To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna Markov Analysis

To accompany Quantitative Analysis for Management, 8e by Render/Stair/HannaIntroduction Markov analysis is a technique that deals with the probabilities of future occurrences by analyzing presently known probabilities. The technique has numerous applications in business, including market share analysis, bad debt prediction, university enrollment predictions, and determining whether a machine will break down in the future.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna Introduction Markov analysis makes the assumption that the system starts in an initial state or condition. For example, two competing manufacturers might have 40% and 60% of the market sales, respectively, as initial states. Perhaps in two months the market shares for the two companies will change to 45% and 55% of the market, respectively. Predicting these future states involves knowing the system's likelihood or probability of changing from one state to another.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna Introduction For a particular problem, these probabilities can be collected and placed in a matrix or table. This matrix of transition probabilities shows the likelihood that the system will change from one time period to the next. This is the Markov process, and it enables us to predict future states or conditions.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna Assumptions of Markov Analysis There are a limited or finite number of possible states. The probability of change remains the same over time. Future state predictable from current state. Size of system remains the same.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna States and State Probabilities States are used to identify all possible conditions of a process or a system. For example, a machine can be in one of two states at any point in time. It can be either functioning correctly or not functioning correctly. We can call the proper operation of the machine the first state, and we can call the incorrect functioning the second state. Indeed, it is possible to identify specific states for many processes or systems.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna States and State Probabilities In Markov analysis we also assume that the states are both collectively exhaustive and mutually exclusive. Collectively exhaustive means that we can list all of the possible states of a system or process. Our discussion of Markov analysis assumes that there is a finite number of states for any system. Mutually exclusive means that a system can be in only one state at any point in time. A student can be in only one of the three management specially areas and not in two or more areas at the same time. It also means that a person can only be a customer of one of the three grocery stores at any point in time.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna States and State Probabilities After the states have been identified, the next step is to determine the probability that the system is in this state. Such information is then placed into a vector of the probabilities.  (i) = vector of State probabilities = (  1,  2,  3, …  n) Where n = number of states  1,  2,  3, …  n = probability of being in state 1, state 2, ….., state n.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna States and State Probabilities In some cases, where we are only dealing with one item, such as one machine, it is possible to know with complete certainty what state this item is in. For example, if we are investigating only one machine, we may know that at this point in time the machine is functioning correctly. Then the vector of states can be represented as follows:  (1)=(1,0) Where  (1)= vector of states for the machine in period 1  1=1= probability of being in the first state  2=0= probability of being in the second state

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna States and State Probabilities The last one machine example shows that the probability the machine is functioning correctly, state 1, is 1, and the probability that the machine is functioning incorrectly, state 2, is 0 for the first period. In most cases, however, we are dealing with more than one item.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna Grocery Store Example Let's look at the vector of states for people in the small town with the three grocery stores. There could be a total of 100,000 people that shop at the three grocery stores during any given month. Forty thousand people may be shopping at American Food Store, which will be called state 1. Thirty thousand people may be shopping at Food Mart, which will be called state2, and 30,000 people may be shopping at Atlas Foods, which will be called state 3. The probability that a person will be shopping at one of these three grocery stores is as follows: State 1 – American Food Store: 40,000/100,000=0.40=40% State 2– Food Mart: 30,000/100,000=0.30=30% State 3 – Atlas Foods: 30,000/100,000=0.30=30%

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna Grocery Store Example These probabilities can be placed in the vector of state probabilities shown as follows:  (1)=(0.4, 0.3, 0.3) Where  (1)= vector of state probabilities for the three grocery stores for period 1  1= 0.4= probability that a person will shop at American Food, state1  2= 0.3= probability that a person will shop at Food Mart, state2  3= 0.3= probability that a person will shop at Atlas Food, state3

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna Grocery Store Example You should also notice that the probabilities in the vector of states for the three grocery stores represent the market shares for these three stores for the first period. Thus, American Food has 40% of the market, Food Mart has 30%, and Atlas Foods has 30% of the market in period 1. when we are dealing with market shares, the market shares can be used in place of probability values. When the initial states and state probabilities have been determined, the next step is to find the matrix of transition probabilities. This matrix is used along with the state probabilities in predicting the future.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna Matrix of Transition Probabilities The concept that allows us to get from a current state, such as market shares, to a future state is the matrix of transitions probabilities. This is a matrix of conditional probabilities of being in a future state given a current state. The following definition is helpful: Let P ij = conditional probability of being in state j in the future given the current state of i. For example, P 12 is the probability of being in state 2 in the future given the event was in state 1 in the period before.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna Matrix of Transition Probabilities Let P = matrix of transition probabilities Individual P ij values are usually determined empirically. For example, if we have observed over time that 10% of the people currently shopping at store 1 (or state 1) will be shopping at store 2 (state 2) next period, then we know that P 12 = 0.1 or 10%.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna Transition Probabilities for the Three Grocery Stores Let's say we can determine the matrix of transition probabilities for the three grocery stores by using historical data. The results of our analysis appear in the following matrix: Recall that American Food represents state 1, Food Mart is state 2, and Atlas Foods is state 3. The meaning of these probabilities can be expressed in terms of the various states, as follows:

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna Transition Probabilities for the Three Grocery Stores Row = P11 = probability of being in state 1 after being in state 1 the preceding period. 0.1 = P12 = probability of being in state 2 after being in state 1 the preceding period. 0.1 = P13 = probability of being in state 3 after being in state 1 the preceding period. Row = P21 = probability of being in state 1 after being in state 2 the preceding period. 0.7 = P22 = probability of being in state 2 after being in state 2 the preceding period. 0.2 = P23 = probability of being in state 3 after being in state 2 the preceding period.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna Transition Probabilities for the Three Grocery Stores Row = P31 = probability of being in state 1 after being in state 3 the preceding period. 0.2 = P32 = probability of being in state 2 after being in state 3 the preceding period. 0.6 = P33 = probability of being in state 3 after being in state 3 the preceding period. Note that the three probabilities in the top row sum to 1. The probabilities for any row in a matrix of transition probabilities will also sum to 1. After the state probabilities have been determined along with the matrix of transition probabilities, it is possible to predict future state probabilities.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna Predicting Future Markets Shares Given the vector of state probabilities and the matrix of transition probabilities, it is not very difficult to determine the state probabilities at a future date. With this type of analysis, we are able to compute the probability that a person will be shopping at one of the grocery stores in the future. Because this probability is equivalent to market share, it is possible to determine future market shares for American Food, Food Mart, and Atlas Foods.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna Predicting Future Markets Shares When the current period is 1, calculating the state probabilities for the next period (period 2) can be accomplished as follows:  (2)=  (1)P ………… (1) Furthermore, if we are in any period n, we can compute the state probabilities for period n+1 as follows:  (n+1)=  (n)P ………… (2) Equation (1) can be used to answer the question of next period's market shares for the grocery stores. The computations are:

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna Predicting Future Markets Shares  (2)=  (1)P = [(0.4)(0.8) + (0.3)(0.1) + (0.3)(0.2), (0.4)(0.1) + (0.3)(0.7)+ (0.3)(0.2), (0.4)(0.1) + (0.3)(0.2) + (0.3)(0.6)] = (0.41, 0.31, 0.28) As you can see, the market share for American Food and Food Mart has increased while the market share for Atlas Foods has decreased.