Markov Chains Applications

Brand Switching  100 customers currently using brand A 84 will stay with A, 9 will switch to B, 7 will switch to C  100 customers currently using brand B 78 will stay with B, 14 will switch to A, 8 will switch to C  100 customers currently using brand C 90 will stay with c, 4 will switch to A, 6 will switch to B

Brand Switching  States Brands of product  Time Period Interval between purchases  Transition Probabilities

Brand Switching  States Brands of product  Time Period Interval between purchases  Transition Probabilities ABC P  F H G G I K J J

Brand Switching If the current market share is given by: a.What is the market share after one time period ? b.What is long term market share ? P () (...) 

Brand Switching Market share in 1 time period P () (...) (...)  F H G G I K J J 

Brand Switching Long term market share

Brand Switching Long term market share P n () (...) (.  F H G G I K J J  )

Brand Switching PPP P nn ABC ABC ABC ()()() ()   F H G G I K J J 0 0    P A n AAA A A ().().().() (...)      

Brand Switching  Assumptions  Markovian Property Brand switching might take into consideration more than just the past brand; e.g. customer dissatisfaction with brand A led to B, marketing lead in led to C  Stationarity Property Marketing strategies may change transition probabilities over time

Stock Market Analysis  States Price of Stock: $0, 5, 10, 15, 20  Time increment Opening price on successive trading days  Transition Probabilities Prob going up 5 = a i = ¼ Prob going down 5 = b i = ¼ Prob staying same = c i = ½

Stock Market Analysis  Transition Probabilities FI P  H G G G G G G G G G G G K J J J J J J J J J J J

Stock Market Analysis  Assumptions  Markovian Property  Stationarity Property

Equipment Replacement  States State 1: New Filter State 2: One year old, no repairs State 3: Two years old, no repairs State 4: Repaired once  Time Period One year  Transition Probabilities Repaired filter will be replaced after one year Filters scrapped after 3 years use

Equipment Replacement  Transition Probabilities

Equipment Replacement  Transition Probabilities P  F H G G G G I K J J J J

Equipment Replacement  Steady State Probabilities P  F H G G G G I K J J J J        ......

Equipment Replacement  Steady State Probabilities P  F H G G G G I K J J J J                ....

Equipment Replacement

 Suppose that the following costs apply  New Filters$ 500  Repair Filter$ 150  Scrap filters ($ 50)(scrap salvage)  For a pool of 100 filters, on average, what is the expected cost of our repair policy ?

Equipment Replacement  Suppose that the following costs apply  New Filters$ 500  Repair Filter$ 150  Scrap filters ($ 50)(scrap salvage) ECostCj buyrepairsalvage j []() ()()()().().()(..)() $201.        / filter

Population Mobility  Forest consists of 4 major species  Aspen  Birch  Oak  Maple ABOM P  F H G G G G I K J J J J ABOMABOM

Population Mobility  States State 1: No movement State 2: Movement within region State 3: Movement out of a region State 4: Movements into a region  Time period  Transition Probabilities

Population Mobility  States: Movement at SDSM&T State 1: Student remains in major State 2: Student switches major State 3: Student leaves school (transfer out, matriculates) State 4: Student enters school (transfer in, first year studs.)  Time period Semester  Transition Probabilities

Population Mobility  States: Movement at SDSM&T State 1: Student in IE State 2: Student in other major State 3: Student leaves school (transfer out, matriculates) State 4: Student enters school (transfer in, first year studs.)  Time period Semester  Transition Probabilities P  F H G G G G I K J J J J