1. operation research model types, problems and solutions 2/21/2019 BY: Mequanint Birhan

2. Introduction  Analytical Modeling/M anagement Science/OR was developed & used during world war II, when the British military asked scientists to solve military problems.  Decision making for complex systems are very complicated and is out of human’s mind capability to solve problems with many variables.  Operations Research and optimization methods try to find “The Best” solution for a problem.

3. Cont.… Operations research (OR) is a discipline explicitly devoted to aiding decision makers in scarce resource. A model is always an abstraction that is of necessity simpler than the real situation. Elements that are irrelevant or unimportant to the problem are to be ignored, hopefully leaving sufficient detail so that the solution obtained with the model has value with regard to the original problem.

4. Cont.… A typical mathematical program consists of a single objective function, representing either a profit to be maximized or a cost to be minimized, and a set of constraints that circumscribe the decision variables. A Countless real-world applications have been successfully modeled and solved using linear programming techniques. In the case of a linear program (LP) the objective function and constraints are all linear functions of the decision variables.

5. Cont… The founder of the field is George B. Dantzig around 1947 who invented Simplex method for solving Linear Programming (LP) problems. His first paper entitled programming in linear structure. Simplex method also published in 1949 The work of Kantorovich(1939) was published in1959 Koopmans coined the term linear programming in 1948

6. Cont.… There are 3 main parts that forms an optimization problem: Decision Variables: Variables that represent the decision that can be made. Objective function: Each optimization problem is trying to optimize (maximize/minimize) some goal such as costs, profits, revenue. Constraints: Set of real restricting parameters that are imposed in real life or by the structure of the problem. Limited budget for a project, manpower or resources and Being limited to choose only one option out of many options.

7. Formulation of LP A small manufacturer making two products A and B, two resources R1 and R2 are required to make these products. 1.Each unit of product A requires 1unit of R1 and 3units of R2 2.Each unit of product B requires 1unit of R1 and 2 unit of R2 3.The manufacturer has 5units of R1 and 12units of R2 available.

8. Cont… The manufacturer also makes a profit of 6birr per unit of product A sold and 5birr per unit of product B sold. Manufacturer would like to produce in such a way that the profit is maximize Lets X-number of A unit of product Y-number of B unit of product

9. Cont… Max z = 5x+6y st. x+y ≤ 5 3x+2y ≤ 12 x, y ≥ 0

10. Con… Feasible region is defined by the set of constraints of the problem, which is all the possible points that satisfy the all the constraints. sensitivity analysis duality analysis degenerated solution optimal solution branch and bound algorithm

11. Model classifications 1)Function(predictive, descriptive & normative) 2)Structure(iconic, symbolic & analoji) 3)Dimensionality(2, & multi dimensional) 4)Degree of certainty(certain, conflict,& risk) 5)Time reference(static and dynamic) 6)Degree of generality(general & specific) 7)Degree of closure (closed & opened) 8)Degree of quantification (qualitative/ mental/ verbal & quantitative/statistical/ heuristic/ simulation)

12. Example A firm that assembles computers and computer equipment's is about to start production of two new microcomputers will require assembly time, inspection time and storage space. The amount of each of these resources that can be devoted to the production of the micro computers is limited. The manager of the firm would like to determine the quantity of each micro computers to produce in order to maximize the profit generated by sales of these microcomputers.

13. Cont….

15. Required; I.What is the objective function? II.What are the constraints? III.What are the decision variables? IV.Formulate the mathematical model? V.What is the optimal combination?

16. Solution  Maximization problem  Time and space  Number of type I microcomputers let it be x1  Number of type II microcomputers let it be x2  Max z = 60x1+50x2 St. 4x1+10x2 ≤ 100 2x1+ x2 ≤ 22 3x1+3x2 ≤ 39 X1,x2 ≥0

17. Example 2 A steel company operates two steel mills with different production capacities. Mill I can produce 1000tons/day of AAA steel, 3000 tons/day of AA steel and 5000tons/day of A steel. Mill ‘F’ can produce 2000tons/day of each grade of steel. The company has made a contract with the construction firm to provide 24000tons of AAA steel, 32000 tons of AA steel and 40000tons of A steel. For each of the following costs determine the number of days the company should operate each mill in order to meet the terms of the above contract most economically, the minimum cost and & also what grades of steel would be over produced.

18. Cont… a.The cost of running mill I is $1400/day and mill F is $1200/day b.The cost of running mill I is $1500/day and mill F is $2900/day. Additionally the construction firm wants to deliver in no more than 14 days.

19. 1. Linear programming Graphing and simplex method Steps; 1.Standard form 2.Simultaneous equation to get roots 3.Table 4.Graph 5.Optimum solution 6.Checking optimality

20. 2. Decision theory Under certainty Under uncertainty Under risk Decision tree EMV EML EMIP

21. 3. Non linear programming Quadratic solution When expressions defining the objective function or constraints of an optimization model are not linear, one has a nonlinear programming model.

22. Cont.… There are many different classes of nonlinear programming models. The specific form has much to do with how easily the problem is solve, but in general a nonlinear programming model is much more difficult to solve than a similarly sized linear programming model.

23. 4. Integer programming Ip is concerned with optimization problems in which some of the variables are required to take on discrete values. Rather than allow a variable to assume all real values in a given range, only predetermined discrete values within the range are permitted. In most cases, these values are the integers, giving rise to the name of this class of models. Models with integer variables are very useful. Situations that cannot be modeled by linear programming are easily handled by integer programming. Primary among these involve binary decisions such as yes-no, build-no build or invest-not invest.

24. Cont.. Although one can model a binary decision in linear programming with a variable that ranges between 0 and 1, there is nothing that keeps the solution from obtaining a fractional value such as 0.5, hardly acceptable to a decision maker. Integer programming requires such a variable to be either 0 or 1, but not in-between.

25. 5. Goal programming model positive negative

26. 6. Markov chain Discrete Say a system is observed at regular intervals such as every day or every week. Then the stochastic process can be described by a matrix which gives the probabilities of moving to each state from every other state in one time interval. Assuming this matrix is unchanging with time, the process is called a Discrete Time Markov Chain (DTMC). Computational techniques are available to compute a variety of system measures that can be used to analyze and evaluate a DTMC model.

27. Continuous Here we consider a continuous time stochastic process in which the duration of all state changing activities are exponentially distributed. Time is a continuous parameter. The process satisfies the Markovian property and is called a Continuous Time Markov Chain (CTMC). The process is entirely described by a matrix showing the rate of transition from each state to every other state. The rates are the parameters of the associated exponential distributions. The analytical results are very similar to those of a DTMC.

28. 7. Dynamic programming Dynamic programming (DP) models are represented in a different way than other mathematical programming models. Rather than an objective function and constraints, a DP model describes a process in terms of states, decisions, transitions and returns. The process begins in some initial state where a decision is made. The decision causes a transition to a new state. Based on the starting state, ending state and decision a return is realized. N=1, N=2, N=3, N=n

29. Cont.…. The process continues through a sequence of states until finally a final state is reached. The problem is to find the sequence that maximizes the total return. The models considered here are for discrete decision problems. Although traditional integer programming problems can be solved with DP, the models and methods are most appropriate for situations that are not easily modeled using the constructs of mathematical programming.

30. Cont.… Objectives with very general functional forms may be handled and a global optimal solution is always obtained. The price of this generality is computational effort. Solutions to practical problems are often stymied by the "curse of dimensionally" where the number of states grows exponentially with the number of dimensions of the problem.

31. 8. Queuing model Single server Multiple server

32. 9. Transportation problem TO FIND OPTIMAL SOLUTION North west corner method Least cost method Vogel's approximation method OPTIMALITY TEST Stepping stone method Modified distribution method

33. 10. Assignment problem Hungarian method

34. 11. Forecasting model Qualitative Quantitative

35. 12. Inventory model PURCHASING - without shortage -----with shortage PRODUCTION - without shortage -----with shortage

36. 13. project management CPM and PERT Networking The term network flow program describes a type of model that is a special case of the more general linear program. The class of network flow programs includes such problems as the transportation problem, the assignment problem, the shortest path problem, the maximum flow problem, the pure minimum cost flow problem, and the generalized minimum cost flow problem.

37. Cont.. It is an important class because many aspects of actual situations are readily recognized as networks and the representation of the model is much more compact than the general linear program. When a situation can be entirely modeled as a network, very efficient algorithms exist for the solution of the optimization problem, many times more efficient than linear programming in the utilization of computer time and space resources.

38. 14. Game programming model Agenda:  Introduction  Game development  Game Architecture  Project Introduction  Game development  Game Architecture  Project Introduction

39. 15. Stochastic programming The mathematical programming models, such as linear programming, network flow programming and integer programming generally neglect the effects of uncertainty and assume that the results of decisions are predictable and deterministic. This abstraction of reality allows large and complex decision problems to be modeled and solved using powerful computational methods.

40. Cont.. Stochastic programming explicitly recognizes uncertainty by using random variables for some aspects of the problem. With probability distributions assigned to the random variables, an expression can be written for the expected value of the objective to be optimized. Then a variety of computational methods can be used to maximize or minimize the expected value.

41. The end

42. . T H A N K Y o u