Acme Soap Company markets one brand of soap, called Standard Acme (SA), and Best Soap Company markets two brands, Standard Best (SB) and Deluxe Best (DB). Currently, Acme has 40% of the market, and the remainder is divided equally between the two Best brands. A proposed new Acme brand, called brand X, was test-marketed in several large cities, producing the transition matrix below for the consumers' weekly buying habits. SB DB SA X SB 0.2 0.3 0.4 0.1 DB 0.3 0.5 0.1 0.1 P = SA 0.2 0.3 0.3 0.2 X 0.5 0.1 0.1 0.3 Assuming that P represents the consumers buying habits over a long period of time, use this transition matrix and the initial-state matrix So = [0.3 0.3 0.4 0 ] to compute successive state matrices in order to approximate the elements in the stationary matrix. If Acme decides to market this new soap, what is the long-run expected total market share for their two soaps? C Let S₁ represent the state matrix one week after introducing brand X to the market. Write the equation in terms of P and So that can be used to calculate S₁. Find the stationary matrix. S= (Type an integer or decimal for each matrix element. Round to two decimal places as needed.) In the long-run, the total market share for Acme's two soaps is expected to be %. (Round to the nearest percent as needed.)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Acme Soap Company markets one brand of soap, called Standard
Acme (SA), and Best Soap Company markets two brands,
Standard Best (SB) and Deluxe Best (DB). Currently, Acme has
40% of the market, and the remainder is divided equally between
the two Best brands. A proposed new Acme brand, called brand X,
was test-marketed in several large cities, producing the transition
matrix below for the consumers' weekly buying habits.
SB DB SA X
SB
0.2 0.3 0.4 0.1
DB
0.3 0.5 0.1 0.1
P =
SA
0.2 0.3 0.3 0.2
X 0.5 0.1 0.1 0.3
Assuming that P represents the consumers buying habits over a
long period of time, use this transition matrix and the initial-state
matrix So = [0.3 0.3 0.4 0 ] to compute successive state
matrices in order to approximate the elements in the stationary
matrix. If Acme decides to market this new soap, what is the
long-run expected total market share for their two soaps?
Let S₁ represent the state matrix one week after
introducing brand X to the market. Write the equation in
terms of P and So that can be used to calculate S₁.
Find the stationary matrix.
S=
(Type an integer or decimal for each matrix element.
Round to two decimal places as needed.)
In the long-run, the total market share for Acme's two
soaps is expected to be %. (Round to the nearest
percent as needed.)
Transcribed Image Text:Acme Soap Company markets one brand of soap, called Standard Acme (SA), and Best Soap Company markets two brands, Standard Best (SB) and Deluxe Best (DB). Currently, Acme has 40% of the market, and the remainder is divided equally between the two Best brands. A proposed new Acme brand, called brand X, was test-marketed in several large cities, producing the transition matrix below for the consumers' weekly buying habits. SB DB SA X SB 0.2 0.3 0.4 0.1 DB 0.3 0.5 0.1 0.1 P = SA 0.2 0.3 0.3 0.2 X 0.5 0.1 0.1 0.3 Assuming that P represents the consumers buying habits over a long period of time, use this transition matrix and the initial-state matrix So = [0.3 0.3 0.4 0 ] to compute successive state matrices in order to approximate the elements in the stationary matrix. If Acme decides to market this new soap, what is the long-run expected total market share for their two soaps? Let S₁ represent the state matrix one week after introducing brand X to the market. Write the equation in terms of P and So that can be used to calculate S₁. Find the stationary matrix. S= (Type an integer or decimal for each matrix element. Round to two decimal places as needed.) In the long-run, the total market share for Acme's two soaps is expected to be %. (Round to the nearest percent as needed.)
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