13.1 OVERVIEWThe Markov processes m

13.1 OVERVIEW
The Markov processes module of The Management Scientist will analyze prob¬lems with up to 10 states. Input to the program is the matrix of transition prob¬abilities for the states. The solution provides the steady-state probabilities for the states.
In some applications it may not be possible to make a transition out of one or more states once the state has been reached. Such states are referred to as absorbing states. The Markov processes module will solve problems for which the total number of absorbing and nonabsorbing states is 10 or less. In applications with absorbing states, the solution provides the probability that units currently in each of the nonabsorbing states will eventually end up in each of the absorbing states.
13.2 AN EXAMPLE PROBLEM
Two grocery stores in a small town compete for customers. Each customer makes one shopping trip per week to one of the two stores. A survey of store loyalty among the customers shows that for customers who shop at Murphy’s Foodliner one week, 90% will shop at Murphy’s the next week and 10% will switch to Ashley’s Supermarket. For customers who shop at Ashley’s Super¬market one week, 20% will switch to Murphy’s the next week and 80% will shop again at Ashley’s. These transition probabilities are summarized as follows:
Current Weekly Next Weekly Shopping Period
Shopping Period Murphy’s Foodliner Ashley’s Supermarket
Murphy’s Foodliner 0.9 0.1
Ashley’s Supermarket 0.2 0.8
What are the steady-state probabilities for the two grocery stores? If there are 1,000 customers that make weekly shopping trips to one of the two stores, how many customers can be expected to shop at each of the stores?
13.3 CREATING AND SOLVING A PROBLEM
To determine the steady-state probabilities for the two grocery stores in our example problem, we begin by selecting the Markov processes module and choosing New from the File menu. When the Markov Processes dialog box appears, enter 2 for the Number of States and choose OK; the Transition Matrix data input screen will then appear. Figure 13.1 shows the Transition Matrix after entering the transition probabilities for the grocery store example problem. After selecting Solve from the Solution menu, we obtain the output shown in Figure 13.2.


Figure 13.1 Transition Matrix Data Input Screen


As you can see, Murphy’s Foodliner (state 1) has the higher steady-state probability. Thus, we conclude that in the long run, Murphy’s Foodliner will have a 66.7% share of the market and Ashley’s Supermarket will have the remaining 33.3% share of the market. With 1,000 weekly customers, 667 should shop at Murphy’s and 333 should shop at Ashley’s.


Figure 13.2 Output for the Grocery Store Market Share Problem
13.4 AN EXAMPLE PROBLEM
WITH ABSORBING STATES
Heidman’s Department Stores has two aging categories for its accounts receiv¬able: (1) accounts that are classified as 0 to 30 days old and (2) accounts that are classified as 31 to 90 days old. If any portion of an account balance exceeds 90 days, that portion is written off as a bad debt. The total account balance for each customer is placed in the age category corresponding to the oldest unpaid amount; hence, this method of aging accounts receivable is called the total balance method.
Let us assume that Heidman’s shows a total of $3,000 in its accounts receivable and the firm’s management would like an estimate of how much of the $3,000 will eventually be collected and how much will result in bad debts.
To see how we can view the accounts receivable operation as a Markov process, consider what happens to one dollar currently in accounts receivable. As the firm continues to operate into the future, we can consider each week as a trial

Results (Indonesian) 1:

13.1 IKHTISARModul proses Markov The manajemen ilmuwan akan menganalisis prob¬lems dengan Serikat hingga 10. Input ke program adalah matriks prob¬abilities transisi untuk Serikat. Solusi ini memberikan probabilitas mapan untuk Serikat. Dalam beberapa aplikasi mungkin tidak mungkin untuk membuat transisi dari satu atau lebih negara ketika negara telah tercapai. Keadaan seperti itu disebut sebagai negara menyerap. Modul proses Markov akan memecahkan masalah yang jumlah negara menyerap dan nonabsorbing 10 atau kurang. Dalam aplikasi dengan menyerap Serikat, solusi ini memberikan kemungkinan bahwa unit saat ini di tiap negara nonabsorbing akhirnya akan berakhir di tiap negara menyerap.13.2 CONTOH MASALAHDua Toko di sebuah kota kecil bersaing untuk pelanggan. Setiap pelanggan membuat perjalanan belanja satu per minggu untuk salah satu dari dua toko. Sebuah survei toko loyalitas antara pelanggan menunjukkan bahwa untuk pelanggan yang berbelanja di Murphy Foodliner satu minggu, 90% akan berbelanja di Murphy minggu berikutnya dan 10% akan beralih ke Ashley Supermarket. Untuk pelanggan yang berbelanja di Ashley Super¬market satu minggu, 20% akan beralih ke Murphy minggu berikutnya dan 80% akan berbelanja lagi di Ashley. Probabilitas transisi ini diringkas sebagai berikut:Saat ini unduhan mingguan berikutnya belanja periodeBelanja Supermarket periode Murphy Foodliner AshleyMurphy’s Foodliner 0.9 0.1Ashley’s Supermarket 0.2 0.8What are the steady-state probabilities for the two grocery stores? If there are 1,000 customers that make weekly shopping trips to one of the two stores, how many customers can be expected to shop at each of the stores?13.3 CREATING AND SOLVING A PROBLEMTo determine the steady-state probabilities for the two grocery stores in our example problem, we begin by selecting the Markov processes module and choosing New from the File menu. When the Markov Processes dialog box appears, enter 2 for the Number of States and choose OK; the Transition Matrix data input screen will then appear. Figure 13.1 shows the Transition Matrix after entering the transition probabilities for the grocery store example problem. After selecting Solve from the Solution menu, we obtain the output shown in Figure 13.2.Figure 13.1 Transition Matrix Data Input Screen As you can see, Murphy’s Foodliner (state 1) has the higher steady-state probability. Thus, we conclude that in the long run, Murphy’s Foodliner will have a 66.7% share of the market and Ashley’s Supermarket will have the remaining 33.3% share of the market. With 1,000 weekly customers, 667 should shop at Murphy’s and 333 should shop at Ashley’s.Figure 13.2 Output for the Grocery Store Market Share Problem13.4 AN EXAMPLE PROBLEM WITH ABSORBING STATESHeidman’s Department Stores has two aging categories for its accounts receiv¬able: (1) accounts that are classified as 0 to 30 days old and (2) accounts that are classified as 31 to 90 days old. If any portion of an account balance exceeds 90 days, that portion is written off as a bad debt. The total account balance for each customer is placed in the age category corresponding to the oldest unpaid amount; hence, this method of aging accounts receivable is called the total balance method. Let us assume that Heidman’s shows a total of $3,000 in its accounts receivable and the firm’s management would like an estimate of how much of the $3,000 will eventually be collected and how much will result in bad debts. To see how we can view the accounts receivable operation as a Markov process, consider what happens to one dollar currently in accounts receivable. As the firm continues to operate into the future, we can consider each week as a trial