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Applying Preemptive Goal Programming to Heat Wave Events in Newark, NJ

Alex Liao aliao817@gmail.com

Ava Chen

xachenx@gmail.com

Eric Gan xylazan@gmail.com

Chetna Johri

chetnakjohri@yahoo.com

NJ Governor’s School of Engineering & Technology 2011

1 Abstract

Public health interventions require city-specific data to effectually manage finite fiscal and material resources. In this technical paper, we found relative risks and odds ratios to identify elevated probabilities in Newark for cardiovascular diseases, respiratory illnesses, and acute renal failure in the elderly between heat waves and respective reference periods. We employed preemptive goal programming, a form of optimization where several functions with constraints are solved with ranked goals. Preemptive goal programming attempts to find a solution which may not necessarily exist for all relevant functions and goals. The goals, then, prioritize the functions and allow the optimal solution to satisfy the most pertinent ones. Using LINGO – software that can solve optimization functions – we manipulated hospitalization and mortality data from 1997 to 2006 for the city of Newark to optimize locations for acute care centers during future heat wave events. Along with Markov chain analyses calculated in MATLAB, which confirmed probabilities of our model’s success, our optimization model reduces 89% of excess mortality during the worst-case, weeklong heat wave scenario by providing disease-specific and severity-specific treatment options for Newark.

2 Introduction

Every summer, heat-related illnesses sweep across the globe, fomenting public health crises, especially in cities. The implications of heat waves for public health are enormous; rising healthcare costs, cuts in government spending, and turbulence in the private donation market have combined to engender a discouraging outlook for the urban citizen. In particular, the elderly, ages 65 and older, have been connected to a higher mortality risk during heat wave episodes.

Heat waves occur, on average, 4 times in Newark every year for approximately 5 days each [1]. The local government, struggling with continuous deficits and political infighting, must find a precarious balance between austerity measures and public health. Preemptive goal programming, a derivative of linear programming, and optimization models can more efficaciously allocate Newark’s precious fiscal and material resources. Once elevated risks of cardiovascular diseases, respiratory illnesses, and acute renal failure are found, and cost and capacity constraints established, the optimization problem can be solved to identify which patients should be sent to each treatment option.

However, hospitals, acute care centers and cooling shelters cannot be randomly situated across the city. Every city has its own unique hospital locations and population densities. Hence, acute care centers must be optimally

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placed across the city with these factors taken into account to minimize the excess mortality rate caused by a predicted heat wave event.

2.1 Existing Research

Studies have been conducted on the 1995 and 1999 heat waves in Chicago, the 1993 heat wave in Philadelphia, and the 2006 heat wave in California. These studies all attempted to find a correlation between certain demographic factors and their impacts on either mortality or morbidity rates during each heat wave. In the context of this issue, mortality refers to deaths that occurred during the heat waves, while morbidity refers to the development of a disease.

Semenza et al. (1996) conducted a study on the 1995 heat wave in Chicago, interviewing individuals whose relatives or friends died during the heat wave and comparing the answers of the interviewees to those of a control group. Using the collected data, they calculated odds ratios – estimates of the probability that an event will occur – to determine the risk of mortality among those exposed to the specified factors as opposed to the risk of mortality in the control group. The results showed that older age, medically related bed confinement, lack of air conditioning, and social isolation were all major factors that increased the risk of either a cardiovascular attack or death during a heat wave [2]. Later, Naughton et al. (2002) studied the 1999 Chicago heat wave, collecting similar demographic data regarding housing and medical conditions as well as calculating the odds ratios for different ethnicities. The study showed that Asians, Pacific Islanders, and Hispanics were least prone to heat-related deaths, and that a working air conditioner proved a key protective factor against heat-induced illnesses. It also confirmed the results posited by Semenza that social isolation and old age increased the risk of death. Furthermore, the study revealed that mortality rates peaked after several consecutive days of high temperature and humidity [3].

In a paper on the 1993 Philadelphia heat wave, Johnson et al. (2008) analyzed demographic factors, such as age and ethnicity, along with the correlations between the urban heat island (UHI) effect and risk, and between

socioeconomic factors and risk during a heat wave. The results of this study associated a greater risk of death with a high concentration of the UHI effect and with higher poverty levels [4]. More recently, Knowlton et al. (2009) conducted a study on the 2006 California heat wave, focusing mainly on morbidity instead of mortality. The study calculated relative risks (RRs), comparing hospitalization rates during a heat wave with those during a reference period. The RRs were significantly higher for electrolyte imbalances, acute renal failure, respiratory illnesses, and general heat-related illnesses during the heat wave [5].

3 Theory

3.1 Relative Risk and Odds Ratio The relative risk (RR) is a proportion which

measures the probability of an event occurring in an exposure group in relation to a control group. A relative risk of 1 denotes no difference in risk between the two groups, which implies that the exposure group had no significant effect on the occurrence of an event.

If RR < 1, the event is less likely to occur in the exposure group than in the control group; if RR > 1, the event is more likely to occur in the exposure group than in the control group.

In a similar vein, the odds ratio (OR) is a ratio of the probability of an event occurring relative to the probability of it not occurring. In this way, the odds ratio indicates the strength of the probability of occurrence. Thus, relative risks and odds ratios provide two divergent perspectives from which to evaluate the probability of an elderly person being hospitalized during a heat wave in Newark.

Confidence intervals for each provide a range of estimated values from the data. 95% confidence intervals determine the amount of certainty that the parameter – either relative risk or odds ratio – will fall in the range given the sampling method associated with the data. Likewise, if the value 1 does not exist in a 95% confidence interval, then the outcome is statistically significant in one direction, in terms of increased or decreased likelihood.

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3.2 Markov Chains Certain past events may occasionally alter

future outcomes. A Markov chain represents the states at discrete time epochs of a dynamical system that takes its values randomly and satisfies the Markov property. The Markov property holds when the future state depends only on the present state and not on the past. As such, Markov chains allow each state to be conditionally independent from the past. This forms a stochastic process model in that it accounts for randomness based on the model’s present state. Consider a Markov chain that takes its values from a set of states, i.e., the state space, S = {s1, s2, s3,….,sm} such that the probability of each state occurring depends on a transition probability pij that is defined between each state. In formal terms, if we define Sn as the state of the system at time n, then the set {Sn for n >= 0} is a Markov chain if:

P(Sn+1 = j | Sn = i) = P(i,j) where P(i,j) is a determined for each state (i,j) ∈ S x S in period n ∈ {0,1,…}.

Conditional probabilities between each state

are organized into a transition matrix. Each row represents the probability of entering all possible states in the state-space from the state associated with the row, while each column represents the probability of moving from all possible states in the state-space into the state associated with the column. Each row becomes a probability vector that sums to 1, i.e.,

P(i, j) =1

j!S" for all i ∈ S

If P is the transition matrix and the process

is in state Si, then the probability that the Markov chain will be in state Sj after n transitions is pij

(n), expressed as the ijth entry of the matrix Pn. Nonetheless, the initial state need not be fixed. Given a probability vector d of the set of all initial states that sums to 1, then the probability that the Markov chain is in state Si after n steps is the jth entry in the vector dP(n). A sample three-state Markov chain can then be visualized as:

As the diagram shows, each state has a

transition probability with other states and itself. These, of course, can be set to 0 if certain states do not interact with each other [6][7]. Hence, Markov chains provide a pragmatic process that includes randomness to analyze probabilities in social science.

3.3 Linear and Preemptive Goal Programming Linear programming is a method used to

optimize a linear function containing several factors that limit the possible inputs of the function. Optimization problems are formulated as mathematical models, one of which includes linear programming models. Linear programming models include two main components: a linear objective function, and linear constraints. The objective function denotes the performance measure of a specific problem; for instance, the monthly profit, daily cost of a manufacturing company, and distance to travel. Its output represents the value that should either be maximized or minimized. The constraints provide limitations on the inputs of the objective function. If not for constraints, one could make infinite profit or reduce its cost to negative infinity. Linear programming can be applied to various real life problems; it is most commonly used by businesses to maximize profits or yields and to minimize costs.

To better illustrate the concept of linear programming, we posit the following potential

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scenario: Suppose that a company builds both toy soldiers and toy trains. This is the information about the costs and labor time associated with each:

Soldiers Revenue: $27

Cost: $14 Carpentry Hours Required: 2 Finishing Hours Required: 1 Trains Revenue: $21 Cost: $9 Carpentry Hours Required: 1 Finishing Hours Required: 1

The workshop’s total weekly carpentry

hours are limited to 80 hours, and finishing hours per week are limited to 100 hours. Furthermore, only a maximum of 40 toy soldiers can be built each week. Using this information and the assumptions that all soldiers and trains built will be sold, the company must determine the number of soldiers and trains that it should manufacture in order to maximize its profit.

The first step of this problem would be to develop an objective function. Since the value that should be maximized is the company’s profit, the objective function should be equal to the profit. If the amount of soldiers built is denoted by the variable x1 and the amount of trains built is designated as x2, the objective function would then be [8]: Profit = Revenue – Cost = 27x1 + 21x2 – 14x1 – 9x2 = 13x1 + 12x2.

Several constraints can be placed in optimizing this problem. Sign constraints determine whether an input can be only positive or negative. Constraints can also be placed on time and on the number of manufactured soldiers. The feasible region, or the set of all possible input values that satisfy the constraints, can then be computed by programming the objective function and constraints into LINGO.

Preemptive goal programming differs slightly from linear programming. In linear programming, there exists an ideal set of values within a set of constraints that optimizes a single objective function. However, preemptive goal

programming optimizes more than one objective function and can also account for multiple numerical goals. Furthermore, there may not be a feasible region with solutions that can satisfy every goal and function. In this case, relative priorities are established for each of the goals. The objective function of the first priority goal must be satisfied, and subsequent attempts are then made to minimize the deviation of the rest of the objective function outputs from the goals that are of lesser priority.

The following scenario presents an application of preemptive goal programming. Suppose that a company produces three types of products – the amount of product 1 can be represented by the variable x1, the amount of product 2 by x2, and the amount of product 3 by x3. The following table summarizes the objective functions for the company’s profit, number of employees, and investment, as well as the cost for violations of each function:

The company has four goals. The first two

are of top priority: to preserve all invested capital and to ensure that no more than 40 employees are hired. The second two are of a lower priority in comparison to the top two goals, but of equal priority to each other: to earn

Objective Function

Cost of Violation of Objective Function

Profit 12x1 + 9x2 + 15x3 > 125

$5 million for every per-unit loss when profit < 125

Number of employees

5x1 + 3x2 + 4x3 = 40

$4 million for every per-unit loss when the number of employees is < 40 $2 million per-unit loss when the number of employees > 40

Investment 5x1 + 7x2 + 8x3 < 55

$3 million per-unit loss when investment > 55

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at least a $125 million profit and to keep the employment level at no less than 40 people. Variables are created to represent the error for each objective function. The error for having too low of a profit is represented by y1-, the error for having too many employees is represented by y2+, the error for having too few employees is represented by y2-, and the error for having too much capital investment is represented by y3+. The equations are then rewritten with the error variables from the top priorities; the error variables from the lower priority objectives are left out: Error to be Minimized = 2y2+ + 3y3+

12x1 + 9x2 + 15x3 > 125 5x1 + 3x2 + 4x3 = 40 + y2+ 5x1 + 7x2 + 8x3 < 55 + y3+ y2+, y3+ > 0

At this point, the problem can be solved using optimization software. The solution yields that x1, x2, and x3 are all equal to 0. However, the problem can then be written again, this time including the violations for the lower priority goals: E = 5y1- + 2y2+ + 4y2- + 3y3+

12x1 + 9x2 + 15x3 > 125 – y1- 5x1 + 3x2 + 4x3 = 40 + y2+ – y1- 5x1 + 7x2 + 8x3 < 55 + y3+ y2+, y3+ = 0 y1-, y2- > 0

Plugging these constraints into optimization software gives the values (x1, x2, x3) = (5, 0, 3.75). Only the first constraint is violated, with the profit being 125 − y1- = 125 − 8.75 = 116.25. This solution would allow the company to maximize its profits by efficiently utilizing its employees and investment to produce three different products [8]. In sum, preemptive goal programming provides an optimal solution when multiple functions of diverging importance exist.

4 Methodology 4.1 Definitions and Data Sources

The Office of the New Jersey Climatologist

defines heat waves as consecutive periods of at least three days of above 90 degrees Fahrenheit temperature [1]. This definition excludes any mention of humidity or wind speed because we did not have sufficient data to calculate more robust measurements of heat. This definition yielded 25 heat wave events from 1997 to 2006.

We obtained data from the Centers for Medicare and Medicaid Services, courtesy of Elena Naumova of Tufts University, on hospitalization counts for patients in Newark who were at least 65 years old when checked into the hospital. The counts exclude emergency room visits unless the patients were later hospitalized. The data was divided by ICD9-CM, or the International Classification of Diseases, revealing hospitalization counts by disease for every day from January 1st, 1991 to December 31st, 2006. Alongside the data were the minimum and maximum temperatures for each day, measured at Newark Airport.

Corresponding with the current research on heat waves, we chose three groups of diseases to analyze: cardiovascular diseases, respiratory illnesses, and acute renal failure. In line with previous heat wave hospitalization studies, (including Knowlton et al. 2009) we coded cardiovascular diseases as ICD-9-CM 390–398, 402, 404–429, and 440–448, respiratory illnesses as 460–519, and acute renal failure as 584. Acute renal failure was chosen to be indicative of dehydration cases that warranted further attention than in-home care. The capacity of all hospitals in Newark of 194 beds was estimated by obtaining the number of beds in each hospital. The U.S. Department of Health & Human Services’ Public Emergency Preparedness program estimates that approximately 15–25% of a hospital’s capacity could be made available for emergency events [9]. We took a conservative approach and estimated that 10% of each hospital’s 2000 census capacity could be made available during a heat wave event, given that heat waves

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generally appear less dangerous than state-declared emergencies.

While the hospitalization dataset was primarily used for statistical analysis, we also obtained mortality data for optimization in Newark from 1997 to 2006 from the Centers for Disease Control and Prevention’s Morbidity and Mortality Weekly Report [10].

Likewise, we were only able to employ apparent temperature for the optimization portion of our research. It was calculated according to an equation formulated by researchers in the field [11]:  Tapp = -2.653 + 0.994 temp + 0.0153 (dew)2 As the equation shows, the temperatures used earlier were combined with dew point measurements extracted from the National Oceanic and Atmospheric Administration’s Quality Controlled Local Climatological Data [12]. These measurements were taken from the same station at Newark Airport.

4.2 Calculating Relative Risk and Odds Ratio With this data, we identified all heat waves

from 1997 to 2006, forming a 10-year research period. We calculated relative risks (RR) by comparing hospitalizations by disease in each of 25 heat wave periods with a reference period of the same length. Reference periods were the days following the heat wave, unless another heat wave occurred during that time frame. In these cases, we chose reference periods with the days preceding each heat wave. Moreover, 95% confidence intervals for each relative risk were calculated according to the formula [13]: CI = exp(log(RR)± z*! /2 ! SE)  where

RR = a / (a+ c)b / (b+ d)

SE = 1a!1

a+ c+1b!

1b+ d

andz* =1.96

The variables a and b were defined as the sum of hospitalizations of the elderly for each disease during the heat wave and reference period, respectively. The variables c and d served as estimators of the population of elderly in Newark who would be at risk of contracting the disease. As such, the sum of a and c, and of b and d were standardized to 10,000 for cardiovascular and respiratory illnesses, and 1,000 for acute renal failure, in line with previous research.

Furthermore, we calculated the odds ratios (OR) by disease between each heat wave and its reference period. The 95% confidence intervals were calculated according to the formula [13]: CI = exp(log(OR)± z*! /2 ! SE)  where

OR = adbc

SE = 1a+1b+1c+1d

andz* =1.96

 

 The variables c and d, computed the same way as in the relative risk, have a greater effect in odds ratio since they estimate the relative probabilities of contracting a disease between each heat wave group and the reference group. Both the relative risks and odds ratios were calculated in line with those done in the field literature on the Chicago and California heat waves.

4.3 Markov Chain Probabilities We set up a transition matrix M to determine

the conditional probabilities of heat-related illnesses (ICD9-CM 992) occurring on the nth day of a heat wave. This drew data out of the hospitalization dataset from all heat waves from 1991 to 2006.

On the nth day for each heat wave, we calculated the probability that heat-related illnesses would occur on each of the following days n + 1, n + 2, n + 3…etc. In other words, n

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represented the initial state, which transitioned by addition of integer values up to a maximum of 10 days. The conditional probabilities were extracted from M and analyzed in a table for n + 1 change between every n and subsequent day.

After the optimization, we employed more rigorous Markov chains to estimate the probability of a person dying if they contracted the severe or non-severe form of one of the three disease groups during our simulation. The states included severe and non-severe forms of cardiovascular diseases, respiratory illnesses, and acute renal failure. The data was collected from the weeklong, worst-case scenario simulation. Moreover, additional states were added for the healthy state and untreated states for each treatment option. The untreated states represented the excess deaths that our model could not prevent. Healthy and untreated states were defined as absorbing states in that they only transitioned to themselves with a probability of 1. This meant that they were the only end states. Using MATLAB, we manipulated the initial transition matrix to find the Markov chain’s fundamental matrix, or (I – Q)-1, where I is an identity matrix from the absorbing states and Q includes part of the transition matrix. The fundamental matrix determines the transition probabilities for an absorbing chain and was output to a table comparing the probabilities of survival for each disease group and severity level.

4.4 Preemptive Goal Programming   Using LINGO, we programmed an objective function to minimize excess mortality with the following goals:

1. Goal 1: There should be enough capacity for hospitals to treat all patients with severe cases of each disease.

2. Goal 2: There should be enough capacity for acute care centers to treat all patients with non-severe cases of respiratory and cardiovascular diseases

3. Goal 3: There should be enough capacity for cooling shelters to treat all patients with non-severe cases of acute renal failure.

The function and constraints are outlined below: min z =m1s.t.

SiCii=1

3

! + Tii=1

3

! ki " B

S1 "194S2 "100

S1 # qjE +m11 # n1j=1

3

! = 0

S2 # pjE +m2 # n2j=1

2

! = 0

S3 # p3E +m3 # n3 = 0Si,mk,nk $ 0, Integersi, j,k =1,2,3

where • Si is the capacity of each treatment

option i, where S1 is the vacant capacity of all hospitals, 194, determined with adjusted census data, S2 is the total capacity or upper limit of acute care centers estimated to be less than 200, and S3 is the capacity of cooling shelters, determined to be unlimited.

• Ci is the cost per patient of each treatment option i, where C1 is the cost of hospital care, which was $2,271 in 1996 when adjusted to 2006 dollars by the Agency for Healthcare Research and Quality [14]. C2 is the cost of acute care centers per patient, which was estimated by the Red Cross for our research to be $50. C3, the cost of cooling shelters, was set to $0 because cooling shelters are generally located in public buildings, which are already open.

• Ti is the setup cost for each treatment option i, where T1 is $0 because the setup cost for hospitals is $0. T2 is set at $9,600, defined by the Mobile Health Clinic Network as the setup cost of a mobile health clinic for one week [15]. T3 was estimated at $100, or the amount of money a public building would need to accommodate extra citizens.

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• ki is the number of each facility which • needs to be set up. This variable

becomes relevant if and only if acute care centers need to be set up.

• B is the total budget, set at $25,000. • E is the excess mortality during a

weeklong heat wave, defined by the multivariate regression of maximum apparent temperature, Julian date, and consecutive heat days. The multivariate regression yielded:

Excess Mortality = 0.70 + 0.0622 Tappmax - 0.0061 JulianDate - 0.729 CHD

We simulated the heat wave scenario with the largest excess mortality with data input from 1997 to 2006, and

predicted 27 people would die in a heat wave absent any optimization.

• pj is the proportion of cases for each disease j which are not severe. These were estimated using statistics from the

• 1995 Chicago heat wave compiled and analyzed by Dematte et al. [16]

• qj is the proportion of cases for each disease j which are severe. These were computed as 1 – pj, where j1 represented cardiovascular diseases, j2 represented respiratory illnesses, and j3 represented acute renal failure.

• mk and nk are deviational variables for each goal k, or quantities which account for the optimal solution exceeding or falling short of a goal. m accounts for falling short of a goal, while n accounts for exceeding one.

Table 1: Relative Risks Hospitalizations RR (95% CI)

Heat Wave Period (Reference Period) Cardiovascular Diseases Respiratory Illnesses Acute Renal Failure 13–18 July 1997 (19–24) 0.90 (0.70 – 0.93) 1.40 (1.08 – 1.80) 0.75 (0.27 – 2.08) 20–23 July 1998 (24–27) 1.97 (1.68 – 2.31) 1.31 (1.01 – 1.70) 3.50 (0.75 – 16.44) 24–26 August 1998 (27–29) 1.22 (1.00 – 1.48) 1.28 (0.92 – 1.79) 1.00 (0.39 – 2.56) 29–31 May 1999 (26–28) 0.74 (0.60 – 0.91) 0.74 (0.53 – 1.04) 0.67 (0.25 – 1.80) 26 June–7 July 1999 (14–25) 0.94 (0.89 – 0.99) 0.91 (0.78 – 1.06) 1.46 (0.76 – 2.80) 16–19 July 1999 (12–15) 0.99 (0.84 – 1.17) 1.00 (0.74 – 1.36) 2.00 (0.62 – 6.43) 23 July–7 August 1999 (8–23) 0.95 (0.93 – 0.98) 1.03 (0.91 – 1.16) 2.90 (1.49 – 5.63) 6–9 May 2000 (10–13) 0.73 (0.62 – 0.86) 1.23 (0.95 – 1.60) 2.67 (0.73 – 9.76) 9–11 June 2000 (6–8) 0.68 (0.55 – 0.84) 0.61 (0.43 – 0.86) 1.50 (0.26 – 8.79) 2–4 May 2001 (5–7) 1.02 (0.83 – 1.24) 1.57 (1.16 – 2.14) 0.73 (0.31 – 1.73) 26 June–1 July 2001 (2–7) 0.80 (0.70 – 0.92) 0.91 (0.74 – 1.13) 1.50 (0.64 – 3.51) 23–25 July 2001 (20–22) 2.08 (1.67 – 2.59) 1.65 (1.16 – 2.36) 1.00 (0.30 – 3.35) 6–10 August 2001 (11–15) 1.18 (1.03 – 1.35) 1.33 (1.04 – 1.70) 1.64 (0.82 – 3.29) 2–9 July 2002 (24–1) 0.98 (0.89 – 1.08) 0.90 (0.76 – 1.06) 0.95 (0.53 – 1.70) 15–19 July 2002 (20–24) 1.03 (0.90 – 1.18) 1.03 (0.81 – 1.30) 1.25 (0.62 – 2.53) 29 July–5 August (21–28) 0.92 (0.84 – 1.01) 0.95 (0.80 – 1.13) 1.12 (0.62 – 2.02) 10–19 August 2002 (20–29) 0.96 (0.89 – 1.03) 1.08 (0.93 – 1.26) 0.60 (0.34 – 1.07) 23–27 June 2003 (28–2) 1.26 (1.12 – 1.42) 1.12 (0.89 – 1.40) 1.58 (0.81 – 3.09) 4–8 July 2003 (9–13) 0.87 (0.76 – 1.01) 1.00 (0.79 – 1.27) 0.76 (0.39 – 1.49) 13–22 August 2003 (23–1) 1.13 (1.05 – 1.21) 1.08 (0.92 – 1.26) 1.39 (0.87 – 2.20) 7–14 June 2005 (15–22) 0.95 (0.88 – 1.01) 1.08 (0.92 – 1.26) 1.42 (0.98 – 2.05) 18–27 July 2005 (8–17) 1.04 (0.98 – 1.10) 1.05 (0.91 – 1.22) 1.78 (1.21 – 2.60) 2–5 August 2005 (6–9) 1.41 (1.20 – 1.65) 1.19 (0.92 – 1.53) 1.63 (0.70 – 3.75) 11–14 August 2005 (7–10) 1.01 (0.86 – 1.19) 0.80 (0.60 – 1.06) 2.14 (0.91 – 5.03) 16–21 July 2006 (22–27) 1.02 (0.92 – 1.14) 1.14 (0.93 – 1.40) 0.91 (0.53 – 1.56) 27 July–5 August 2006 (6–15) 0.92 (0.86 – 0.98) 1.03 (0.89 – 1.21) 1.03 (0.73 – 1.43)

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The objective function assumes that all excess mortality contracted during a heat wave is either a form of cardiovascular disease, respiratory disease, or acute renal failure. Moreover, the model implicitly assumes that providing the proper treatment will successfully prevent a death. In reality, this can only reduce the risk of death. However, this model still remains useful for allocating resources. With preemptive goal programming, we were able to prioritize hospitals and acute care centers over cooling shelters.

In addition, the excess mortality function was estimated using a multivariate regression model in Minitab. Regression attempts to fit available data into trends expressed by a mathematical function. The regression model included all weeks from 1997 to 2006, arranged by approximated Julian Date numbers, with average

maximum apparent temperatures and counts of consecutive heat days above 90 degrees Fahrenheit for each week. Julian Date number refers to the day of the year, approximated as the Wednesday number for each week. Excess mortality was calculated by first determining a threshold apparent temperature beyond which we saw spikes in mortality levels. The average number of deaths, 21, on days below the apparent threshold temperature of 110 was subtracted from the mortality levels of each day above 110 degrees. In this way, we were able to estimate the excess mortality caused by heat waves. We also utilized integer programming to restrict Si to integer values. This allowed us to obtain integer numbers of hospital beds, acute care center spaces, and cooling shelter spaces. We then performed a sensitivity analysis based

Table 2: Odds Ratios Hospitalizations OR (95% CI)

Heat Wave Period (Reference Period) Cardiovascular Diseases Respiratory illnesses Acute Renal Failure 13–18 July 1997 (19–24) 0.86 (0.71 – 1.05) 1.45 (1.09 – 1.93) 0.73 (0.25 – 2.20) 20–23 July 1998 (24–27) 2.47 (2.01 – 3.05) 1.35 (1.01 – 1.80) 3.69 (0.75 – 18.21) 24–26 August 1998 (27–29) 1.26 (1.00 – 1.60) 1.30 (0.91 – 1.86) 1.00 (0.36 – 2.78) 29–31 May 1999 (26–28) 0.70 (0.54 – 0.89) 0.72 (0.50 – 1.04) 0.65 (0.22 – 1.89) 26 June–7 July 1999 (14–25) 0.81 (0.66 – 0.98) 0.88 (0.72 – 1.08) 1.57 (0.73 – 3.38) 16–19 July 1999 (12–15) 0.99 (0.80 – 1.22) 1.00 (0.72 – 1.40) 2.09 (0.61 – 7.17) 23 July–7 August 1999 (8–23) 0.59 (0.43 – 0.81) 1.05 (0.87 – 1.26) 3.68 (1.68 – 8.05) 6–9 May 2000 (10–13) 0.67 (0.54 – 0.83) 1.26 (0.94 – 1.69) 2.81 (0.72 – 10.92) 9–11 June 2000 (6–8) 0.64 (0.50 – 0.81) 0.59 (0.41 – 0.85) 1.52 (0.25 – 9.27) 2–4 May 2001 (5–7) 1.02 (0.81 – 1.30) 1.63 (1.17 – 2.28) 0.73 (0.29 – 1.81) 26 June–1 July 2001 (2–7) 0.74 (0.61 – 0.89) 0.90 (0.70 – 1.16) 1.57 (0.61 – 4.02) 23–25 July 2001 (20–22) 2.37 (1.84 – 3.06) 1.71 (1.17 – 2.49) 1.00 (0.28 – 3.57) 6–10 August 2001 (11–15) 1.26 (1.04 – 1.53) 1.38 (1.04 – 1.82) 1.78 (0.79 – 3.98) 2–9 July 2002 (24–1) 0.96 (0.81 – 1.15) 0.87 (0.71 – 1.08) 0.94 (0.46 – 1.91) 15–19 July 2002 (20–24) 1.04 (0.86 – 1.26) 1.03 (0.79 – 1.35) 1.29 (0.57 – 2.93) 29 July–5 August (21–28) 0.86 (0.72 – 1.02) 0.94 (0.75 – 1.17) 1.15 (0.56 – 2.36) 10–19 August 2002 (20–29) 0.91 (0.76 – 1.08) 1.11 (0.90 –1.36) 0.53 (0.26 – 1.08) 23–27 June 2003 (28–2) 1.43 (1.19 – 1.72) 1.14 (0.88 – 1.47) 1.72 (0.79 – 3.76) 4–8 July 2003 (9–13) 0.83 (0.68 – 1.01) 1.00 (0.76 – 1.31) 0.73 (0.33 – 1.60) 13–22 August 2003 (23–1) 1.36 (1.14 – 1.63) 1.10 (0.90 – 1.35) 1.58 (0.84 – 2.95) 7–14 June 2005 (15–22) 0.86 (0.72 – 1.03) 1.10 (0.90 – 1.36) 1.75 (0.98 – 3.12) 18–27 July 2005 (8–17) 1.13 (0.93 – 1.36) 1.08 (0.88 – 1.31) 2.50 (1.38 – 4.50) 2–5 August 2005 (6–9) 1.57 (1.28 – 1.93) 1.21 (0.91 – 1.61) 1.72 (0.68 – 4.35) 11–14 August 2005 (7–10) 1.02 (0.82 – 1.26) 0.78 (0.58 – 1.07) 2.34 (0.91 – 6.03) 16–21 July 2006 (22–27) 3.44 (2.79 – 4.24) 1.17 (0.91 – 1.49) 0.89 (0.45 – 1.75) 27 July–5 August 2006 (6–15) 0.78 (0.65 – 0.94) 1.04 (0.85 – 1.28) 1.04 (0.59 – 1.83)

10    

on budget to determine the variability of our model outputs.

4.5 Geographical Information Systems In order to optimally place acute care centers,

we utilized ArcGIS, a geographical information system, to optimize locations for additional acute care centers subject to the constraints in the objective function. Geographical information systems are programs designed to visualize data on a spatial plane.

First, we attempted a linear programming model to determine the optimal block for the acute care center. The model was written as follows:

where • pi is the total population in each block i. • e(h,s) is the proportion of the elderly

with disease h and severity level s. • dij is the distance from block i to facility

j, where j is defined from 1-5 for each hospital in Newark, 6 for acute care centers, and 7 for cooling shelters.

• cj is the capacity of facility j. • fhjs is 1 if facility j is not effective for

each patient with disease h in severity level s, and 0 otherwise.

• h is the type of disease, where cardiovascular diseases are 1, respiratory illnesses are 2, and acute renal failure is 3.

• s is the severity level and equal to 1 if severe, 2 otherwise.

However, this minimization function requires

input data from more than 2000 linear lengths between each block and hospital. Thus, solving this optimization problem would require more than 2000! computations and a method for inputting each linear length, forcing us to find

alternative methods for optimizing acute care center locations.

Therefore, using the ArcGIS program, we entered the hospital information for Newark and Essex County. This allowed us to see where the most severely affected patients could receive hospital treatment. With the data on vacant beds in each hospital, we estimated the area that each hospital could cover.

We then looked at 2000 census data of Newark’s population count by age per city block, and found the densest block by counts of the elderly. We decided to place the acute care center here to minimize the excess deaths in this area.

5 Results

Table 1 summarizes the relative risks for hospitalization counts between each heat wave and reference period in the elderly population (older than 65 years old). The RRs and 95% confidence intervals point to elevated levels of hospitalizations during heat wave periods for all three classes of diseases. Table 3. Markov chain analysis of the probability of heat-related illnesses.

Likewise, Table 2 enumerates the data in terms of odds ratios with 95% confidence intervals. These numbers indicate strong probabilities of higher hospitalization levels for cardiovascular diseases, respiratory illnesses, and acute renal failure during heat waves.

Day Probability Conditional Probability

n+1 Change

1 0.0294 0.0025 9.5 2 0.0882 0.0246 2.17 3 0.2794 0.0534 0.74 4 0.1911 0.0393 0.46 5 0.2058 0.0181 0.29 6 0.0882 0.0051 0.33 7 0.0588 0.0017 0.50 8 0.0294 0.0008 0.00 9 0.0294 0.00 0.00 10 0.00 0.00 0.00

∑∑

∑∑∑

=

= =

≤

=

s hjhjsi

si h jhjsiji

cfshep

ts

fdshepz

3

1

,

3

1

7

1

),(

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),(min

11    

Table 3 illustrates the results of the first Markov chain analysis. These indicate a significant increase in heat-related illnesses on the 2nd and 3rd days of a heat wave. Table 4 describes the results of the second Markov chain analysis, showing the probability of survival for each disease and severity group during our simulation. This demonstrates that severe cases of cardiovascular diseases require additional attention.

Table 4. Markov chain analysis of optimization.

Our optimization model proved to be

successful. In the worst-case, weeklong heat wave scenario, we were able to save 89% of those who would have died under our excess mortality model by assigning them to hospitals, acute care centers, and cooling shelters. This was completed with only a budget of $25,000.

As can be seen in Figure 1, our optimal placement of the acute care center allows the city to extend its medical coverage over the block with the highest elderly population. This block is located south of Harvey Street and north of 3rd Avenue, between Broadway and Summer Avenue. Figure 2 displays the census block.

6 Discussion

6.1 Statistical Analysis The relative risks in Table 1 show that

during heat waves, elevated levels of hospitalization due to cardiovascular diseases, respiratory illnesses, and acute renal failure occur. The ratios are stronger for acute renal failure and respiratory illnesses, which have greater numbers of heat waves where the RR > 1. While the association does not hold itself to be true in many heat waves for cardiovascular diseases, the presence of several heat waves where the RR > 1, combined with past research confirming this linkage, suggests that the elderly of Newark indeed suffer from increased levels of cardiovascular diseases during heat waves. The large majority of all confidence intervals contained an upper limit that was greater than 1, demonstrating an increased risk of higher counts of hospitalizations. While the lower limits

Disease Group (Severity) Healthy Untreated Cardiovascular (S) 0.9185 0.0815 Cardiovascular (NS) 1 0 Respiratory (S) 0.9361 0.0639 Respiratory (NS) 1 0 Acute Renal Failure (S) 0.9986 0.0014 Acute Renal Failure (NS) 1 0

Figure 1. Hospital and acute care center coverage map. Note Newark Airport in the Southeast.

Figure 2. The optimal block for the acute care center as defined by the 2000 census data [17].

12    

dropped below 1 on several of the heat waves, we see this as mainly a result of small sample sizes for each heat wave. Past research has mainly been conducted on much longer heat waves and over a much larger population.

Likewise, the odds ratios in Table 2 express the notion that the increased probabilities of cardiovascular diseases, respiratory illnesses, and acute renal failure were somewhat strong. For instance, the odds ratio for acute renal failure was as high as 3.68 in the heat wave which occurred between 23 July and 7 August of 1999. From 16–21 July of 2006, the odds ratio was 3.44 for cardiovascular disease, showing a large spike in hospitalizations compared to the reference period. Respiratory illnesses had their largest odds ratio during 23–25 July of 2001, when it was 1.71. In each of these heat waves, the lower bounds of the 95% confidence intervals were greater than 1, which establishes statistical significance of elevated hospitalizations in the elderly.

The high variability of temperature and hospitalization data, combined with the fact that some heat waves only lasted 3 days while others lasted for more than a week, contributed to wide confidence intervals due to small sample sizes. This makes it difficult to determine whether heat waves truly caused an elevated risk of hospitalizations across the three diseases, or whether the increases were simply due to random variation. Additionally, the temperature data from Newark may not have been consistent over the ten-year period, and our definition of a heat wave in this section did not include any indicator of humidity or wind speed. Apparent temperature, or an estimate of the true temperature, which combines the dew point and temperature, was only used in the optimization portion of the research. While these deficiencies exist, the relative risks and odds ratios do point to elevated levels of hospitalizations for all three groups of diseases when heat waves are compared to reference periods. These trends mirror those found in existing research, further reinforcing our thesis.

Similarly, the Markov chain analyses offers deeper insight into how heat waves affect the urban populace. The n+1 changes in Table 3 for days 1 and 2 – which elucidate the increases in heat-related illnesses between days 1 and 2, and

days 2 and 3 – are 9.5 and 2.17, respectively, intimating that heat waves cause the most damage after the first day. After the third day, the conditional probability of transitioning into a state with heat-related illnesses slowly decreases to 0 on the 10th day. This implies that public health officials must mobilize on the first day of an expected heat wave or the extensive increases in heat-related illnesses will not be tamed.

6.3 Optimization

Existing research suggests that acute care centers and cooling shelters would help minimize excess mortality by reducing social isolation and the lack of air conditioning. Thus, our optimization model should, in theory, be able to reduce stress on hospitals while directing the elderly towards cost-saving palliative services.

The optimization solution was able to prevent 89% of the excess deaths among the elderly by allocating 7 hospital beds, 16 spaces in acute care centers, and 1 space in cooling shelters. This proved to be the most cost effective and life saving allocation of services during the simulated weeklong, worst-case heat wave event which would have caused 27 deaths absent optimization.

The second Markov chain analysis in Table 4 provides an additional perspective on the optimization model’s viability. Severe acute renal failure patients were most likely out of all severe forms of diseases to transition into a healthy state in our absorbing chain, with a 99.86% survival rate. On the other hand, severe cardiovascular disease patients were the least likely to receive the proper treatment, with only 91.85% reaching a hospital and surviving. This suggests that public health programs should be directed towards patients suffering from severe forms of cardiovascular diseases. Indeed, our model was able to provide treatment for all non-severe patients in all three disease groups, implying that the estimated resources available today in Newark are sufficient for those groups.

Additionally, our sensitivity analysis suggests that cost is not an extremely prohibitive factor. From our preemptive goal programming model, under a $25,000 budget, we found that we fall

13    

short from reaching Goal 1 by 3 patients. If we raised our budget to $30,000, then we only fall short by 1 patient. At a $35,000 budget, we would be able to meet all our goals with $4,056 leftover, while at $40,000 we would have $9,056. We find, therefore, that the break-even point is at $30,944, and any budget above this value will be superfluous; the extra money may be invested in facility services, employees, or emergency funding. Moreover, since the total number of excess mortality tends to fluctuate depending on the temperature, humidity, and a variety of other circumstances, the number of expected deaths can exceed that of our worst-case scenario. At 30 excess deaths per week, we fall short from Goal 1 by 5 patients. At 35 deaths per week, we fall short by 7 patients. This implies the need for more rigorous models for excess mortality.

Our multivariate regression model for excess mortality estimates a positive relationship between excess mortality (which includes both severe and non-severe cases) and maximum apparent temperature. In addition, the coefficient for Julian Date suggests that heat waves occur earlier in the summer, while the coefficient for consecutive heat days paradoxically suggests that the shortest heat waves can actually cause the most residual deaths in a week. Combined with data suppression, or missing mortality data for several weeks, it is clear that our excess mortality model should be improved for future work. Notwithstanding, our optimization model minimizes costs while successfully allocating resources to treat 89% of patients who would otherwise die during a heat wave.

Combined with optimal placement of the acute care center, we are confident that our model can improve public health officials’ heat wave planning initiatives.

7 Conclusion

We were able to identify statistically significant increases in elderly hospitalization counts during heat waves compared to reference periods from 1997 to 2006. Several of the relative risks and odds ratios for each disease group were greater than 1 during heat waves,

while the Markov chain analysis demonstrated increases in heat-related illnesses on the 2nd and 3rd day of heat waves. From these findings, we concluded that an optimization model to prevent excess mortality stemming from cardiovascular diseases, respiratory illnesses, and acute renal failure would substantially aid Newark’s public health programs.

Our preemptive goal programming model was able to reduce excess mortality by 89% during a simulated worst-case, week-long heat wave. We pragmatically allocated treatment for 24 of 27 expected mortality patients with hospital beds, acute care centers, and cooling shelters. This would cost Newark $25,000. Nonetheless, a sensitivity analysis confirmed that cost was not a significant factor in determining the amount of excess deaths prevented. A second Markov chain analysis confirmed that emphasis should be placed on the most severely afflicted patients for cardiovascular diseases, and that Newark’s current resources are sufficient for those who suffer non-severe cases of each disease.

Throughout the past several years, Newark has suffered multiple heat waves which have disproportionately affected the elderly aged 65 and older. By optimally placing an acute care center in the densest census block of the city’s elderly population, we were able to satisfy our model and provide a cogent public heath intervention for the city of Newark in the future.

7.1 Recommendations for Future Work Our foremost recommendation would be to

test our models on more specific datasets, specifically ones that focus on age distribution in Newark. This could reveal alternative methods for assisting various cross-sections of society. Realistically, the model would also have to take into account costs for transportation to hospitals, acute care centers, and cooling shelters. Population density models could specifically aid the elderly by finding the shortest routes from their homes to hospitals, acute care centers, and cooling shelters.

Moreover, additional data is required to more accurately determine excess mortality. We

14    

utilized a multivariate regression model based on time and apparent temperature, but more robust methods such as logistic regression should be explored. While we obtained mortality data per week, daily mortality counts would create a denser picture of the implications of heat wave events.

In addition, taking into account diseases other than respiratory illnesses, cardiovascular diseases, and acute renal failure would also help determine a more holistic picture of effects of heat waves on populations. There are several other conditions and illnesses that can occur as a result of sustained exposure to heat, including nephritis, general dehydration, and heatstroke. While we were limited to integer ICD disease numbers as a result of time constrictions, future work could look into more specific disease codes as well. Since the excess mortality rate during a heat wave includes deaths resulting from these diseases, subsequent models could then better estimate excess mortality during future heat wave periods.

Because statistics is an estimated representation of reality, it does not consider the fluctuations that may occur in reality. Our model is a static goal programming model, so more research needs to be done in order to take into account the dynamic fluctuations of the variables, such as the number of deaths per week and the budget available. More complex optimization models for treatment allocation options by location should also be explored in the future.

8 Acknowledgements

We would like to thank our project mentors, Professor Melike Baykal-Gürsoy and graduate student Colman Cheung. We are also grateful for the continued assistance of our RTA, Joshua Binder. In addition, we greatly appreciate the opportunity given to us by Dean Ilene Rosen, Program Coordinator Jean Patrick Antoine, Head Counselor Daniel Cobar, and the New Jersey Governor’s School of Engineering & Technology and its Board of Overseers. We would finally like to extend our gratitude to the sponsors of the 2011 Governor’s School:

Rutgers University, the Rutgers University School of Engineering, the State of New Jersey, Morgan Stanley, Lockheed Martin, Automated Control Concepts Inc., Silver Line Building Products, and private sponsors Sharon Ma, Nan Yao, Laura Overdeck, and the Tomasetta family for their ongoing support of the program.

9 References [1] David A. Robinson, Mathieu R. Gerbush, and Jacob Carlin. “An evaluation of excessive heat at Newark, NJ: 1997-2010.” Unpublished, March 2010. [2] Semenza, Jan C., Carol H. Rubin, Kenneth H. Falter, Joel D. Selanikio, Dana Flanders, Holly L. Howe, John H. Wilhelm. “Heat-Related Deaths During the July 1995 Heat Wave in Chicago.” The New England Journal of Medicine (1996) pp. 84-90. [3] Naughton, Mary P., Alden Henderson, Maria C. Mirabelli, Reinhard Kaiser, John L. Wilhelm, Stephanie M. Kieszak, Carol H. Rubin, Michael A. McGeehin. “Heat-Related Mortality During a 1999 Heat Wave in Chicago.” American Journal of Preventive Medicine (2002) pp. 221-227. [4] Johnson, Daniel P., Jeffrey S. Wilson. “The socio-spatial dynamics of extreme urban heat events: The case of heat-related deaths in Philadelphia.” Applied Geography (2008) pp. 419-434. [5] Knowlton, Kim, Miriam Rotkin-Ellman, Galatea King, Helene G. Margolis, Daniel Smith, Gina Solomon, Roger Trent, and Paul English. “The 2006 California Heat Wave: Impacts on Hospitalizations and Emergency Department Visits.” Environmental Health Perspectives (2009). [6] Grinstead, Charles Miller, James Laurie Snell. “Markov Chains.” Introduction to Probability. American Mathematical Society, 1997. Print. [7] Montgomery, James. “Markov Chains.” Unpublished. 2009.

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http://www.ssc.wisc.edu/~jmontgom/markovchains.pdf. [8] Winston,Wayne Leslie. "The Simplex Algorithm and Goal Programming." Operations Research: Applications and Algorithms. Belmont, CA: Duxbury, 2001. Print. [9] Agency for Healthcare Research and Quality. “Public Health Emergency Preparedness.” U.S. Department for Health & Human Services. 2011. http://archive.ahrq.gov/prep/havbed2/havbed2-5.htm. [10] Centers for Disease Control and Prevention. “Morbidity and Mortality Weekly Report.” Centers for Disease Control and Prevention. 2011. http://wonder.cdc.gov/mmwr/mmwrmort.asp. [11] Baccini, Michela et al. “Heat Effects on Mortality in 15 European Cities.” Epidemiology (2008) pp. 711-719. [12] National Oceanic and Atmospheric Administration. “Quality Controlled Local Climatological Data.” NOAA Satellite and Information Service. 2008. http://www.ncdc.noaa.gov/oa/samples/qclcdsample.html. [13] Altman, Douglas, David Machin, Trevor Bryant, and Stephen Gardner. Statistics with Confidence: Confidence Intervals and Statistical Guidelines, 2nd edition. BMJ Books, 2000. Print. [14] Machlin, Steven R. “Trends in Health Care Expenditures for the Elderly Age 65 and Older: 2006 versus 1996.” U.S. Department for Health & Human Services. 2009. http://www.meps.ahrq.gov/mepsweb/data_files/publications/st256/stat256.pdf [15] Santana, Julieta. “Going Out to the Community.” Mobile Health Clinics Network. 2005. http://www.mobilehealthclinicsnetwork.org/featured.html.

[16] Dematte, Jane E, et al. “Near-Fatal Heat Stroke during the 1996 Heat Wave in Chicago.” Annals of Internal Medicine (1998). [17] The United States Federal Government. “American FactFinder.” U.S. Census Bureau. 2000. http://factfinder.census.gov/.

10 Appendix 10.1 Lingo Programming Input

model: data: EXCESS=27; LIFE=8500000; BUD=20000; C OST1=2271; COST2=50; COST3=5; HOSCOST=0; ACCCOST=9600; SHELCOST=100; PROP1=0.688; PROP2=0.293; PROP3=0.020; NONSEV1=0.464; NONSEV2=0.117; NONSEV3=0.016; SEV1=0.224; SEV2=0.1758; SEV3=0.003871; enddata min = UND1; CAP1*COST1 + CAP2*COST2 + CAP3*COST3 + HOSCOST + ACCCOST*CAP2/20 + SHELCOST <= BUD; CAP1 <= 194; CAP2 <= 100; CAP1 - (SEV1 + SEV2 + SEV3)*EXCESS + UND1 - OVR1 = 0; CAP2 - (NONSEV1 + NONSEV2)*EXCESS + UND2 - OVR2 = 0; CAP3 - NONSEV3*EXCESS + UND3 - OVR3 = 0; UND1 >= 0; UND2 = 0; UND3 = 0; OVR1 >= 0; OVR2 >= 0; OVR3 >= 0; @gin(CAP1); @gin(CAP2); @gin(CAP3); end

10.2 Lingo Programming Output

16    

Global optimal solution found. Objective value: 3.495446 Objective bound: 3.495446 Infeasibilities: 0.000000 Extended solver steps: 0 Total solver iterations: 0 Model Class: MILP Total variables: 7 Nonlinear variables: 0 Integer variables: 3 Total constraints: 11 Nonlinear constraints: 0 Total nonzeros: 17 Nonlinear nonzeros: 0 Variable Value Reduced Cost EXCESS 27.00000 0.000000 LIFE 8500000. 0.000000 BUD 25000.00 0.000000 COST1 2271.000 0.000000 COST2 50.00000 0.000000 COST3 5.000000 0.000000 HOSCOST 0.000000 0.000000 ACCCOST 9600.000 0.000000 SHELCOST 100.0000 0.000000 PROP1 0.6880000 0.000000 PROP2 0.2930000 0.000000 PROP3 0.2000000E-01 0.000000 NONSEV1 0.4640000 0.000000 NONSEV2 0.1170000 0.000000 NONSEV3 0.1600000E-01 0.000000 SEV1 0.2240000 0.000000 SEV2 0.1758000 0.000000 SEV3 0.3871000E-02 0.000000 UND1 3.495446 0.000000 CAP1 7.000000 -1.000000 CAP2 16.00000

0.000000 CAP3 1.000000 0.000000 OVR1 0.000000 1.000000 UND2 0.000000 0.000000 OVR2 0.8940000 0.000000 UND3 0.000000 0.000000 OVR3 0.5840000 0.000000 Row Slack or Surplus Dual Price 1 3.495446 -1.000000 2 518.0000 0.000000 3 187.0000 0.000000 4 184.0000 0.000000 5 0.000000 -1.000000 6 0.000000 0.000000 7 0.000000 0.000000 8 3.495446 0.000000 9 0.000000 0.000000 10 0.000000 0.000000 11 0.000000 0.000000 12 0.8940000 0.000000 13 0.5840000 0.000000

10.3 MATLAB 2nd Markov Chain Input

I = [1,0,0,0 0,1,0,0 0,0,1,0 0,0,0,1] R = [0,0,0,0 0.9185, .0815, 0, 0 1, 0, 0, 0 .9361, 0.0639, 0,0 1,0,0, 0 .99859, 0.00141, 0,0 1,0,0,0]

17    

Q = [0,.224, .464, .176, .107, .004, 0.016 0,0,0,0,0,0,0 0,0,0,0,0,0,0] I = eye(size(Q)) E = I – Q inv(E)*R