Markov Processes ManualComputer-Based Homework Solution MGMT E-5070

Machine Operation Problem A manufacturing firm has developed a transition matrix containing the probabilities that a particular machine will operate or break down the probabilities that a particular machine will operate or break down in the following week, given its operating condition in the present week. This Week Next Week OperateNext Week Breakdown Operate.4.6 Break Down.8.2 REQUIREMENT: Assuming that the machine is operating in week 1, that is, the initial state is (.4,.6 ) : 1.Determine the probabilities that the machine will operate or break down in weeks 2, 3, 4, 5, and 6. 2, 3, 4, 5, and 6. 2.Determine the steady-state probabilities for this transition matrix algebraically and indicate the percentage of future weeks in which the machine will break down. indicate the percentage of future weeks in which the machine will break down.

Machine Operation Problem Week No. 2 (.4,.6 )

Machine Operation Problem Week No. 3 (.64,.36 )

Machine Operation Problem Week No. 4 (.544,.456 )

Machine Operation Problem Week No. 5 (.5824,.4176 )

Machine Operation Problem Week No. 6 (.56704, )

Machine Operation Problem.4X 1.6X 1.4X 1.6X 1.8X 2.2X 2.8X 2.2X 2 P(O) = 1X 1 P(B) = 1X 2 OPERATE BREAKDOWN P (O) =.4X 1 +.8X 2 = 1X 1 ( dependent equation ) P (B) =.6X 1 +.2X 2 = 1X 2 ( dependent equation ) 1X 1 + 1X 2 = 1 ( independent equation ) 1X 1 + 1X 2 = 1 ( independent equation )

Machine Operation Problem.6X 1 +.2X 2 – 1.0X 2 = 0 becomes…… becomes…….6X 1 -.8X 2 = 0.6X 1 -.8X 2 = 0 SET DEPENDENT EQUATIONS EQUAL TO ZERO.4X 1 +.8X 2 – 1.0X 1 = 0 becomes…… becomes…… -.6X 1 +.8X 2 = 0 -.6X 1 +.8X 2 = 0

Machine Operation Problem.6X 1 -.8X 2 = 0.6X 1 -.8X 2 = 0.6 ( 1X 1 + 1X 2 = 1 ).6X 1 +.6X 2 =.6.6X 1 +.6X 2 = X 2 = X 2 = -.6 X 2 =.4285 = P ( BREAKDOWN ) X 2 =.4285 = P ( BREAKDOWN ) Since X 1 + X 2 = 1, then: 1 – X 2 = X 1 1 – X 2 = X =.5715 = P ( OPERATION ) =.5715 = P ( OPERATION ) STEADY-STATE PROBABILITIES

Newspaper Problem A city is served by two newspapers – The Tribune and the Daily News. Each Sunday, readers purchase one of the newspapers at a stand. The following transition matrix contains the probabilities of a customer’s buying a particular newspaper in a week, given the newspaper purchased the previous Sunday.

Newspaper Problem ( This Sunday ) Tribune ( Next Sunday ) Daily News ( Next Sunday ) Tribune Daily News REQUIREMENT: 1.Determine the steady-state probabilities for the transition matrix algebraically, and explain what they mean.

Newspaper Problem.65 X 1.35 X 1.65 X 1.35 X 1.45 X 2.55 X 2.45 X 2.55 X 2 P(T) = X 1 P(DN) = X 2 Tribune Daily News Tribune Daily News

Newspaper Problem P ( T ) =.65X X 2 = 1X 1 ( dependent equation ) P ( DN ) =.35X X 2 = 1X 2 ( dependent equation ) 1X 1 + 1X 2 = 1 ( independent equation ) 1X 1 + 1X 2 = 1 ( independent equation )

Newspaper Problem SET DEPENDENT EQUATIONS EQUAL TO ZERO.65X X 2 = 1X 1.65X X 2 = 1X 1.65X X 2 – 1X 1 = X X 2 = X X 2 = 0.35X X 2 = 1X 2.35X X 2 – 1X 2 = 0.35X X 2 = 0

Newspaper Problem STEADY - STATE PROBABILITIES.35X X 2 = 0.35 ( 1X 1 + 1X 2 = 1 ).35X X 2 = X 2 = X 2 = -.35 X 2 =.4375 = P ( Daily News ) X 2 =.4375 = P ( Daily News ) Since X 1 + X 2 = 1, then: 1 – X 2 = X =.5625 = P ( Tribune )

Fertilizer Problem In Westville, a small rural town in Maine, virtually all shopping and business is 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.

Fertilizer Problem Plant PlusCrop ExtraGro-Fast Plant Plus.4.3 Crop Extra Gro-Fast NEXT SPRING THIS SPRING PROBABILITY TRANSITION MATRIX

Fertilizer Problem The number of customers presently using each brand of fertilizer is shown below:BrandCustomers Plant Plus3,000 Crop Extra4,000 Gro-Fast2,000

Fertilizer Problem REQUIREMENT: 1.Determine the steady-state probabilities for the fertilizer brands. 2.Forecast the customer demand for each brand of fertilizer in the long run and the changes in customer demand.

Fertilizer Problem Transition Matrix Plant Plus Crop Extra Gro Fast

Fertilizer Problem Transition Matrix Plant Plus Crop Extra Gro Fast.4X 1.3X 1.3X 1.4X 1.3X 1.3X 1.5X 2.1X 2.4X 2.5X 2.1X 2.4X 2.4X 3.2X 3.4X 3.4X 3.2X 3.4X 3 P (PP) = 1X 1 P(CE) = 1X 2 P(GF) = 1X 3 P (PP) = 1X 1 P(CE) = 1X 2 P(GF) = 1X 3

Fertilizer Problem THE EQUATIONS P (PP) =.4X 1 +.5X 2 +.4X 3 = 1X 1 ( DEPENDENT ) P (CE) =.3X 1 +.1X 2 +.2X 3 = 1X 2 ( DEPENDENT ) P (GF) =.3X 1 +.4X 2 +.4X 3 = 1X 3 ( DEPENDENT ) 1X 1 + 1X 2 + 1X 3 = 1 ( INDEPENDENT ) 1X 1 + 1X 2 + 1X 3 = 1 ( INDEPENDENT )

Fertilizer Problem SET DEPENDENT EQUATIONS EQUAL TO ZERO P (PP) =.4X 1 +.5X 2 +.4X X 1 = 0 P (CE) =.3X 1 +.1X 2 +.2X X 2 = 0 P (GF) =.3X 1 +.4X 2 +.4X 3 – 1.0X 3 = 0 1X 1 + 1X 2 + 1X 3 = 1 ( INDEPENDENT ) 1X 1 + 1X 2 + 1X 3 = 1 ( INDEPENDENT )

Fertilizer Problem SET DEPENDENT EQUATIONS EQUAL TO ZERO P (PP) = -.6X 1 +.5X 2 +.4X 3 = 0 P (CE) =.3X 1 -.9X 2 +.2X 3 = 0 P (GF) =.3X 1 +.4X 2 -.6X 3 = 0

Fertilizer Problem.3X 1 -.9X 2 +.2X 3 = 0.3X 1 +.4X 2 -.6X 3 = X 2 +.8X 3 = X 2 +.8X 3 = 0.3 ( 1X 1 + 1X 2 + 1X 3 = 1.0 ).3 ( 1X 1 + 1X 2 + 1X 3 = 1.0 ).3X 1 +.3X 2 +.3X 3 =.3.3X 1 +.3X 2 +.3X 3 =.3.3X 1 +.4X 2 -.6X 3 = 0.3X 1 +.4X 2 -.6X 3 = 0 -.1X 2 +.9X 3 =.3 -.1X 2 +.9X 3 =.3 CANCEL OUT VARIABLE X 1

Fertilizer Problem CANCEL OUT VARIABLE X X 2 +.8X 3 = X 2 +.8X 3 = (.1X 2 +.9X 3 =.3 ) -13 (.1X 2 +.9X 3 =.3 ) - 1.3X X 3 = X X 3 = X 3 = X 3 = X 3 = X 3 =

Fertilizer Problem -1.3X (.358 ) = X 2 = X 2 = X 2 =.220 X 2 =.220 X = 1.0 X 1 = X 1 = X 1 =.422 X 1 =.422 SOLVING FOR THE REMAINING VARIABLES

Fertilizer Problem Fertilizer Brand Present Customers Long-Term Market Share Long-Term Customers Plant Plus3, ,798 Crop Extra4, ,980 Gro-Fast2, ,222 Σ = 9, ,000 Σ = 9, ,000