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Brandon Blaydes Jonathan Gutierrez Ismael Reyes IE 417 Operations Research II Winter 2011

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Dr. Parisays comments are in red. I modified some, but needs some more. In our example we are interested in the population size in different groups. It is useful to be able to predict how many people, classified in groups are present in the steady state.

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In steady state census, the idea is: Number of people entering group during each period = Number of people leaving group during each period.

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H i = number hired for group i at the beginning of each period. N i = number in group i at steady state. P ki = probability of those entering and leaving a certain state.

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Number of people entering group i during each period = H i + Σ ki N k p ki Number of people leaving group i during each period = N i Σ ki p ki Hence: H i + Σ ki N k p ki = N i Σ ki p ki

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Suppose that each American can be classified into one of these four groups: Children Working adults Retired people Dead

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During a one-year period,.959 of all children remain children,.04 of all children become working adults, and.001 of all children die. During any given year,.96 of all working adults remain working adults,.03 of all working adults retire, and.01 of all working adults die. Also,.95 of all retired people remain retired, and.05 of all retired people die. One thousand children are born each year.

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It is important to notice that for this example we have a type of isolated population since the only group that can enter the system every year are the 1000 new born children. Also notice how the only absorbing state is Dead, but it can be disregarded since none of the states communicate, and it affects no other state including itself, meaning you can not die twice or be reborn.

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Group 1 = children,H 1 =1000 Group 2 = working adults, H 2 =0 Group 3 = retired people, H 3 =0 Group 4 = dead, H 4 =0

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In steady state situation we should have: 1000 = (.04 +.001)N 1 (children).04N 1 = (.03 +.01)N 2 (working adults).03N 2 =.05N 3 (retired people)

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Probabilities across the matrix are complimentary

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Solving the linear system of equations can give us the number of people in each group in each year in steady state situation. Please notice that it is possible that these equations do not have an answer which means steady state does not exists. N 1 = 24, 390 N 2 = 24, 390.24 N 3 = 14, 634.14

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Since in the steady state, there are 14, 634.14 retired people, in the steady state they receive 14, 634.14(5,000) dollars per year. Hence, each working adult must pay tax as much as 14, 634.14(5,000)/ 24, 390.24 = $3,000 per year

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Suppose that advances in medical science have reduced the annual death rate for retired people from 5% to 3%. By how much would this increase the annual tax that a working adult would pay to cover the pension fund?

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1000 = (.04 +.001)N 1 (Children).04N 1 = (.03 +.01)N 2 (Working adults).03N 2 =.03N 3 (Retired people)

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N 1 =24,390 N 2 =24,390.24 N 3 =24,400 24,400(5,000)/24,390.24 = $5,000 per year

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We believe $5000 a year tax to cover retiree pension fund is too expensive. Suppose we wanted to reduce the amount of tax an average working adult has to pay by $1000. We are planning to allow more children into the system and to exile (how?) working adults and retirees out of the system in order to reduce the retiree pension. How many children would we have to allow and how many adults and retirees would we have to exile per year to reach our goal? (changes are usually made one by one)

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We will perform analysis by assuming the planned steady state numbers of population in each groups are to be as below. Children (N 1 )= 35,000. Working Adults (N 2 )= 30,500 Retired People (N 3 )= 25,000

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H 1 = (.04 +.001)N 1 H 1 = (.04 +.001)35000 H 1 = 1435 H 2 +.04N 1 = (.03 +.01)N 2 H 2 +.04(35000) = (.03 +.01)30500 H 2 = -180 H 3 +.03N 2 =.03N 3 H 3 +.03(30500) =.03(25000) H 3 = -165

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New Pension per adult: 25000(5,000)/30500 = $4,098 Our answers indicate that we have to allow 435 more children be born every year. The negative number in H 2 indicate that we will have to kick out 180 adults every year!! The negative number in H 3 indicate that we have to kick out 165 retirees every year!!

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With our analysis and our new set of numbers, we have discovered that one of our options to decrease the yearly pension fund by almost $1000 we can: Allow 435 more children born every year Exile 180 adults every year Exile 165 retirees every year This would allow us to maintain a steady state census with a yearly retiree pension fund of $4098 per working adult.

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This type of solution is practical for our problem purposes but not in a real life scenario. More applicable ways to use this type of problem approach are for example: if we wanted to determine how many people to hire/lay off from a company. reduce/increase costs, etc.

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