CU-BBA-SEM-V-Operation Research-Second Draft

Consider the simple game called odds and evens. Suppose that player 1 takes evens and player 2 takes odds. Then, each player simultaneously shows either one finger or two fingers. If the number of fingers matches, then the result is even, and player 1 wins the bet ($2). If the number of fingers does not match, then the result is odd, and player 2 wins the bet ($2). Each player has two possible strategies: show one finger or show two fingers. The payoff matrix shown below represents the payoff to player 1. Basic Concepts of Two-Person Zero-Sum Games This game of odds and evens illustrates important concepts of simple games.  A two-person game is characterized by the strategies of each player and the payoff matrix.  The payoff matrix shows the gain (positive or negative) for player 1 that would result from each combination of strategies for the two players. Note that the matrix for player 2 is the negative of the matrix for player 1 in a zero-sum game.  The entries in the payoff matrix can be in any units as long as they represent the utility (or value) to the player.  There are two key assumptions about the behaviour of the players. The first is that both players are rational. The second is that both players are greedy meaning that they choose their strategies in their own interest (to promote their own wealth). 7.4 PURE AND MIXED STRATEGY GAMES If a game has no saddle point, then the game is said to have mixed strategies. To demonstrate the determination of mixed strategies, consider the 2 × 2 payoff matrix with respect to Player A which has no saddle point. Definition of saddle point 151 CU IDOL SELF LEARNING MATERIAL (SLM)

A saddle point of a matrix is the position of such an element in the payoff matrix, which is minimum in its row and the maximum in its column. Procedure to find the saddle point Select the minimum element of each row of the payoff matrix and mark them with circles. Select the maximum element of each column of the payoff matrix and mark them with squares. If there appears an element in the payoff matrix with a circle and a square together then that position is called saddle point and the element is the value of the game. A Game with mixed strategies Algorithm to determine mixed strategies Step 1: Find the absolute value of a – b (i.e. |a – b|) and write it against Row 2. Step 2: Find the absolute value of c – d (i.e. |c – d|) and write it against Row 1. Step 3: Find the absolute value of a – c (i.e. |a – c|) and write it against Column 2. Step 4: Find the absolute value of b – d (i.e. |b – d|) and write it against Column 1. The absolute values are called oddments. Payoff matrix with oddments Step 5: Compute the probabilities of selection of the alternatives of Player A (p1 and p2) and that of Player B (q1 and q2). 152 CU IDOL SELF LEARNING MATERIAL (SLM)

The value of the game can be computed using any one of the following formulae: Example: Consider the payoff matrix with respect of Player A and solve it optimally. Solution: The maximum and minimax values of the given problem are shown in below: 153 CU IDOL SELF LEARNING MATERIAL (SLM)

In this problem, the maximum value (6) is not equal to the minimax value (8). Hence, the game has no saddle point. Under this situation, the formulae given in this section are to be used to find the mixed strategies of the players and also the value of the game. The computations of oddments of the game are summarized in Table below: Payoff matrix with Oddments Let p1 and p2 be the probabilities of selection Alternative 1 and of Alternative 2, respectively of Player A. Also q1 and q2 be the probability of selection of Alternative 1 and of Alternative 2 of Player B, respectively. Then, we have where, the value of the game is Hence, the strategies of Player A are: A (4/7, 3/7) and of Player B: B (5/7, 2/7) 7.5 ODDS METHOD Use of odds method is possible only in case of games with 2 x 2 matrix. Here it should be ensured that the sum of column odds and row odds is equal. Method of Finding Out Odds 154 CU IDOL SELF LEARNING MATERIAL (SLM)

Step 1. Find out the difference in the value of in cell (1, 1) and the value in the cell (1,2) of the first row and place it in front of the second row. Step 2. Find out the difference in the value of cell (2, 1) and (2, 2) of the second row and place it in front of the first row. Step 3. Find out the differences in the value of cell (1, 1) and (2, 1) of the first column and place it below the second column. Step 4. Similarly find the difference between the value of the cell (1, 2) and the value in cell (2, 2) of the second column and place it below the first column. The above odds or differences are taken as positive (ignoring the negative sign). The value of the game is determined with the help of the following equation. Example: Solve the following game by odds method. 155 CU IDOL SELF LEARNING MATERIAL (SLM)

Since the game does not have saddle points, the players will use mixed strategy. We apply odds methods to solve the game. Probabilities of Selecting Strategies 7.6 DOMINANCE METHOD Dominance method is also applicable to pure strategy and mixed strategy problems. In pure strategy the solution is obtained by itself while in mixed strategy it can be used for simplifying the problem. 156 CU IDOL SELF LEARNING MATERIAL (SLM)

Principle of Dominance- The Principle of Dominance states that if the strategy of a player dominates over the other strategy in all conditions, then the later strategy is ignored because it will not affect the solution in any way. From the gainer point of view, if a strategy gives more gain than another strategy, then the first strategy dominates over the other and the second strategy can be ignored altogether. Similarly from a loser point of view, if a strategy involves lesser loss than others in all conditions then second can be ignored. So determination of superior or inferior strategy is based upon the objective of the player. Since each player is to select his best strategy, the inferior strategies can be eliminated. In other words, ineffective rows & columns can be deleted from the game matrix and only effective rows & columns of the matrix are retained in the reduced matrix. For deleting the ineffective rows & columns the following general rules are to be followed. 1. If all the elements of a row (say ith row) of apayoff matrix are less than or equal to (≤) the corresponding each element of the other row (say jth row) then the player A will never choose the ith strategy OR ith row is dominated by jth row. Then delete ith row. 2. If all the elements of a column (say jth column are greater than or equal to the corresponding elements of any other column (say jth column) then ith column is dominated by the jth column. Then delete ith column. 3. A pure strategy of a player may also be dominated if it is inferior to some convex combination of two or more pure strategies. As a particular case, if all the elements of a column are greater than or equal to the average of two or more other columns then this column is dominated by the group of columns. Similarly if all the elements of a row are less than or equal to the average of two or more rows then this row is dominated by another group of rows. 4. By eliminating some of the dominated rows and columns and if the game is reduced to 2 x 2 form it can be easily solved by the odds method. Example: Solve the game. 157 CU IDOL SELF LEARNING MATERIAL (SLM)

Since there is no saddle point, so we apply the dominance method. Here Row II dominates Row II so we will delete Row II. Since column III dominates column I, we delete column I we get: Probability of Selecting Strategies 158 CU IDOL SELF LEARNING MATERIAL (SLM)

Using the dominance property obtains the optimal strategies for both the players and determines the value of the game. The payoff matrix for player A is given. Solution Since the question desires to show the dominance property, that is why we are using it, even if the question may have saddle points. Since all elements in Row IV are less than each element in Row III. Row III dominates Row IV. Hence we delete Row IV & we get Now all the elements of column I are less than or equal to respective each elements of column IV, so we can delete column IV now we get 159 CU IDOL SELF LEARNING MATERIAL (SLM)

Repeating the above rules now each element of column I is less than the respective elements of column V, we can delete column V and we get. Now as each element of Row I and Row II are less than the respective elements of III, we can delete both Rows I & II. Value of Game (v) = 6 Strategy for player A → III Strategy for player B → I 7.7 SUB GAMES METHOD The concept of a 2 x n zero-sum game When the first player A has exactly two strategies and the second player B has n (where n is three or more) strategies, there results a 2 x n game. It is also called a rectangular game. Since A has two strategies only, he cannot try to give up any one of them. However, since B has many strategies, he can make out some choice among them. He can retain some of the advantageous strategies and discard some disadvantageous strategies. The intention of B is to give as minimum payoff to A as possible. In other words, B will always try to minimize the loss to himself. Therefore, if some strategies are available to B by which he can minimize the payoff to A, then B will retain such strategies and give such strategies by which the payoff will be very high to A. 160 CU IDOL SELF LEARNING MATERIAL (SLM)

Approaches for 2 x n zero-sum game There are two approaches for such games: (1) Sub game approach and (2) Graphical approach. Sub game approach The given 2 x n game is divided into 2 x 2 sub games. For this purpose, consider all possible 2 x 2 submatrices of the payoff matrix of the given game. Solve each sub game and have a list of the values of each sub game. Since B can make out a choice of his strategies, he will discard such of those sub games which result in more payoff to A. On the basis of this consideration, in the long run, he will retain two strategies only and give up the other strategies. Problem Solve the following game Solution: Let us consider all possible 2x2 sub games of the given game. We have the following sub games: 161 CU IDOL SELF LEARNING MATERIAL (SLM)

Let E be the expected value of the payoff to player A. Let p be the probability that player A will use his first strategy. Then the probability that he will use his second strategy is 1-p. We form the equations for E in all the sub games as follows: 162 CU IDOL SELF LEARNING MATERIAL (SLM)

Solve the equations for each sub game. Let us tabulate the results for the various sub games. We have the following: 163 CU IDOL SELF LEARNING MATERIAL (SLM)

Interpretation: Since player A has only 2 strategies, he cannot make any choice on the strategies. On the other hand, player B has 4 strategies. Therefore he can retain any 2 strategies and give up the other 2 strategies. This he will do in such a way that the pay-off to player A is at the minimum. The pay-off to A is the minimum in the case of sub game 4. i.e., the sub game with the matrix Therefore, in the long run, player B will retain his strategies 2 and 3 and give up his strategies 1 and 4. In that case, the probability that A will use his first strategy is p = 5/ 9 and the probability that he will use his second strategy is 1-p = 4 /9.i.e., out of a total of 9 trials, he will use his first strategy five times and the second strategy four times. The value of the game is 10/ 9. The positive sign of V shows that the game is favourable to player A. 7.8 EQUAL GAINS METHOD This is a method that provides a comprehensible approach for obtaining optimal strategies for every player in a 2x2 pay-off matrix with no saddle point. If the payoff matrix is lengthier 164 CU IDOL SELF LEARNING MATERIAL (SLM)

than 2x2, then the dominance method would be employed and finally the algebraic procedure to help obtain the optimal strategies and also the value of the game. This method or procedure is also called the Method of the Oddments. Given a pay-off matrix as in the above table, the procedures followed in the arithmetic method are as given below: i. (i) Find the absolute value of the difference between the two values of row 1 (i.e. | a-b|) and put this value against row 2. ii. (ii) Find the absolute value of the difference between the two values of row 2 (i.e. | c-d |) and put this value against row 1. iii. (iii) Find the absolute value of the difference between the two values of column 1 (i.e. | a-c |) and put this value against column 2. iv. (iv) Find the absolute value of the difference between the two values of column 2 (i.e.,| b-d |) and put this value against column 1. v. (v) Find the probabilities of each entry as follows: Steps (i) to (v) therefore yield the resulting matrix as given in the following table. 165 CU IDOL SELF LEARNING MATERIAL (SLM)

vi. (vi) Obtain the value of the game, V by using the algebraic method as follows: 7.9 SUMMARY  A game represents a competitive or conflicting situation between two or more players. Each player has a number of choices, called moves (or pure strategies).  A player selects his moves without any knowledge of the moves chosen by the other players.  The simultaneous choices of all players lead to the respective payoffs of the game. If the sum of the payoffs to all players is zero, then it is called a zero-sum game.  A game played by n persons is called an n-person game. We shall consider only two- person zero-sum games, which are also called matrix games because the gain of one player signifies an equal loss to the other that it suffices to express the outcomes (as a result of the selection of moves) in terms of the payoffs to one player (player I) in a matrix.  A strategy for a given player (I or II) is a plan that specifies which of the available choices he should make with what probabilities.  Hence a strategy for player I is the specification of xi (xi ≥ 0, Pxi = 1) and for player II, yj (yj ≥ 0, Pyj = 1).  A strategy is generally a mixed strategy to distinguish it from a pure strategy in which xi (or yj) is 1 for some i (or j) and all other xk (or yk) being 0, k 6= i, j.  Furthermore, we explicitly adopt the “rationality assumptions” about the players, who are then assumed to be “intelligent” players that act so as to maximize his expected payoff (or minimize his expected loss). 166 CU IDOL SELF LEARNING MATERIAL (SLM)

 Game theory deals with the determination of the “optimal” strategies for each player. In view of the conflicting nature and lack of information about the specific strategies selected by each player, optimality is based on a rather conservative criterion, namely, each player selects his strategy (mixed or pure) which guarantees a payoff that can never be worsened by the selections of his opponent. This criterion is known as the “minimax criterion” (or “maximin criterion”).  Equal gains method is a method that provides a comprehensible approach for obtaining optimal strategies for every player in a 2x2 pay-off matrix with no saddle point.  If the payoff matrix is lengthier than 2x2, then the dominance method would be employed and finally the algebraic procedure to help obtain the optimal strategies and also the value of the game. This method or procedure is also called the Method of the Oddments. 7.10 KEYWORDS  Two-person Zero-Sum Game- A two player game is called a zero-sum game if the sum of the payoffs to each player is constant for all possible outcomes of the game. More specifically, the terms (or coordinates) in each payoff vector must add up to the same value for each payoff vector. Such games are sometimes called constant-sum games instead.  Pure Strategy- A pure strategy provides a complete definition of how a player will play a game. Pure strategy can be thought about as a plan subject to the observations they make during the course of the game of play. In particular, it determines the move a player will make for any situation they could face.  Payoff Matrix- In game theory, a payoff matrix is a table in which strategies of one player are listed in rows and those of the other player in columns and the cells show payoffs to each player such that the payoff of the row player is listed first.  Saddle Point- In a zero-sum matrix game, an outcome is a saddle point if the outcome is a minimum in its row and maximum in its column. The argument that players will prefer not to diverge from the saddle point leads us to offer the following principle of game theory.  Dominance Method- The principle of dominance in Game Theory (also known as dominant strategy or dominance method) states that if one strategy of a player dominates over the other strategy in all conditions then the later strategy can be ignored. A strategy dominates over the other only if it is preferable over others in all conditions. 167 CU IDOL SELF LEARNING MATERIAL (SLM)

7.11 LEARNING ACTIVITY 1. Two computer firms, A and B, are planning to market network systems for office information management. Each firm can develop either a fast, high-quality system (High), or a slower, low-quality system (Low). Market research indicates that the resulting profits to each firm for the alternative strategies are given by the following payoff matrix If both the firms make their decisions at the same time and follow maximum (low- risk) strategies, what will be the outcome? ___________________________________________________________________________ ___________________________________________________________________________ 2. Solve the game with the following pay-off matrix. Player B ___________________________________________________________________________ ___________________________________________________________________________ 7.12 UNIT END QUESTIONS A. Descriptive Questions Short Questions 168 CU IDOL SELF LEARNING MATERIAL (SLM)

1. What is a mixed strategy? 2. Brief on the concept of a 2 x n zero-sum game. 3. Write a short note on the sub-game approach? 4. What is the odds method? 5. Write a few properties of the game? Long Questions 1. Write down the procedure to identify the saddle point? 2. What are the concepts behind two-person zero-sum games? 3. Write about the prominent characteristics of game theory? 4. What is the method to find out odds? 5. What is the principle of dominance? B Multiple Choice Questions 1. What would be the result if the first player A has exactly two strategies and the second player B has n (where n is three or more) strategies? a. 2 x n game b. 1xn game c. 3xn game d. 4xn game 2. What is the term used for a condition in which the strategy of a player dominates over the other strategy? a. Zero-sum b. Dominance c. Pure strategy d. Mixed strategy 3. Which is not a step to find out odds? a. Find out the difference in the value of in cell (1, 1) and the value in the cell (1,2) of the first row and place it in front of the second row b. Find out the difference in the value of cell (2, 1) and (2, 2) of the second row and place it in front of the first row c. Find out the differences in the value of cell (1, 2) and (2, 1) of the first column and place it below the second column 169 CU IDOL SELF LEARNING MATERIAL (SLM)

d. Find the difference between the value of the cell (1, 2) and the value in cell (2, 2) of the second column and place it below the first column 4. Which is/are algorithms to determine mixed strategies? a. Find the absolute value of a – b (i.e. |a – b|) and write it against Row 2 b. Find the absolute value of c – d (i.e. |c – d|) and write it against Row 1 c. Find the absolute value of a – c (i.e. |a – c|) and write it against Column 2 d. All of these 5. What would be the right term to describe a condition in which a player knows exactly what the other player is going to do, a deterministic situation is obtained and the objective function is to maximize the gain? a. Pure strategy b. Mixed strategy c. Dominance strategy d. Gain strategy Answers 1-a 2-b 3-c 4-d 5-a 7.13 REFERENCES References  Egwald Operations Research - Game Theory - Introduction  A Class of Two-Person Zero-Sum Matrix Games with Rough Payoffs (hindawi.com)  A Comprehensive Review of Solution Methods and Techniques for Solving Games in Game Theory (sapub.org) Textbooks  KantiSwarup, P.K.Gupta and Man Mohan, “Operations Research, Eighth Edition”, Sultan Chand & Sons, New Delhi, 1999.  S.Hillier and J.Liebermann, Operations Research, Sixth Edition, McGraw Hill Company, 1995.  Operations Research problems and solution, Third edition, J.K Sharma, Macmillan Publishers India Ltd, 2012. Websites 170 CU IDOL SELF LEARNING MATERIAL (SLM)

 Two-Person Zero-Sum Games: Basic Concepts | NEOS (neos-guide.org)  Game Theory Operation Research (acsce.edu.in)  Chp_48_Competitive Strategies (Games Theory).pdf  Microsoft Word - Unit 8 Lecturer notes of Games theory of OR by Narhari RVCE, Ba.doc (azureedge.net)  Microsoft Word - QT for Managers (acsce.edu.in) 171 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 8 – SIMULATION STRUCTURE 8.0 Learning Objectives 8.1 Introduction 8.2 Simulation 8.3 Summary 8.4 Keywords 8.5 Learning Activity 8.6 Unit End Questions 8.7 References 8.0 LEARNING OBJECTIVES After studying this unit, you will be able to:  Grasp the overview of simulation and its objectives  Define Monte- Carlo simulation and its purpose  Generate random variables for the simulation 8.1 INTRODUCTION Simulation is the most important technique used in analysing a number of complex systems. There are many real world problems which cannot be represented by a mathematical model due to the stochastic nature of the problem, the complexity in problem formulation and many values of the variables are not known in advance and there is no easy way to find these values. Simulation has become an important tool for tackling the complicated problem of managerial decision-making. Simulation determines the effect of a number of alternate policies without disturbing the real system. Recent advances in simulation methodologies, technical development and software availability have made simulation as one of the most widely and popularly accepted tools in Operation Research. Simulation is a quantitative technique that utilizes a computerized mathematical model in order to represent actual decision-making under conditions of uncertainty for evaluating alternative courses of action based upon facts and assumptions. 172 CU IDOL SELF LEARNING MATERIAL (SLM)

John Von Newman and StanislawUlam made the first important application of simulation for determining the complicated behaviour of neutrons in a nuclear shielding problem, being too complicated for mathematical analysis. After the remarkable success of this technique on neutron problems, it has become more popular and has many applications in business and industry. The development of digital computers has further increased the rapid progress in the simulation technique. Designers and analysts have long used the techniques of simulation by physical sciences. Simulation is the representative model of a real situation. For example, in a city, a children's park with various signals and crossings is a simulated model of city traffic. A planetarium is a simulated model of the solar system. In laboratories we perform a number of experiments on simulated models to predict the behaviours of the real system under a true environment. For training a pilot, flight simulators are used. The simulator under the control of computers gives the same readings as the pilot gets in real flight. The trainee can intervene when there is a signal, like engine failure etc. Simulation is the process of generating values using random numbers without really conducting an experiment. Whenever the experiments are costly or infeasible or time-consuming simulation is used to generate the required data. 8.2 SIMULATION Definition 1. Simulation is a representation of reality through the use of a model or other device, which will react in the same manner as reality under a given set of conditions. 2. Simulation is the use of a system model that has the designed characteristic of reality in order to produce the essence of actual operation. 3. According to Donald G. Malcolm, simulation model may be defined as one which depicts the working of a large scale system of men, machines, materials and information operating over a period of time in a simulated environment of the actual real world conditions. 4. According to Naylor, et al. simulation is a numerical technique for conducting experiments on a digital computer, which involves certain types of mathematical and logical relationships necessary to describe the behaviour and structure of a complex real world system over an extended period of time. There are two types of simulation, they are: Analog Simulation: Simulating the reality in physical form (e.g.: Children's park, planetarium, etc.) is known as analogue simulation. 173 CU IDOL SELF LEARNING MATERIAL (SLM)

Computer Simulation: For problems of complex managerial decision-making, the analogue simulation may not be applicable. In such a situation, the complex system is formulated into a mathematical model for which a computer programme is developed. Using high-speed computers then solves the problem. This type of simulation is known as computer simulation or system simulation. Classification of Simulation Models Simulation models are classified as: i. Simulation of Deterministic models: In the case of these models, the input and output variables are not permitted to be random variables and models are described by exact functional relationship. ii. Simulation of Probabilistic models: In such cases a method of random sampling is used. The techniques used for solving these models are termed as Monte-Carlo technique. iii. Simulation of Static Models: These models do not take variable time into consideration. iv. Simulation of Dynamic Models: These models deal with time varying interaction. Advantages of Simulation Simulation is a widely accepted technique of operations research due to the following reasons:  It is straightforward and flexible.  It can be used to analyse large and complex real world situations that cannot be solved by conventional quantitative analysis models.  It is the only method sometimes available.  It studies the interactive effect of individual components or variables in order to determine which ones are important.  Simulation model, once constructed, may be used over and again to analyse all kinds of different situations.  It is the valuable and convenient method of breaking down a complicated system into subsystems and their study. Each of these subsystems works individually or jointly with others. Limitations of Simulation Technique 174 CU IDOL SELF LEARNING MATERIAL (SLM)

 Since the simulation model mostly deals with uncertainties, the results of simulation are only reliable approximations involving statistical errors, optimum results cannot be produced by simulation.  In many situations, it is not possible to identify all the variables, which affect the behaviour of the system.  In very large and complex problems, it is very difficult to make the computer program in view of the large number of variables and the inter–relationship among them.  For problems requiring the use of computers, simulation may be comparatively costlier and time consuming in many cases.  Each solution model is unique and its solutions and inferences are not usually transferable to other problems, which can be solved by other techniques. Monte-Carlo Simulation The Monte-Carlo method is a simulation technique in which statistical distribution functions are created using a series of random numbers. Working on the digital computer for a few minutes we can create data for months or years. The method is generally used to solve problems which cannot be adequately represented by mathematical models or where solution of the model is not possible by analytical method. Monte-Carlo simulation yields a solution, which should be very close to the optimal, but not necessarily the exact solution. But this technique yields a solution, which converges to the optimal solution as the number of simulated trials tends to infinity. The Monte-Carlo simulation procedure can be summarized in the following steps: Step 1: Clearly define the problem: i. Identify the objectives of the problem. ii. Identify the main factors, which have the greatest effect on the objective of the problem. Step 2: Construct an approximate model: i. Specify the variables and parameters of the mode. ii. Formulate the appropriate decision rules, i.e. state the conditions under which the experiment is to be performed. iii. Identify the type of distribution that will be used. Models use either theoretical distributions or empirical distributions to state the patterns of the occurrence associated with the variables. iv. Specify the manner in which time will change. Problem 175 CU IDOL SELF LEARNING MATERIAL (SLM)

With the help of a single server queuing model having inter-arrival and service times constantly 1.4 minutes and 3 minutes respectively, explain discrete simulation technique taking 10 minutes as the simulation period. Find from this average waiting time and percentage of idle time of the facility of a customer. Assume that initially the system is empty and the first customer arrives at time t = 0. Solution Data: System is initially empty. Service starts as soon as the first customer arrives. First customer arrives at t = 0. The departure time of the first customer = 0 + 3 i.e. arrival time + service time = 3 minutes (Dep) in the table. The second customer arrives at 1.4 minutes and third arrives at 2.8 minutes (Arr). Until the first customer leaves the system, second and third customers have to wait for service. We can calculate the waiting time for the second customer by taking the difference of time of departure of first customer and the time of arrival of second customer i.e. 3 – 1.4 = 1.6 minutes. The procedure is shown in the table below: Table 8.1: Procedure 176 CU IDOL SELF LEARNING MATERIAL (SLM)

Average waiting time per customer for those who must wait = Sum of waiting time of all customers / number of waiting times taken = (1.4 + 2.8 + 4.2 + 5.6 + 7.0 + 8.4 + 9.8) / 7 = 18.8 / 7 = 2.7 minutes. Percentage of idle time of server = Sum of idle time of server / total time = 0%. Random Numbers Random number is a number in a sequence of numbers whose probability of occurrence is the same as that of any other number in that sequence. Pseudo-Random Numbers Random numbers which are generated by some deterministic process but which satisfy statistical tests for randomness are called Pseudo-random numbers. Generation of Random Numbers Using some arithmetic operation one can generate Pseudo-random numbers. These methods most commonly specify a procedure, where starting with an initial number called seed generates the second number and from that a third number and so on. A number of recursive procedures are available, the most common being the congruence method or the residue method. This method is described by the expression: ri + 1 = (ari + b) (modulo m), Where a, b and m are constants, ri and ri +1 are the ith and (i + 1) th random numbers. The expression implies multiplication of a by ri and addition of b and then division by m. Then ri +1 is the remainder or residue. To begin the process of random number generation, in addition to a, b and m, the value of r0 is also required. It may be any random number and is called seed. Problem With the help of an example explain the additive multiplicative and mixed types of the congruence random number generators. Solution The congruence random number generator is described by the recursive expression ri + 1 = (ari + b) (modulo m), Where a, b and m are constants. The selection of these constants is very important as it determines the starting of a random number, which can be obtained by this method. The above expression is for a mixed type congruential method as it comprises both multiplication of a and ri and addition of ari and b. 177 CU IDOL SELF LEARNING MATERIAL (SLM)

If a = 1, the expression reduces to ri + 1 = (ri + b) (modulo m). This is known as additive type expression. When b = 0, the expression obtained is rI + 1 = (ar I) (modulo m), this is known as the multiplicative method. To illustrate the different types of the congruence methods, let us take a = 16, b = 18 and m = 23 and let the starting random number or seed be r0 = 1.  (a) Mixed Congruential method: ri + 1 = (ari + b) (modulo m), therefore, Table 8.2: Mixed Congruential method The random numbers generated by this method are: 1, 11, 10, 17, 14, 12, 3, 20, 16, 21, and 9.  (b) Multiplicative Congruential Method: ri + 1 = ari (modulo m) 178 CU IDOL SELF LEARNING MATERIAL (SLM)

Table 8.3: Multiplicative Congruential method The string of random numbers obtained by the multiplicative congruential method is 1, 16, 3, 2, 9, 6, 4, 18, 12, 8, and 13.  (c) Additive Congruential Method: ri + 1 = (ri + b) (modulo m). Table 8.4: Additive Congruential method 179 The random numbers generated are: 1, 19, 14, 19, 4, 22, 17, 12, 7, 2, 20, and 15. CU IDOL SELF LEARNING MATERIAL (SLM)

8.3 SUMMARY  Simulation is a very powerful problem solving technique. Its applicability is so general that it would be hard to point out disciplines or systems to which simulation has not been applied.  The basic idea behind simulation is simple, namely model the given system by means of some equations and then determine its time dependent behaviour.  In Simulation we make a model of a conceptual model and then results are compared with the real system.  Normally simulation is used when either an exact analytic expression for the behaviour of the system under investigation is not available or the analytic solution is too time consuming.  Simulation is considered an interdisciplinary subject because it uses concepts from mathematics, computer science and the application field.  A model is a representation of an actual system.  Models can be of different kinds. Discrete-Event Simulation Model is defined as one in which the state variables change only at discrete points in time at which events occur.  A deterministic system is defined as in which randomness does not affect the behaviour of the system.  A stochastic system is defined as in which randomness affects the behaviour of the system. In order to process a simulation of an event first we have to perform formulation of the problem, and then we have to implement the defined model in any suitable programming language.  Finally we have to perform verification and validation and then analysis of output results.  The Monte-Carlo method is a simulation technique in which statistical distribution functions are created using a series of random numbers. Working on the digital computer for a few minutes we can create data for months or years.  The method is generally used to solve problems which cannot be adequately represented by mathematical models or where solution of the model is not possible by analytical method. Monte-Carlo simulation yields a solution, which should be very close to the optimal, but not necessarily the exact solution. But this technique yields a solution, which converges to the optimal solution as the number of simulated trials tends to infinity. 180 CU IDOL SELF LEARNING MATERIAL (SLM)

 Random number is a number in a sequence of numbers whose probability of occurrence is the same as that of any other number in that sequence.  Pseudo-Random Numbers are random numbers which are generated by some deterministic process but which satisfy statistical tests for randomness are called Pseudo-random numbers. 8.4 KEYWORDS  Simulation- By simulation is meant, the technique of setting up a stochastic model of a real situation, and then performing sampling experiments upon the model.  Deterministic models- In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.  Probabilistic models- Probabilistic models incorporate random variables and probability distributions into the model of an event or phenomenon. While a deterministic model gives a single possible outcome for an event, a probabilistic model gives a probability distribution as a solution.  Static Model- A static model is one in which the decision variables do not involve sequences of decisions over multiple periods.  Dynamic Model- A dynamic model is a model in which the decision variables do involve sequences of decisions over multiple periods. 8.5 LEARNING ACTIVITY 1. The following table gives the arrival pattern at a coffee counter for one minute interval. The service is taken as two persons in one minute in one counter. Using Monte Carlo simulation technique and the following random numbers, generate the pattern of arrivals and the queue formed when following 20 random numbers are given: 8,28,19,83,38,51,70,82,93,95,12,17,4,58,23,74,34,45,63 and 88 181 CU IDOL SELF LEARNING MATERIAL (SLM)

___________________________________________________________________________ ___________________________________________________________________________ 2. Redo the Monte Carlo Simulation for the United Motors example, using the following setting: Do 200 trials. Plot the cumulative distribution and histogram for annual cost. Calculate the average and standard deviation for the annual cost. ___________________________________________________________________________ ___________________________________________________________________________ 8.6 UNIT END QUESTIONS A. Descriptive Questions Short Questions 1. What are the advantages of simulation? 2. Categorize all the models of simulation? 3. What is computer simulation? 4. Give a few definitions of simulation. 5. What is simulation of deterministic models? Long Questions 1. Write a note on Monte-Carlo simulation procedure? 2. How does simulation help in solving complicated problems? 3. Explain all the types of simulation. 4. What are the limitations of simulation technique? 5. Write both advantages and disadvantages of simulation? B. Multiple Choice Questions 1. What would be the type of simulation if simulating takes place in reality in physical form? 182 CU IDOL SELF LEARNING MATERIAL (SLM)

a. Analog b. Computer c. Monte Carlo d. Numerical 2. Which type of simulation would you use to solve complex managerial problems? a. Analog b. Computer c. Monte Carlo d. None of these 3. Which are categories of simulation models? a. Deterministic a. Probabilistic b. Dynamic c. All of these 4. Which is not an advantage of simulation? a. It is straightforward and flexible a. It can be used to analyse large and complex real-world situations b. It is the only method always available c. It studies the interactive effect of individual components 5. Which is not a procedure to construct an approximate model involving Monte Carlo simulation? a. Doesn’t specify the variables and parameters of the mode b. Formulate the appropriate decision rules c. Identify the type of distribution to be used d. Specify the manner in which time will change Answers 183 1-a 2-b 3-d 4-c 5-a 8.7 REFERENCES References CU IDOL SELF LEARNING MATERIAL (SLM)

 Microsoft PowerPoint - Research Seminar v1.07.ppt (nott.ac.uk)  Simulation Techniques in Operations Research—A Review | Operations Research (informs.org)  Proceedings of the 1999 Winter Simulation Conference, Jerry Banks, Introduction to Simulation Textbooks  Operations Research, Second Edition (bbau.ac.in)  Bernard P. Zeigler, Herbert Praehofer, and Tag Gon Kim. Theory of Modelling and Simulation: Integrating Discrete Event and Continuous Complex Dynamic Systems. Academic Press, second edition.  Giordano, Frank R., Maurice D. Weir, and William P. Fox. 2003. A First Course in Mathematical Modelling. 3rd ed. Pacific Grove, Calif.: Brooks/Cole-Thompson Learning. Websites  Microsoft PowerPoint - Chapter1.ppt (wmich.edu)  Simulation Model in Operations Research - CouponED (couponed12.com)  (PDF) MONTE CARLO SIMULATION (researchgate.net) 184 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 9 – INVENTORY MANAGEMENT TECHNIQUES STRUCTURE 9.0 Learning Objectives 9.1 Introduction 9.2 Inventory Management Techniques 9.3 Summary 9.4 Keywords 9.5 Learning Activity 9.6 Unit End Questions 9.7 References 9.0 LEARNING OBJECTIVES After studying this unit, you will be able to:  Know how inventory management helps in encouraging growth of a business  Overcome the hurdles of supply chain and make it hassle free  Learn many essential inventory management techniques 9.1 INTRODUCTION The word ‘inventory’ means simply a stock of idle resources of any kind having an economic value. In other words, inventory means a physical stock of goods, which is kept in hand for smooth and efficient running of future affairs of an organization. It may consist of raw materials, work-in-progress, spare parts/consumables, finished goods, human resources such as unutilized labour, financial resources such as working capital, etc. It is not necessary that an organization has all these inventory classes but whatever may be the inventory items, they need efficient management as generally a substantial amount of money is invested in them. The basic inventory decisions include: 1. 1) How much to order? 2. 2) When to order? 3. 3) How much safety stock should be kept? The problems faced by different organizations have necessitated the use of scientific techniques in the management of inventories known as inventory control. Inventory control is 185 CU IDOL SELF LEARNING MATERIAL (SLM)

concerned with the acquisition, storage, and handling of inventories so that the inventory is available whenever needed and the associated total cost is minimized. Reasons for Carrying Inventory Inventories are carried by organisations because of the following major reasons:  Improve customer service- An inventory policy is designed to respond to individual customer’s or organization’s requests for products and services.  Reduce costs- Inventory holding or carrying costs are the expenses that are incurred for storage of items. However, holding inventory items in the warehouse can indirectly reduce operating costs such as loss of goodwill and/or loss of potential sale due to shortage of items. It may also encourage economies of production by allowing larger, longer and more production runs.  Maintenance of operational capability- Inventories of raw materials and work in- progress items act as a buffer between successive production stages so that downtime in one stage does not affect the entire production process.  Irregular supply and demand- Inventories provide protection against irregular supply and demand; an unexpected change in production and delivery schedule of a product or a service can adversely affect operating costs and customer service level.  Quantity discount- Large size orders help to take advantage of price-quantity discount. However, such an advantage must keep a balance between the storage cost and costs due to obsolescence, damage, theft, insurance, etc.  Avoiding stock outs (shortages)- Under situations like, labour strikes, natural disasters, variations in demand and delays in supplies, etc., inventories act as a buffer against stock out as well as loss of goodwill. Costs Associated with Inventories Various costs associated with inventory control are often classified as follows:  Purchase (or production) cost: It is the cost at which an item is purchased, or if an item is produced.  Carrying (or holding) cost: The cost associated with maintaining inventory is known as holding cost. It is directly proportional to the quantity kept in stock and the time for which an item is held in stock. It includes handling cost, maintenance cost, depreciation, insurance, warehouse rent, taxes, etc.  Shortage (or stock out) cost: It is the cost which arises due to running out of stock. It includes the cost of production stoppage, loss of goodwill, loss of profitability, special orders at higher price, overtime/idle time payments, loss of opportunity to sell, etc. 186 CU IDOL SELF LEARNING MATERIAL (SLM)

 Ordering (or set up) cost: The cost incurred in replenishing the inventory is known as ordering cost. It includes all the costs relating to administration (such as salaries of the persons working for purchasing, telephone calls, computer costs, postage, etc.), transportation, receiving and inspection of goods, processing payments, etc. If a firm produces its own goods instead of purchasing the same from an outside source, then it is the cost of resetting the equipment for production. 9.2 INVENTORY MANAGEMENT TECHNIQUES Inventory management is a step in the supply chain where inventory and stock quantities are tracked in and out of your warehouse. The goal of inventory management systems is to know where your inventory is at any given time and how much of it you have in order to manage inventory levels correctly. Some companies may opt to scan in inventory via a barcode scanner to increase efficiency along pick routes and accuracy. Unlike an ERP system, an inventory management system focuses on one supply chain process. They often come with the ability to integrate with other software systems – point of sale, channel management, shipping – so you can build a personalized integration stack to the needs of your unique business. Why is Inventory Management Important? Inventory management is the fundamental building block to longevity. When an inventory is properly organized, the rest of supply-chain management will fall into place. Without it, one risks a litany of mistakes like mis-shipments, out of stocks, overstocks, mis-picks, and so on. Proper warehouse management is key. Mis-picks result from incorrect paper pick lists, disorganized shelf labels, or just a messy warehouse in general. Mis-shipments are a direct result of mis-picks at the beginning of the inventory process, and are also a result of a lack in quality control procedures. Out of stocks and overstocks occur when a company uses manual methods to place orders without having a full grasp on the state of their inventory. This is not a good predictor for inventory forecasting and results in too much stock or too little. All of these mistakes will not only cost money, but also cost in wasted labour spent correcting the mistakes later. When management tools are not implemented, it leads to risk of human error mistakes going up by the minute. And customer reviews and loyalty take a negative hit as well. Inventory Management Techniques Economic order quantity 187 CU IDOL SELF LEARNING MATERIAL (SLM)

Economic order quantity, or EOQ, is a formula for the ideal order quantity a company needs to purchase for its inventory with a set of variables like total costs of production, demand rate, and other factors. The overall goal of EOQ is to minimize related costs. The formula is used to identify the greatest number of product units in order to minimize buying. The formula also takes the number of units in the delivery of and storing of inventory unit costs. This helps free up tied cash in inventory for most companies. The concept of economic ordering quantity (EOQ) was first developed by F. Harris in 1916. The concept is as follows: Management of inventory is confronted with a set of opposing costs. As the lot size increases, the carrying cost increases while the ordering cost decreases. On the other hand, as the lot size decreases, the carrying cost decreases but the ordering cost increases. The two opposite costs can be shown graphically by plotting them against the order size as shown in figure below: Figure 9.1: Graphic representation of two opposite costs Economic ordering quantity (EOQ) is that size of order which minimizes the average total cost of carrying inventory and ordering under the assumed conditions of certainty and the total demand during a given period of time is known. Economic order quantity can be determined by two methods: 1. Tabulation method. 2. Algebraic method. 188 CU IDOL SELF LEARNING MATERIAL (SLM)

Determination of EOQ by Tabulation (Trial & Error) Method. This method involves the following steps: a. Select the number of possible lot sizes to purchase. b. Determine average inventory carrying cost for the lot purchased. c. Determine the total ordering cost for the orders placed. d. Determine the total cost for each lot size chosen which is the summation of inventory carrying cost and ordering cost. e. Select the ordering quantity, which minimizes the total cost. The data calculated in a tabular column can be plotted showing the nature of total cost, inventory cost and ordering cost curve against the quantity ordered. Example: The XYZ Ltd. carries a wide assortment of items for its customers. One of its popular items has an annual demand of 8000 units. Ordering cost per order is found to be Rs. 12.5. The carrying cost of average inventory is 20% per year and the cost per unit is Re. 1.00. Determine the optimal economic quantity and make your recommendations. Solution Table 9.1: Determination of EOQ by Tabulation 189 CU IDOL SELF LEARNING MATERIAL (SLM)

Figure 9.2: Inventory cost curve The table and the graph indicate that an order size of 1000 units will give the lowest total cost among the different alternatives. It also shows that minimum total cost occurs when carrying cost is equal to ordering cost. Determination of EOQ by Analytical Method In order to derive an economic lot size formula following assumptions are made: 1. Demand is known and uniform. 2. Let D denotes the total number of units purchased/produced and Q denotes the lot size in each production run. 3. Shortages are not permitted, i.e., as soon as the level of the inventory reaches zero, the inventory is replenished. 4. Production or supply of commodities is instantaneous. 5. Lead-time is zero. 6. Set-up cost per production run or procurement cost is C3. 7. Inventory carrying cost is C1= CI, where C is the unit cost and I is called inventory carrying cost expressed as a percentage of the value of the average inventory. This fundamental situation can be shown on an inventory-time diagram, with Q on the vertical axis and the time on the horizontal axis. The total time period (one year) is divided into n parts. Figure 9.3: Inventory-time diagram 190 The most economic point in terms of total inventory cost exists where, Inventory carrying cost = Annual ordering cost (set-up cost) Average inventory = 1/2 (maximum level + minimum level) = (Q + 0)/2 = Q/2 CU IDOL SELF LEARNING MATERIAL (SLM)

Total inventory carrying cost = Average inventory × Inventory carrying cost per unit i.e., Total inventory carrying cost = Q/2 × C1= QC1/2… (1) Total annual ordering costs = Number of orders per year × Ordering cost per order i.e., Total annual ordering costs = (D/Q) × C3= (D/Q) C3 … (2) Now, summing up the total inventory cost and the total ordering cost, we get the total inventory cost C(Q). i.e., Total cost of production run = Total inventory carrying cost + Total annual ordering costs C(Q) = QC1/2 + (D/Q) C3(cost equation) … (3) But the total cost is minimum when the inventory carrying costs becomes equal to the total annual ordering costs. Therefore, QC1/2 = (D/Q) C3 or QC1= (2D/Q) C3or Q2= 2C3D/C1 Then, optimal quantity (EOQ), Q0= √2C3D C1 D0= √2 ×16× 1800 √8.4 =82.8 or 83 lubricants (approx.) Minimum Order Quantity On the supplier side, minimum order quantity (MOQ) is the smallest amount of set stock a supplier is willing to sell. If retailers are unable to purchase the MOQ of a product, the supplier won’t sell it to you. For example, inventory items that cost more to produce typically have a smaller MOQ as opposed to cheaper items that are easier and more cost effective to make. ABC Analysis. This inventory categorization technique splits subjects into three categories to identify items that have a heavy impact on overall inventory cost.  Category A serves as the most valuable products that contribute the most to overall profit. 191 CU IDOL SELF LEARNING MATERIAL (SLM)

 Category B is the products that fall somewhere in between the most and least valuable.  Category C is for the small transactions that are vital for overall profit but don’t matter much individually to the company altogether. In this analysis, the classification of existing inventory is based on annual consumption and the annual value of the items. Hence we obtain the quantity of inventory item consumed during the year and multiply it by unit cost to obtain annual usage cost. The items are then arranged in the descending order of such annual usage cost. The analysis is carried out by drawing a graph based on the cumulative number of items and cumulative usage of consumption cost. Classification is done as follows: Table 9.2: Classification The classification of ABC analysis is shown by the graph given as follows. Figure 9.4: ABC classification Once ABC classification has been achieved, the policy control can be formulated as follows: 192 CU IDOL SELF LEARNING MATERIAL (SLM)

A-Item: Very tight control, the items being of high value. The control needs to be exercised at a higher level of authority. B-Item: Moderate control, the items being of moderate value. The control needs to be exercised at the middle level of authority. C-Item: The items being of low value, the control can be exercised at gross root level of authority, i.e., by respective user department managers. Just-in-time Inventory Management Just-in-time (JIT) inventory management is a technique that arranges raw material orders from suppliers in direct connection with production schedules. JIT is a great way to reduce inventory costs. Companies receive inventory on an as-needed basis instead of ordering too much and risking dead stock. Dead stock is inventory that was never sold or used by customers before being removed from sale status. Safety Stock Inventory Safety stock inventory management is extra inventory being ordered beyond expected demand. This technique is used to prevent stockouts typically caused by incorrect forecasting or unforeseen changes in customer demand. FIFO and LIFO LIFO and FIFO are methods to determine the cost of inventory. FIFO, or First in, First out, assumes the older inventory is sold first. FIFO is a great way to keep inventory fresh. LIFO, or Last-in, First-out, assumes the newer inventory is typically sold first. LIFO helps prevent inventory from going bad. Reorder Point Formula The reorder point formula is an inventory management technique that’s based on a business’s own purchase and sales cycles that varies on a per-product basis. A reorder point is usually higher than a safety stock number to factor in lead time. Batch Tracking Batch tracking is a quality control inventory management technique wherein users can group and monitor a set of stock with similar traits. This method helps to track the expiration of inventory or trace defective items back to their original batch. Consignment Inventory 193 CU IDOL SELF LEARNING MATERIAL (SLM)

If you’re thinking about your local consignment store here, you’re exactly right. Consignment inventory is a business deal when a consigner (vendor or wholesaler) agrees to give a consignee (retailer like your favourite consignment store) their goods without the consignee paying for the inventory upfront. The consigner offering the inventory still owns the goods and the consignee pays for them only when they sell. Perpetual Inventory Management Perpetual inventory management is simply counting inventory as soon as it arrives. It’s the most basic inventory management technique and can be recorded manually on pen and paper or a spreadsheet. Drop shipping Drop shipping is an inventory management fulfilment method in which a store doesn’t actually keep the products it sells in stock. When a store makes a sale, instead of picking it from their own inventory, they purchase the item from a third party and have it shipped to the consumer. The seller never sees our touches of the product itself. Lean Manufacturing Lean is a broad set of management practices that can be applied to any business practice. Its goal is to improve efficiency by eliminating waste and any non-value-adding activities from daily business. Six Sigma Six Sigma is a brand of teaching that gives companies tools to improve the performance of their business (increase profits) and decrease the growth of excess inventory. Lean Six Sigma Lean Six Sigma enhances the tools of Six Sigma, but instead focuses more on increasing word standardization and the flow of business. Demand Forecasting Demand forecasting should become a familiar inventory management technique to retailers. Demand forecasting is based on historical sales data to formulate an estimate of the expected forecast of customer demand. Essentially, it’s an estimate of the goods and services a company expects customers to purchase in the future. Cross-Docking Cross-docking is an inventory management technique whereby an incoming truck unloads materials directly into outbound trucks to create a JIT shipping process. There is little or no storage in between deliveries. Bulk Shipments 194 CU IDOL SELF LEARNING MATERIAL (SLM)

Bulk shipments are a cost-efficient method of shipping when you palletize inventory to ship more at once. Examples of Successful Inventory Management MB Klein MB Klein, a historic retailer in Maryland of model trains, train sets, and railroad accessories, is a mutual client of SkuVault and BigCommerce. Mat Huffman, digital warehouse lead at MB Klein, told us about his experience using a SkuVault and BigCommerce integration to manage the company’s online store and brick-and-mortar store. “After having implemented SkuVault’s system, the task of maintaining our inventory has become more straightforward and efficient. Our customer satisfaction has improved with accelerated shipping times and the ability to accurately represent our stock in real-time with the BigCommerce integration for our website,” said Huffman. “The ease of only needing to upload new products into BigCommerce that migrate into SkuVault has been a tremendous advantage for dealing with workflow. One of the most beneficial aspects of utilizing SkuVault has been the ability to streamline, not only the inventory counts, but the ordering and receiving processes as well,” he said. “Overall, we have found the use of SkuVault to be invaluable, and the benefits have had an extensive impact on MB Klein’s growth during the past year.” 9.3 SUMMARY  Inventory management is about tracking and controlling a business’ inventory as it is bought, manufactured, stored, and used. It governs the entire flow of goods from purchasing right through to sale and ensuring always to have the right quantities of the right item in the right location at the right time.  Inventory is the goods that a company handles with the intention of selling. It might be raw materials that it buys and turns into something entirely new, or it might be a bulk product that is broken down into its constituent parts and sold separately. It could even be something completely intangible: software, for instance.  There are lots of different types of inventory, and which ones the company deals with depends on the goods it sells. Here’s an overview of some of the types that are more likely to encounter: a. Finished goods/for-sale goods: The products a company sells to its customers b. Raw materials: The inventory use by a company to make finished goods c. Work-in-progress: Essentially, unfinished goods inventory which is part-way through the manufacturing process. 195 CU IDOL SELF LEARNING MATERIAL (SLM)

d. MRO goods: MRO stands for maintenance, repair and operating. This is the inventory used to support the manufacturing process. e. Safety stock: The additional inventory is kept in store to deal with supplier shortages or surges in demand.  At a basic level, inventory management works by tracking products, components and ingredients across suppliers, stock on hand, production and sales to ensure that stock is used as efficiently and effectively as possible.  It can go as deep as required; for example, by examining the difference between dependent and independent demand, or forecasting sales to plan ahead. But at the end of the day, it all goes back to a company’s stock. The inventory management process goes through five stages such as purchasing, production, holding stock, sales and reporting.  Overall, inventory management is a broader concept. It takes the company or business through supply chain, manufacturing, fulfilment, sales and reporting into account. Almost any business will have to get an inventory management system in place. Otherwise, it would find it difficult in managing suppliers, production or sales.  After that, there are countless methods for storing and selling products better. Whether the focus is on optimising purchasing, control, production or sales, it is up to the requirement. However, if the inventory management is in place, then it would help running a business, serving customers and growing sales. 9.4 KEYWORDS  Economic Order Quantity-Economic order quantity is a technique used in inventory management. It refers to the optimal amount of inventory a company should purchase in order to meet its demand while minimizing its holding and storage costs.  Stockout Cost-Stockout cost is the lost income and expense associated with a shortage of inventory. This cost can arise in two ways, which are: Sales-related. When a company needs inventory for a production run and the inventory is not available, it must incur costs to acquire the needed inventory on short notice.  Just-in-time (JIT) Inventory-JIT involves holding as little stock as possible, negating the costs and risks involved with keeping a large amount of stock on hand.  ABC Analysis- This technique aims to identify the inventory, i.e., earning profit, by classifying goods into different tiers.  Consignment- This technique allows a consignor, usually a wholesaler, to give their goods to a consignee, usually a retailer, without the consignee paying for the goods up 196 CU IDOL SELF LEARNING MATERIAL (SLM)

front. The consignor still owns the goods, and the consignee pays for the goods only when they actually sell. This sounds great but it also carries major risks. 9.5 LEARNING ACTIVITY 1. A manufacturing company purchases 9000 parts of a machine for its annual requirements, ordering one month usage at a time. Each part costs Rs.20. The ordering cost per order is Rs. 15 and the carrying charges are 15% of the average inventory per year. Suggest a more economic purchasing policy for the company. How much would it save the company per year? ___________________________________________________________________________ ___________________________________________________________________________ 2. Demand in an inventory system is at a constant and uniform rate of 2400 kg. per year. The carrying cost is Rs. 5 per kg. per year. No shortage is allowed. The replenishment cost is Rs. 22 per order. The lot size can only be in 100 kg. unit. What is the optimal lot size of the system? ___________________________________________________________________________ ___________________________________________________________________________ 9.6 UNIT END QUESTIONS A. Descriptive Questions Short Questions 1. What are the goals of inventory management? 2. Who is responsible for making inventory management effectively? 3. What to determine with regards to ordering frequency? 4. What is demand forecasting? 5. Brief on the technique of cross docking. Long Questions 1. What are the ways to manage inventory successfully? 2. What are the parameters to determine that the inventory manager is performing poorly? 3. How should an inventory manager prepare to tackle peak season? 4. In what circumstance, ABC analysis is used? 197 CU IDOL SELF LEARNING MATERIAL (SLM)

5. What are the assumptions to determine EOQ through an analytical method? B. Multiple Choice Questions 1. Which are basic inventory decisions? a. How much to order? b. When to order? c. How much safety stock should be kept? d. All of these 2. Which is not a reason for carrying inventory? a. Improved customer service b. Increased cost c. Maintenance of operational capability d. Irregular supply and demand 3. What is known as the cost associated with maintaining inventory? a. Product cost b. Holding cost c. Stock out cost d. Set up cost 4. What is the goal of economic order quantity? a. Minimize related costs b. Maximize related costs c. Holding cost d. Enhancing other costs 5. Which is not a step involving EOQ by tabulation method? a. Select the number of possible lot sizes to purchase b. Determine average inventory carrying cost for the lot purchased c. Determine the half ordering cost for the orders placed d. Determine the total cost for each lot size chosen Answers 1-d 2-b 3-b 4-a 5-c 198 CU IDOL SELF LEARNING MATERIAL (SLM)

9.7 REFERENCES References  How to Find the Right Economic Order Quantity | by EazyStock | Medium  (1) New Messages! (bigcommerce.com)  da2a0Paper 4.pdf (amity.edu) Textbooks  1527249935E-textofChapter5Module1.pdf (inflibnet.ac.in)  Disney, S.M., Potter, A.T. and Gardner, B.M. (2003), “The impact of vendor managed inventory  on transport operations”, Transportation Research: Part E, Vol. 39 No. 5, pp. 363-80  Inventory Control and Management, Second Edition (wordpress.com) Websites  Inventory Management 101: Helpful Techniques + Methods (2021) (bigcommerce.com)  INVENTORY CONTROL OR MANAGEMENT in Production and Operations Management Tutorial 01 September 2021 - Learn INVENTORY CONTROL OR MANAGEMENT in Production and Operations Management Tutorial (9599) | Wisdom Jobs India  What is Inventory Management? Techniques for 2021 (unleashedsoftware.com) 199 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 10 – PERT/CPM STRUCTURE 10.0 Learning Objectives 10.1 Introduction 10.2 Process 10.3 Methods to Calculate Network 10.4 Project Crashing 10.5 Summary 10.6 Keywords 10.7 Learning Activity 10.8 Unit End Questions 10.9 References 10.0 LEARNING OBJECTIVES After studying this unit, you will be able to:  Identify the PERT analysis and its purpose  Elaborate on the advantages of CPM  Identify the ways to crash a project successfully 10.1 INTRODUCTION Program Evaluation and Review Technique (PERT) is a method used to examine the tasks in a schedule and determine a Critical Path Method variation (CPM). It analyses the time required to complete each task and its associated dependencies to determine the minimum time to complete a project. It estimates the shortest possible time each activity will take, the most likely length of time, and the longest time that might be taken if the activity takes longer than expected. The US Navy developed the method in 1957 on the Polaris nuclear submarine project. PERT Analysis informs Program Managers and project personnel on the project’s tasks and the estimated amount of time required to complete each task. By utilizing this information a Program Manager will be able to estimate the minimum amount of time required to complete the entire project. This helps in the creation of more realistic schedules and cost estimates. Definition of PERT 200 CU IDOL SELF LEARNING MATERIAL (SLM)