Question

In: Operations Management

Markov Processes (Market Share Analysis) In Westvale, a small rural town in Maine, virtually all shopping...

Markov Processes (Market Share Analysis)
In Westvale, a small rural town in Maine, virtually all shopping and business are done in the town. The town has one farm and garden center that sells fertilizer to the local farmers and gardeners. The center carries three brands of fertilizer--- Plant plus, Crop extra, and Gro-fast--- so every person in the town who uses fertilizer uses one of the three brands. The garden center has 9,000 customers for fertilizer each spring. An extensive market research study has determined that customers switch brands of fertilizer according to the following probability transition matrix

	Next Spring
This Spring	Plant Plus (P)	Crop Extra (C)	Gro-fast (G)
Plant Plus (P)	.4	.3	.3
Crop Extra (C)	.5	.1	.4
Gro-fast (G)	.4	.2	.4

The number of customers presently using each brand of fertilizer is shown in the following table

Fertilizer Brand	Customers
Plant Plus	3000
Crop Extra	4000
Gro-fast	2000

A) Determine the steady-state probabilities for the fertilizer brands.
B) Forecast the customer demand for each brand of fertilizer in the long run and the changes in customer demand.

Expert Solution

A) Steady-state probabilities are determined as below

P = 0.422

C = 0.220

G = 0.358

B) Total number of customers in each spring = 9000

Forecast customer demand for brand P in the long run = 9000*0.422 = 3798

Forecast customer demand for brand C in the long run = 9000*0.220 = 1980

Forecast customer demand for brand G in the long run = 9000*0.220 = 3222

Changes in customer demand for P = 3798 - 3000 = 798 increase

Changes in customer demand for C = 1980 - 4000 = 2020 decrease

Changes in customer demand for G = 3222 - 2000 = 1222 increase

keosha answered 1 year ago

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