CDG2A/CDZ4A/CDC4A/ MBT4A ELEMENTS OF OPERATIONS RESEARCH Unit : I - V
UNIT I Introduction Operations Research Meaning and definition. Origin and History Characteristics and Scope Techniques in Operations research Applications and Limitations CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 2
INTRODUCTION Operations Research (OR) is a discipline that helps to make better decisions in complex scenarios by the application of a set of advanced analytical methods. It couples theories, results and theorems of mathematics, statistics and probability with its own theories and algorithms for problem solving. Applications of OR techniques spread over various fields in engineering, management and public systems. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 3
DEFINITIONS It is the application of scientific methods, techniques and tools to problems involving the operations of a system so as to provide those in the control of the system with optimum solutions to the problems. Operation Research is a tool for taking decisions which searches for the optimum results in parity with the overall objectives and constraints of the organisation. O.R. is a scientific method of providing executive department with a quantitative basis of decisions regarding the operations under their control. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 4
ORIGIN AND HISTORY The ambiguous term operations research (O.R.) was coined during World War II, when the British military management called upon a group of scientists together to apply a scientific approach in the study of military operations to win the battle. The main objective was to allocate scarce resources in an effective manner to various military operations and to the activities within each operation. The effectiveness of operations research in military spread interest in it to other government departments and industry. Due to the availability of faster and flexible computing facilities, it is now widely used in military, business, industry, transportation, public health, crime investigation, etc. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 5
CHARACTERISTICS CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 6
Creating a Model : OR first makes a model. A model is a logical representation of a problem. It shows the relationships between the different variables in the problem. It is just like a mathematical formula. For e.g. Assets - Liabilities = Capital + Accumulated Reserves. Shows Important Variables : OR shows the variables which are important for solving the problem. Many of the variables are uncontrollable. Symbolises the Model : The OR model, its variables and goals are converted into mathematical symbols. These symbols can be easily identified, and they can be used for calculation. Achieving the Goal : The main goal of OR is to select the best solution for solving the problem. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 7
Quantifying the Model : All variables in the OR model are quantified. That is, they are converted into numbers. This is because only quantified data can be put into the model to get results. Using Mathematical Devices : Data is supplemented with mathematical devices to narrow down the margin of error. Use of Computer : The main focus is on decision-making and problem solving. For this purpose computers are widely used. Interdisciplinary : OR is interdisciplinary, because it uses techniques from economics, mathematics, chemistry, physics, etc. Highest Efficiency : The main aim of OR is to make decisions and solve problems. This results in the highest possible efficiency. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 8
METHODOLOGY CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 9
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APPLICATIONS OF OR A few applications of operations research are (a) Location factories and warehouses to minimize transportation costs (b) Work allocation to machines for minimizing production time and costs (c) Inventory problems (d) Material handling (e) Dealing with waiting times (f) Equipment replacements (g) Dividing advertising budget (h) Establishing equitable bonus systems (i) Routing of tankers (j) Traffic control (k) Petrochemical mixes (l) Municipal and hospital administration (m) Marketing, etc. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 11
SCOPE OF OR Operation Research is today recognized as an Applied Science concerned with a large number of diverse human activities. This techniques can be widely used in almost all fields. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 12
1. Finance Budgeting and investments Cash flow analysis long range capital requirements investment portfolios dividend policies, etc. Credit policies credit risks and delinquent account proceduresclaim and complaint procedures. 2. Purchasing Procurement and Exploration. Determining the quantity and timing of purchase of raw materials machinery etc. Rules for buying and supplies under varying prices Equipment replacement policies Determination of quantities and timings of purchases. Strategies for exploration and exploitation of new material sources. 3. Production Management (i) Physical distribution. Location and size of warehouses distribution centers retail outets etc. Distribution policy. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 13
(ii) Manufacturing and facility planning. Production scheduling and sequencing Project scheduling and allocation or resources. Number and location of factories were houses hospitals and their sizes. Determining the optimum production mix. (iii) Manufacturing Maintenance policies and preventive maintenance Maintenance crew sizes. 4. Marketing Management Product selection timing competitive actions. Advertising strategy and choice of different media of advertising Number of salesman frequency of calling of account etc Effectiveness of market research. Size of the stock to meet the future demand CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 14
5. Personal Management Recruitment policies and assignment of jobs. Selection of suitable personnel on minimum salary. Mixes of age and skills. Establishing equitable bonus systems. 6. Research and Development. Determination of areas of concentration of research and development Reliability and evaluation of alternative designs Control of developed projects Co-ordination of multiple research projects Determination of time cost requirements CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 15
ADVANTAGES Better Systems: Often, an O.R. approach is initiated to analyze a particular problem of decision making such as best location for factories, whether to open a new warehouse, etc. It also helps in selecting economical means of transportation, jobs sequencing, production scheduling, replacement of old machinery, etc. Better Control: The management of large organizations recognize that it is a difficult and costly affair to provide continuous executive supervision to every routine work. An O.R. approach may provide the executive with an analytical and quantitative basis to identify the problem area. The most frequently adopted applications in this category deal with production scheduling and inventory replenishment. Better Decisions: O.R. models help in improved decision making and reduce the risk of making erroneous decisions. O.R. approach gives the executive an improved insight into how he makes his decisions. Better Co-ordination: An operations-research-oriented planning model helps in co-ordinating different divisions of a company. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 16
LIMITATIONS Dependence on an Electronic Computer: O.R. techniques try to find out an optimal solution taking into account all the factors. In the modern society, these factors are enormous and expressing them in quantity require voluminous calculations that can only be handled by computers. Non-Quantifiable Factors: O.R provide a solution only when all the elements related to a problem can be quantified. All relevant variables do not lend themselves to quantification. Factors that cannot be quantified find no place in O.R. models. Distance between Manager and Operations Researcher: O.R. being specialist's job requires a mathematician or a statistician, who might not be aware of the business problems. Similarly, a manager fails to understand the complex working of O.R. Thus, there is a gap between the two. Money and Time Costs: When the basic data are subjected to frequent changes, incorporating them into the O.R. models is a costly affair. Moreover, a fairly good solution at present may be more desirable than a perfect O.R. solution available after sometime. Implementation: Implementation of decisions is a delicate task. It must take into account the complexities of human relations and behavior. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 17
UNIT II LINEAR PROGRAMMING PROBLEM ASSUMPTIONS APPLICATIONS FORMULATION OF LPP ADVANTAGES LIMITATIONS CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 18
LINEAR PROGRAMMING Linear programming (LP, also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming. Linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 19
STRUCTURE OF A LP MODEL The general structure of the Linear Programming model essentially consists of three components. i) The activities (variables) and their relationships ii) The objective function and iii) The constraints CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 20
General Mathematical Model of an LPP Let: X 1, X 2, X 3,, X n = decision variables Z = Objective function or linear function Requirement: Maximization of the linear function Z. Z = c 1 X 1 + c 2 X 2 + c 3 X 3 + + c n X n..eq (1) subject to the following constraints: ecstants. where a ij, b i, and c j are given constants. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 21
THE LP MODEL The linear programming model can be written in more efficient notation as: The decision variables, x I, x 2,..., x n, represent levels of n competing activities. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 22
ASSUMPTIONS CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 23
ASSUMPTIONS Proportionality: The basic assumption underlying the linear programming is that any change in the constraint inequalities will have the proportional change in the objective function. Additivity: The assumption of additivity asserts that the total profit of the objective function is determined by the sum of profit contributed by each product separately. Similarly, the total amount of resources used is determined by the sum of resources used by each product separately. This implies, there is no interaction between the decision variables. Continuity: Another assumption of linear programming is that the decision variables are continuous. This means a combination of outputs can be used with the fractional values along with the integer values. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 24
ASSUMPTIONS Certainty: Another underlying assumption of linear programming is a certainty, i.e. the parameters of objective function coefficients and the coefficients of constraint inequalities is known with certainty. Such as profit per unit of product, availability of material and labor per unit, requirement of material and labor per unit are known and is given in the linear programming problem. Finite Choices: This assumption implies that the decision maker has certain choices, and the decision variables assume non-negative values. The non-negative assumption is true in the sense, the output in the production problem can not be negative. Thus, this assumption is considered feasible. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 25
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FORMULATION OF LP Steps Involved: Determine the objective of the problem and describe it by a criterion function in terms of the decision variables. Find out the constraints. Do the analysis which should lead to the selection of values for the decision variables that optimize the criterion function while satisfying all the constraints imposed on the problem. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 27
EXAMPLE Universal Corporation manufactures two products- P 1 and P 2. The profit per unit of the two products is Rs. 50 and Rs. 60 respectively. Both the products require processing in three machines. The following table indicates the available machine hours per week and the time required on each machine for one unit of P 1 and P 2. Formulate this product mix problem in the linear programming form. Machine Product Available Time P 1 P 2 (in machine hours per week) 1 2 1 300 2 3 4 509 3 4 7 812 Profit Rs. 50 Rs. 60 CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 28
Solution. Let x 1 and x 2 be the amounts manufactured of products P 1 and P 2 respectively. The objective here is to maximize the profit, which is given by the linear function Maximize z = 50x 1 + 60x 2 Since one unit of product P 1 requires two hours of processing in machine 1, while the corresponding requirement of P 2 is one hour, the first constraint can be expressed as 2x 1 + x 2 300 Similarly, constraints corresponding to machine 2 and machine 3 are 3x 1 + 4x 2 509 4x 1 + 7x 2 812 In addition, there cannot be any negative production that may be stated algebraically as x 1 0, x 2 0 CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 29
The problem can now be stated in the standard linear programming form as Maximize z = 50x 1 + 60x 2 subject to 2x 1 + x 2 300 3x 1 + 4x 2 509 4x 1 + 7x 2 812 x 1 0, x 2 0 CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 30
For another example on diet mixing problem follow the link https://youtu.be/96wbc6dydxe CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 31
APPLICATIONS OF LPP CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 32
LIMITATIONS OF LPP CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 33
UNIT III OPTIMAL SOLUTION FOR LPP. GRAPHICAL METHOD. SIMPLEX METHOD. MAXIMIZATION PROBLEMS. MINIMIZATION PROBLEMS CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 34
CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 35
PROCEDURE: Graphical Method The determination of the solution space that defines the feasible solution. Step 1: Since the two decision variable x and y are non-negative, consider only the first quadrant of xy-coordinate plane. Step 2: Each constraint is of the form ax + by c or ax + by c; Draw the line ax + by = c (1) For each constraint, the line (1) divides the first quadrant in to two regions say R 1 and R 2, suppose (x 1, 0) is a point in R 1. If this point satisfies the equation ax + by c or ( c), then shade the region R 1. If (x 1, 0) does not satisfy the inequality, shade the region R 2. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 36
PROCEDURE Step 3: Corresponding to each constant, we obtain a shaded region. The intersection of all these shaded regions is the feasible region or feasible solution of the LP. The determination of the optimal solution from the feasible region. Step 1: Find the co-ordinates of each vertex of the feasible region. These co-ordinates can be obtained from the graph or by solving the equation of the lines. Step 2: At each vertex (corner point) compute the value of the objective function. Step 3: Identify the corner point at which the value of the objective function is maximum (or minimum depending on the LP) The co-ordinates of this vertex is the optimal solution and the value of Z is the optimal value CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 37
Solve by graphical method. Max Z = 50x + 18y Subject to the constraints 2x + y 100; x + y 80; x 0; y 0; Solution : EXAMPLE Step 1: Since x 0, y 0, we consider only the first quadrant of the xy - plane Step 2: We draw straight lines for the equation 2x+ y = 100 (2) x + y = 80 CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 38
To determine two points on the straight line 2x + y = 100 Put y = 0, 2x = 100, we get x = 50 (50, 0) is a point on the line (2) Put x = 0 in (2), y =100 (0, 100) is the other point on the line (2) Plotting these two points on the graph paper draw the line which represent the line 2x + y =100. This line divides the 1 st quadrant into two regions, say R 1 and R 2. Choose a point say (1, 0) in R 1. (1, 0) satisfy the inequality 2x + y =100. Therefore R 1 is the required region for the constraint 2x + y = 100. Similarly draw the straight line x + y = 80 by joining the point (0, 80) and (80, 0). Find the required region say R 1 ', for the constraint x + y = 80. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 39
Step 3 : The intersection of both the region R 1 and R 1 ' is the feasible solution of the LPP. Therefore every point in the shaded region OABC is a feasible solution of the LPP, since this point satisfies all the constraints including the non-negative constraints. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 40
FINDING OPTIMAL SOLUTION: Step 1 : In the graph, the corners of the feasible region are O (0, 0), A (0, 80), B(20, 60), C(50, 0) Step 2 : Find the value of the objective function at these corner points. At (0, 0), Z = 0 At (0, 80) Z = 50 (0) + 18(80) = 1440 At (20, 60), Z = 50 (20) +18 (60) = 1000 + 1080 = 2080 At (50, 0) Z = 50 (50 )+ 18 (0) = 2500. Since our object is to maximize Z and Z has maximum at (50, 0) the optimal solution is x = 50 and y = 0. The optimal value is 2500. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 41
SIMPLEX METHOD CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 42
SIMPLEX METHOD PROCEDURE : CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 43
FOR SOLVED PROBLEMS, FOLLOW THE LINKS http://www.phpsimplex.com/en/simplex_method_example.htm http://www.universalteacherpublications.com/univ/ebooks/or/ch3/si mplex.htm CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 44
PROCEDURE - BIG M METHOD CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 45
For solved problems follow the links http://www.universalteacherpublications.com/univ/ebooks/or/ch3/m method.htm https://youtu.be/mz843vvia0a CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 46
UNIT IV TRANSPORTATION PROBLEMS. OBJECTIVE. BASIC DEFINITIONS. INITIAL BASIC FEASIBLE SOLUTION. VOGEL S APPROXIMATION METHOD. ASSIGNMENT PROBLEM. HUNGARIAN METHOD. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 47
TRANSPORTATION PROBLEM OBJECTIVE : Suppose that the sources S i (i = 1,2,..m) produce the non-negative quantities a i (i = 1,2,..m) of a product and the non-negative quantities b j (j = 1,2,..n) of same product are required at n places called the destinations D j such that the total quantity produced is equal to the total quantity required. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 48
Suppose that c ij is the transportation cost of a unit from the i th source to the j th destination. Then the objective is to find x ij, the quantity transported from i th source to the th destination, in such a way that the total transportation cost is minimized CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 49
BASIC DEFINITIONS Feasible Solution : A feasible solution to a transportation problem is the set of all non-negative allotments (x ij ) which satisfy the constraints Basic Feasible Solution : A feasible solution to a m source, n- destination transportation problem is called a basic feasible solution, if the number of positive allocations (x ij ) is equal to m+n-1. Optimal Solution : A (basic or non-basic) solution is called an optimal solution if it minimizes the total transportation cost. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 50
Independent positions : The allocations are said to be in independent positions if it is impossible to form a closed loop from any allocation back to itself by horizontal and vertical lines only. Non-degenerate basic feasible solution : A feasible solution is called a non-degenerate basic feasible solution if it satisfies the following conditions. (i) Total number of allocation is equal to m+n-1 (ii) The allocations are all in independent positions. Degenerate basic feasible solution : A basic feasible solution is called a degenerate basic feasible solution if it contains less than m+n-1 non-negative allocations. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 51
Solution To A Transportation Problem An Optimal solution to a Transportation problem is obtained in two stages. Stage 1 To obtain an initial basic feasible solution. Stage 2 To obtain an optimal solution from the basic initial solution. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 52
METHODS TO FIND BASIC FEASIBLE SOLUTION There are five basic methods : Northwest Corner Method Row Minima Method Column Minima Method Least Cost Method Vogel s Approximation Method CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 53
PROBLEMS North West Corner Rule : For procedure and solved problems, refer http://www.universalteacherpublications.com/univ/ebooks/or/ch5/nw. htm Row minima Method For solved problem, refer https://youtu.be/1kzu2h z-q CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 54
PROBLEMS COLUMN MINIMA METHOD: For procedure and solved problems, refer http://www.finance-assignment.com/column-minima-method MATRIX MINIMA METHOD : For procedure and solved problems, refer http://www.universalteacherpublications.com/univ/ebooks/or/ch5/matrix.htm CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 55
VOGEL S APPROXIMATION METHOD CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 56
For Solved Examples refer PROBLEMS http://www.universalteacherpublications.com/univ/ebooks/or/ch5/vo gel.htm CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 57
OPTIMUM SOLUTION For solved problems refer http://www.universalteacherpublications.com/univ/ebooks/or/ch5/mod iex.htm CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 58
ASSIGNMENT PROBLEM DEFINITION : Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do ach job at a time, though with varying degree of efficiency. Let C ij be the cost if the i th person is assigned to the j th job. The problem is to find an assignment so that the total cost of performing all jobs is minimum. Problems of this kind are known as assignment problem. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 59
Step 1 : HUNGARIAN METHOD Prepare a cost matrix. If the cost matrix is not a square matrix then add a dummy row (column) with zero cost element. Step 2 : Subtract the minimum element in each row from all the elements of the respective rows. Step 3 : Subtract the minimum element of each column from all the elements of the respective columns. Thus, obtain the modified matrix. Step 4 : Draw minimum number of horizontal and vertical lines to cover all zeros in the resulting matrix. Let the minimum number of lines be N. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 60
Now there are two possible cases. Case i : If N = n, where n is the order of matrix, then an optimal assignment can be made. So make the assignment to get the required solution. Case ii : If N < n, then proceed to step 5. Step 5: Determine the smallest uncovered element in the matrix (element not covered by N lines). Subtract this minimum element from all uncovered elements and add the same element at the intersection of horizontal and vertical lines. Thus, the modified matrix is obtained. Step 6 : Repeat step (3) and step (4) until we get the case (i) of step 3. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 61
ASSIGNMENT PROBLEM 1) Assignment means allocating various jobs to various people. It should be done in such a way that the overall processing time is less, overall efficiency is high, overall productivity is high, etc. 2) In an assignment problem only one allocation can be made in particular row or a column 3) Assignment problem aims at assignment of jobs to various people. 4 ) When no. of jobs is not equal to no. of workers, it is a unbalanced problem TRANSPORTATION PROBLEM 1) A transportation problem is concerned with transportation method or selecting routes in a product distribution network among the manufacture plant and distribution warehouses 2) Many allocations can be done in a particular row or particular column. 3) Transportation problem aims for a distribution route, which can lead to minimization of cost and maximization of profit. 4) When the total demand is not equal to total supply it is unbalanced problem. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 62
UNIT V GAME THEORY TYPES OF GAMES BASIC ASSUMPTIONS PURE STRATEGY MIXED STRATEGY INDETERMINATE MATRIX PAYOFF MATRIX CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 63
GAME THEORY In business management, economics and commerce we come across competitive situations involving conflicting interests. To tackle such situations a special discipline called game theory has been developed. The aim of the theory of the games is to analyze the different situations each player has to face and different situation he has to choose according to those of the opponent. The applications of game theory is not limited to games in ordinary sense of it but also includes in economics, business, ware fare and social behavior, etc. To analyze the theory of game we introduce the terms player, strategy, pay off and saddle point. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 64
BASIC DEFINITIONS Game: A competitive situation is called a game. (i) (ii) (iii) (iv) A situation is termed a game when it possesses the following properties. The number of competitors is finite. There is a conflict in interests between the participants. Each of the participants has a finite set of possible courses of action The outcome of the game is affected by the choices made by all the players. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 65
BASIC DEFINITIONS CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 66
BASIC DEFINITIONS Players: A strategic decision maker within the context of the game Payoff: The payout a player receives from arriving at a particular outcome. The payout can be in any quantifiable form, from dollars to utility. Information Set: The information available at a given point in the game. The term information set is most usually applied when the game has a sequential component. Equilibrium: The point in a game where both players have made their decisions and an outcome is reached. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 67
TYPES OF GAMES CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 68
TYPES OF GAMES CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 69
TYPES OF GAMES (i) (ii) (iii) (iv) Two person game: If the game has only two players it is called two person game. Two person zero sum game: A game with two players where the gain of one player equals the loss to the other is known as a two person zero sum game. Finite game: A game with finite number of players is called finite game. Infinite game: If it is not finite then the game is called infinite game CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 70
BASIC ASSUMPTIONS Assumes that a player can adopt multiple strategies for solving a problem Assumes that there is an availability of pre-defined outcomes Assumes that the overall outcome for all players would be zero at the end of the game Assumes that all players in the game are aware of the game rules as well as outcomes of other players Assumes that players take a rational decision to increase their profit CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 71
THE MAXIMIN AND MINIMAX PRINCIPLE Consider two players A and B. A is a player who wishes to maximize his gain while player B wishes to minimize his losses. Since A would like to maximize his minimum gain, we obtain for player A, the value called maximin value and the corresponding strategy is called the maximin strategy. On the other hand, since player B wishes to minimize his losses, a value called the minimax value which is the minimum of the maximum losses is found. The corresponding strategy is called the minimax strategy Optimum Strategy: When the maximin value is equal to the minimax value, the corresponding strategy is called optimum strategy. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 72
SADDLE POINT CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 73
SADDLE POINT CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 74
PURE STRATEGY In a zero-sum game, the pure strategies of two players constitute a saddle point if the corresponding entry of the payoff matrixes simultaneously a maximum of row minima and a minimum of column maxima. This decision-making is referred to as the minimax-maximin principle to obtain the best possible selection of a strategy for the players. CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 75
MIXED STRATEGY Solve for the mixed strategy Nash equilibrium. Write the probabilities of playing each strategy next to those strategies. For each cell, multiply the probability player 1 plays his corresponding strategy by the probability player 2 plays her corresponding strategy. Write this in the cell. Choose which player whose payoff you want to calculate. Multiply each probability in each cell by his or her payoff in that cell. Sum these numbers together. This is the expected payoff in the mixed strategy Nash equilibrium for that player. For problems refer https://saylordotorg.github.io/text_introduction-to-economicanalysis/s17-03-mixed-strategies.html CDG2A/CDZ4A/CDC4A/MBT4A ELEMENTS OF OPERATION S RESEARCH 76