimum of linear functions in which many variables are subject to constraints.” (dictionary.com)‏ A linear program is a “problem that requires the minimization of a linear form subject to linear constraints...” (Dantzig vii)‏

28. Important Note Linear programming requires linear inequalities In other words, first degree inequalities only! Good: ax + by + cz < 3 Bad: ax2 + log2y > 7

29. Lets look at an example... • Farm that produces Apples (x) and Oranges (y)‏ • Each crop needs land, fertilizer, and time. • 6 acres of land: 3x + y < 6 • 6 tons of fertilizer: 2x + 3y < 6 • 8 hour work day: x + 5y < 8 • Apples sell for twice as much as oranges • We want to maximize profit (z): 2x + y = z • We can't produce negative: x > 0, y > 0

30. Traditional Method Graph the inequalities Look at the line we're trying to maximize. x = 1.71 y = .86 z = 4.29

31. Problems... More variables? Cannot eyeball the answer?

32. Simplex Method George B. Dantzig in 1951 Need to convert equations Slack variables

33. Performing the Conversion -z + 2x + y = 0 (Objective Equation)‏ s1 + x + 5y = 8 s2 + 2x + 3y = 6 s3 + 3x + y = 6 Initial feasible solution

34. More definitions Non-basic: x, y Basic variables: s1, s2, s3, z Current Solution: Set non-basic variables to 0 -z + 2x + y = 0 => z = 0 Valid, but not good!

35. Next step... Select a non-basic variable -z + 2x + 1y = 0 x has the higher coefficient Select a basic variable s1 + 1x + 5y = 8 1/8 s2 + 2x + 3y = 6 2/6 s3 + 3x + y = 6 3/6 3/6 is the highest, use equation with s3

36. New set of equations Solve for x x = 2 - (1/3)s3 -(1/3)y Substitute in to other equations to get... -z – (2/3)s3 +(1/3)y = -4 s1– (1/3)s3 + (14/3)y = 6 s2 – (2/3)s3 +(7/3)y = 2 x + (1/3)s3 +(1/3)y = 2

37. Redefine everything... Update variables Non-Basic: s3 and y Basic: s1, s2, z, and x Current Solution: -z – (2/3)s3 +(1/3)y = -4 => z = 4 x + (1/3)s3 +(1/3)y = 2 => x = 2 y = 0 Better, but not quite there.

38. Do it again! Repeat this process Stop repeating when the coefficients in the objective equation are all negative.

39. Improvements Different kinds of inequalities Minimized instead of maximized L. G. Kachian algorithm proved polynomial

40. Artificial Variable Technique (The Big-M Method)

41. Big-M Method of solving LPP The Big-M method of handling instances with artificial variables is the “commonsense approach”. Essentially, the notion is to make the artificial variables, through their coefficients in the objective function, so costly or unprofitable that any feasible solution to the real problem would be preferred....unless the original instance possessed no feasible solutions at all. But this means that we need to assign, in the objective function, coefficients to the artificial variables that are either very small (maximization problem) or very large (minimization problem); whatever this value,let us call it Big M. In fact, this notion is an old trick in optimization in general; we simply associate a penalty value with variables that we do not want to be part of an ultimate solution(unless such an outcome Is unavoidable).

42. Indeed, the penalty is so costly that unless any of the respective variables' inclusion is warranted algorithmically,such variables will never be part of any feasible solution.This method removes artificial variables from the basis. Here, we assign a large undesirable (unacceptable penalty) coefficients to artificial variables from the objective function point of view. If the objective function (Z) is to be minimized, then a very large positive price (penalty, M) is assigned to each artificial variable and if Z is to be minimized, then a very large negative price is to be assigned. The penalty will be designated by +M for minimization problem and by –M for a maximization problem and also M>0.

43. Example: Minimize Z= 600X1+500X2subject to constraints,2X1+ X2 >or= 80 X1+2X2 >or= 60 and X1,X2 >or= 0Step1: Convert the LP problem into a system of linear equations.We do this by rewriting the constraint inequalities as equations by subtracting new “surplus & artificial variables" and assigning them zero & +M coefficientsrespectively in the objective function as shown below.So the Objective Function would be: Z=600X1+500X2+0.S1+0.S2+MA1+MA2subject to constraints, 2X1+ X2-S1+A1 = 80 X1+2X2-S2+A2 = 60 X1,X2,S1,S2,A1,A2 >or= 0

44. Step 2: Obtain a Basic Solution to the problem.We do this by putting the decision variables X1=X2=S1=S2=0,so that A1= 80 and A2=60. These are the initial values of artificial variables.Step 3: Form the Initial Tableau as shown.

45. It is clear from the tableau that X2 will enter and A2 will leave the basis. Hence 2 is the key element in pivotal column. Now,the new row operations are as follows:R2(New) = R2(Old)/2R1(New) = R1(Old) - 1*R2(New)

46. It is clear from the tableau that X1 will enter and A1 will leave the basis. Hence 2 is the key element in pivotal column. Now,the new row operations are as follows:R1(New) = R1(Old)*2/3R2(New) = R2(Old) – (1/2)*R1(New)

47. Since all the values of (Cj-Zj)are either zero or positive and also both the artificial variables have been removed, an optimum solution has been arrived at with X1=100/3 , X2=40/3 and Z=80,000/3.

48. Unit IIDynamic Programming Characteristics and Examples

49. Overview • What is dynamic programming? • Examples • Applications

50. What is Dynamic Programming? • Design technique • ‘optimization’ problems (sequence of related decisions) • Programming does not mean ‘coding’ in this context, it means ‘solve by making a chart’- or ‘using an array to save intermediate steps”. Some books call this ‘memoization’ (see below) • Similar to Divide and Conquer BUT subproblem solutions are SAVED and NEVER recomputed • Principal of optimality: the optimal solution to the problem contains optimal solutions to the subproblems (Is this true for EVERYTHING?)