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MULTIPLE CRITERIA OPTIMIZATION MODELS FOR SOLVING CONTEMPORARY ETHICAL ISSUES

IN DECISION MAKING

Gabriela Cristescu*, Luciana Neamţiu** and Silviu Gabriel Szentesi*

(*�Aurel Vlaicu� University of Arad � România, **�Ion Chiricuţă� Oncology Institute of Cluj-Napoca � România)

Abstract: The wide range of mathematical methods used in decision making or in decision aid, together with the complexity of our days� decision making problems generate questions referring to the public expectations from mathematics. There are mathematical methods accompanied by uniqueness theorems of the solution, which seem to be, form the point of view of various types of decision makers, dictatorial and not easy to agree with. There are mathematical methods leading to possibilities of interacting with the decision makers in various stages of the process, transforming the decision making process into a more human and acceptable one. We present some examples of problems from our own professional experience, together with more possible mathematical solutions, discussing them from the above described point of view. The decision making problems come from pharmacoeconomics and from the environmental economics. They are meant to solve some ethical issues arising from decision making under budget limitations, by taking into account more non-financial criteria, related to the life and behaviour of the humans involved in the situations under debate. The paper is supported by the Romanian Education and Research Ministry, within the Research Project ID-1239/2007.

1. Introduction Many times, in our days� life, there appear conflict situations between the human common sense, the human ethics and the available decisions of actions for solving a life issue. The budget limitations lead often, under cost-benefit type analysis, which is as dictatorial as finances can be into the human life, to solutions generating ethical conflicts within professional groups having professional ethics. For example, when the cost-benefit analysis discusses the possibility of funding a persons� medical treatment in terms of the quality of life gained by this treatment, the fact that the Oath of Hippocrates asks to protect life itself in spite of the quality of life often leads to an ethical conflict between the management staff and the medical professional group. Such a situation of this type is discussed, showing how mathematics does offer possibilities of ethically approaching this class of problems.

Also, ethical issues appear in the process of selecting an environmental policy to protect the employees within a company and to keep the work productivity at high level, also trying to minimise the costs. The need of economic-ecologic efficiency of an environmental policy of a company is often mentioned both in technical and in scientific literature. But a method of assessing this kind of efficiency is never described. There are specific extra-economic possibilities of describing the efficiency of an ecology activity, as, for example, measuring the concentration of certain substances in the soil, air, water, food, etc. But a general tool, as an efficiency index to characterize an environment policy is totally absent. The relationship between industry and environment should receive considerable attention from two points of view: within the organization and between the organization and the society and nature. An ecological behaviour of the organization in relation both with its employees and with the society should be normal at the end of the first decade of the 21st century, when so many changes in the nature are of great mankind concern. So, a method of assessing the efficiency of the environment policy is necessary. Our aim is

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to study few possibilities of solving this problem and to develop an economic-ecologic effectiveness index for this purpose. The research resulting in this paper is part of the Research Project ID-1239/2007, funded by the Romanian Education and Research Ministry.

2. Multiple criteria optimization to approach the ethical issues expressing the conflict between the medico-economic management and the medical professional group.

The insufficient funding is one of the major issues of all the medical units, all over the world. The budget is often exceeded by the total cost of the applications for various treatments and surgical interventions. As consequence, the very difficult decision problem of choosing those applications that are going to be funded and those to reject or put on hold arises. It is difficult not only from mathematical point of view but also from the point of view of ethics. This was the main topic approached within one the working session of the 2nd International Forum on Oncology and Health Economics CECOG, Wien, April 20-21, 2002.The possibility of using the cost-utility analysis in solving it was discussed, given a practical example. Since we consider that the cost-utility analysis does not use all the available information and the solution provided by this method does not accord to the common sense, we gave another solution to this problem, based on multiple criteria optimization.

2.1. The formulation of the medico - economic problem A medical unit has 7000 money units (m.u.) for solving the following applications: � Case 1: Mr. Harper, 68 years old, retired, having two adult sons, has prostate cancer and his

doctor recommended a surgical intervention. Taking into account his present condition, if this operation is not performed, he would be able to live at most 10 more years, having health state 0.9. If the operation is performed, he would live at most 13 more years, but with 0.6 health state (due to the incontinence and impotence following the surgical intervention). The cost of this operation is 3200 m.u.;

� Case 2: Mrs. Patel, 58 years old, needs a hip prosthesis. She takes care on her 2 years old grand-daughter, while her unique daughter works and takes care on her husband and two sons living with her. Her actual condition indicates the chance of living 20 more years, either with or without hip prosthesis. The hip prosthesis would increase her health state with 0.2 for the next 20 years. The cost of this operation is 4100 m.u.;

� Case 3: Mrs. Hargreaves, 69 years old, no-smoking, no children, has a heart disease which condemns her to death if a surgical intervention is not performed. If the operation is done, she has the chance of living 10 more years, with 0.7 health state. The cost of this operation is 6900 m.u.;

� Case 4: Marta, a 25 years old young girl, has a tattoo on her neck, representing a flower, the name �Dean� being written in the middle. This is the name of a former friend that abused her, causing her relocation from the town she lived and worked before. She complains that the tattoo is psychologically harming her, diminishing her health state at 0.8, which affects her employment possibility. The psychiatrist estimates her health state will increase at 1 if the tattoo is removed. She is estimated to live until 79 years old. The cost of this operation is 3100 u.m.;

� Case 5: Javinder, a 7 years old little boy, has cystic fibrosis. He could live 6 more months without operation, with 0.3 health state. After a heart and lung transplant he can leave until 10 years old, his health state being 0.7. The cost of this operation is 7000 m.u.;

� Case 6: Patrick, physician anaesthetist, 35 years old, did a vasectomy 8 years ago because his first wife did not want to have children. He remarried a 37 years old woman and his

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present wife wishes very much to have children. The doctor considers that his wife risks a psychological depression in case she is not able to have children. Patrick�s health is perfect, with or without operation, and he is estimated to live until 75 years old. The cost of the reverse operation to vasectomy is 900 m.u.;

� Case 7: Mr. Moss, worker, 48 years old, smoked about 40 cigarettes by day during the last 28 years. He has a 10 years old boy at home and pays for the college education of another one. He suffers a severe heart infarct and needs a bypass of the coronary artery (CABG). He is not able to live with no surgical intervention. An extension of his life with 10 more years is estimated in case the operation is done, with 0.8 health state. The cost of this operation is 5300 u.m.

The board of the hospital must decide which application to accept, since the total amount of 7000 m.u. cannot be exceeded.

2.2. The solution by cost-utility analysis The solution given by the above mentioned Forum is based on the cost-utility analysis, taking into account the quality of life of the patient after the surgical intervention (see [6]). The principle of this method consists in the following steps:

� Compute, for each application, QALYs gained and the ratio between the cost of the operation and QALYs gained;

� Decreasingly order the applications with respect to the value of QALYs gained and, for equal values, decreasingly order with respect to the ratios values;

� Analyse each application, taking them into account the order obtained before; if the cost of the intervention does not exceed the difference between the initial amount of money and the sum of costs of the interventions already approved, the application is accepted. Otherwise, it is rejected.

The QALYs gained is computed by the formula QALYs gained = QALYo � QALYw,

where, if hso means the health state after the operation is performed, hsw means the health state if the operation is not performed, then

QALYo = hso × years lived after operation,

QALYw = hsw × years lived without operation. The following table contains the data obtained studying the above formulated problem.

NAME OF THE

APPLICANT

NO. QALYS GAINED

COST OF INTERVENTION

COST / QALYS GAINED

Harper 7 -1.2 3200 m.u. -2666 m.u./QALY

Patrick 6 0 900 m.u. -

Marta 1 15.8 3100 m.u. 196

Moss 2 8 5300 m.u. 662

Hargreaves 3 7 6900 m.u. 986

Javinder 4 6.85 7000 m.u. 1022

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Patel 5 4 4100 m.u. 1025

The aim of the cost-utility analysis is to maximize the value of QALYs gained and to minimize the ratio cost / QALYs gained. The table shows that Mr. Harper has a negative QALYs gained and this is zero in case of Patrick. The maximum value of QALYs gained appears to Marta and equals to 15.8, and the corresponding ratio, equal to 196, represents the minimum expense per unit of quality of life gained. The greatest ratio cost / QALYs gained is reached in case of Mrs. Patel, her hip prosthesis leading to the greatest cost per unit of life quality earned. Therefore, the decision reached using the cost-utility analysis is to accept Marta�s application and to reject all the other. One can remark that this decision would allow to direct money either to Mr. Harper�s operation or to Patrick�s operation. But both cases do not lead to increasing of QALYs gained, while the ratio between the total sum of costs of operations performed and the total sum of QALYs gained would increase.

2.3. Critical remarks on the cost-utility analysis solution The solution of the cost-utility analysis is not ethically acceptable. It let three people die (Mrs. Hargreaves, little Javinder and Mr. Patel) and accepts the funding of a plastic operation for a person that, willingly did tattoo the neck.

This happens because the cost-benefit analysis does not take into account other aspects of patient�s life, as:

� their role in supporting the family, � the influence of their health status on the other members of the family, � their contribution to rendering sick, � the operation is vital.

This solution puts the quality of life above life itself, creating an ethical conflict within the medical staff, consisting from members that already took the Oath of Hippocrates. Protecting life above all is one of the mankind ethical values not depending on the stage of the history, type of society, political regime, a.s.o.

Therefore, the choice should be restricted not only by medical and financial criteria but also by criteria of ethical nature. We intend to show, in what follows, how it is possible to solve such a problem by means of the multiple criteria optimization. A more general formulation of the problem is given, together with a mathematical model, solving it by the pounds method. This method allows us, by specifying the relative importance of criteria, to operate with the above discussed ethical issues.

2.4. Generalisation of the problem � mathematical model

The amount of S m.u. is destined to funding some surgical treatments. There are n applications from the part of patients P1, P2, �, Pn. The problem is to choose the patients such that the budget is not exceeded. For each patient, the following data are known:

� the age ti; � the cost of the treatment ci; � qi = QALYs gained; � if the treatment is vital then let vi = 1; otherwise vi = 0; � if the patient has other persons, having no other support, in ones care and the number of

these persons; � if the patient supports (helps) persons from the family and the number of these persons;

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� if the previous behaviour of the patient is responsible on the disease; � if the surgical treatment changes the health state of persons belonging to the patient�s family

and the number of these persons.

Based on these data, we are able to define three quality variables, similar to the quality of life:

� the social variable, s; � the guilt variable, g;

� the responsibility for other person variable, r. The guilt g and the responsibility r are binary variables, taking the values 0 and 1. The social variable has two dimensions, and its one dimension components are: variable care taken and variable supporter. The values of these variables depend on two coefficients, called the care taken coefficient and the supporter coefficient that may have values between 0 and 10 and express the degree of dependence on the care of the patient:

� 0 means that the supported (care taken) person has another possibility of support, respectively, is able to maintain oneself;

� 1 means that the supported (care taken) person has no income, respectively is not able to take care on oneself.

If a patient Pi supports mi persons having the degrees of maintenance equal to iim21 z,...,z,z , and

takes care of ni persons having the degree of help iin21 a,...,a,a , then

∑∑==

+=ii n

1hih

m

1kiki azs .

In order to formulate the mathematical model, let us consider the binary variables xi, i ∈ {1, 2, �, n}:

• xi := 1 if the application of patient Pi is approved;

• xi := 0 otherwise.

Let us define the functions fi: {0, 1} → R, i ∈ {1, 2, 3, 4, 5, 6}, as it follows: if x = (x1, x2,�, xn) ∈ {0, 1}n, then

( ) ∑=

=n

1iii1 xqxf describes the total QALYs gained ,

( ) ∑=

=n

1iii2 xsxf describes the social contribution of all the patients,

( ) ∑=

=n

1iii3 xvxf expresses the value of the operation for the patients� lives,

( ) ∑=

=n

1iii4 xrxf expresses the contribution of the operation to the health state of other family

members of the patients,

( ) ∑=

=n

1ii5 xxf represents the good will of satisfying as most applications as possible,

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( ) ∑=

=n

1iii6 xgxf expresses the degree of guilt of patients for the illness.

The problem, formulated at the beginning of this paragraph, is mathematically modelled and solved by the following multiple criteria optimization problem:

( )

{ } { }

∈∈≤

→→→→→→

∑∑∑∑∑∑∑

=

=

=

=

=

=

=

.n,...,2,1ifor,1,0xSxc

maxxminxgmaxxrmaxxvmaxxsmaxxq

PV

i

n1i ii

n1i i

n1i ii

n1i ii

n1i ii

n1i ii

n1i ii

Let us solve this problem using the weighting method. Let us introduce the pounds λ1, λ2, λ3, λ4, λ5, λ6 satisfying the condition 16

1i i =∑ = λ . Every solution of the problem

( ) ( )

∈→

,Xgmaxxg

Q

( ) { }{ },Sxc1,0x,...,x,xxX n1i ii

nn21 ≤∈== ∑ =

function g: {0, 1}n → R, being defined by

( ) ( ) ( )xfxfxg 665

1jjj λλ −= ∑

=, for x ∈ {0, 1}n,

is a Pareto point of the multiple criteria optimization (PV).

2.5. Solution to the medico - economic problem Let us come back to the initial problem by means of the method presented in subsection 2.4. The data are in the following table:

I CI VI QI GI RI SI

1 3200 0 -1.2 0 0 0

2 4100 0 4 0 0 5

3 6900 1 7 0 0 0

4 3100 0 15.8 1 0 0

5 7000 1 6.85 0 0 0

6 900 0 0 1 0.2 0

7 5300 1 7 1 0 15

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Taking equal weights, λ, = λ2 = λ3 = λ4 = λ5 = λ6 = 51 , function g is given by

( ) ( )7654321n21 x23x2.0x35.7x8.15x5.7x5.9x7.051x,...,x,xg ++++++−= .

The optimum solution of problem (Q) is

( )1,1,0,0,0,0,0x = ,

meaning that Mrs. Moss and Patrick are selected for operation. According to our opinion, this solution is more correct than that given by the cost-utility analysis, since it saves one of the three lives that should be saved and it solves one more application. We remark that the budget allows saving only one life. On the other hand, it is not in conflict with the medical Oath of Hippocrates, supposing first to protect the life itself and second to improve it. As we can remark, the ethical choice puts life itself before the quality of life all along the mankind history.

3. The characterization of an environmental policy There are many domains that use efficiency indexes to assess various types of activities, economic processes. For example, in the energy domain there are more types of efficiency index, both in terms of costs and in terms of effects, depending on the aim of the researcher (see [1], [7], [8]). The relationship between trade and environmental conditions is very important whenever countries are in the process of negotiating trade agreements. So, an environmental efficiency index for a sample of high income and low and middle income countries was developed [8] allowing to examine the role of trade on the changes in environmental efficiency.

The idea of an efficiency of a legislation system was recently published ([9]). We treat the problem of keeping a healthy environment by similar methods to farmaco-economics, from mathematical point of view (see [2], [4]), looking to an environmental policy like to a vaccine to prevent a disease [5]. The multiple criteria programming is used in order to make the best choice.

3.1. Problem formulation The aim of this section is to obtain a practical and useful possibility of characterizing an environmental policy, using multiple criteria programming. We take into account more points of view in the further research: cost, effectiveness, side effects and their seriousness, etc.

Two homogeneous groups of people in similar environment are necessary for testing a policy. One group is submitted to the environmental policy, while the other one, the control group, is not. One can replace these two groups by the same group living first without the environment policy and after it, changing the environment according to the environment policy. We take into account two different aspects: the employees' health and the environment's health.

First, let us deal with the employees� behaviour. After a known period of time,

� out the group living with the environment policy: � pEN % have no disease; � out of people following the environment policy that have some disease:

• pED % died; • pER % completely recovered; • pEC % have complications;

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� pEA % have negative reactions within the environment policy;

� out of the group living without the environment policy: � pN % have no disease; � out of people not following the environment policy, that have some disease:

• pD % died due to the disease; • pR % have disease with completely recovered; • pC % have disease with complications.

The following costs are known:

� cE the cost of applying the environment policy / person; � cR the cost of the total recovery / person if the environment policy is applied; � cC the cost of treating the health complications and death compensations / person if the

environment policy is applied; � cA the cost of treating the negative reactions / person if the environment policy is applied.

The main purpose is to elaborate a method of choosing an environmental policy such as to have disease, negative reactions and death cases as less as possible and to minimize the total expense with it. The ecologic-economic efficiency of the environment policy should be studied in these conditions. For this purpose, a mathematical model is attached to this problem, in terms of a multiple criteria programming problem in variables 0 and 1. These values are meant to express the preference for a type of action, meaning that two binary variables, x1 and x2 are introduced, having the following significance:

• x1 =1 means that the environment policy is used; • x1 =0 means that the environment policy is not used; • x2 =1 means that no environment policy is preferred; • x2 =0 means that the environment policy is preferred.

Of course, x1 + x2 = 1, since an environment policy may be only accepted or rejected. The objective functions are f1: {0,1} × {0,1} → R, f2: {0,1} × {0,1} → R and f3: {0,1} × {0,1} → R, defined, for every (x1, x2) ∈ {0,1} × {0,1} by:

f1(x1, x2) = (1 - pEN)x1 + (1 - pN)x2,

f2(x1, x2) = pEDx1 + pDx2,

f3(x1, x2) = x1 + E

Nαα

x2,

where αN = pRcR + (pD + pC)cC and αE = 100cE + pERcR + pEAcA + (pED + pEC)cC.

Then, the solution comes from finding the min-efficient points of the following vectorial programming problem, denoted by (PE):

(pENx1 + pNx2 , pEDx1 + pDx2 , x1 + E

Nαα

x2) → v-min

when x1 + x2 = 1, (x1, x2) ∈ {0,1} × {0,1}. 3.2. Basic concepts for solving problem (PE)

First, let us recall few basic concepts that are used in this section. Let X be a nonempty set and let f = (f1, f2, ... , fn): X → Rⁿ.

Definition 3.1. A point a ∈ X is called a min-efficient point of f on X if there is no x∈ X such that

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fi(x) ≤ fi(a), i ∈ {1, 2, ... , n}, and ( ) ( )∑∑ == < n1j j

n1j j afxf .

In order to solve problem (PE), we use the weight method, obtaining a unique synthesis function. The main result of Galperin [3] is used: Theorem 3.2. If λ1>0, λ2>0, ..., λn>0 are n given real numbers, then every minimum point of the function F: X → R, defined by

( ) ( )∑ == n1j jj xfxF λ

for every x ∈ X, is a min-efficient point of the vectorial function f on X. An algorithm for finding the min-efficient points of a vectorial function is published in [2].

3.3 Solution to problem (PE) In order to solve problem (PE), we introduce the synthesis function F: {0, 1} × {0, 1} → R using the pounds λ1>0, λ2>0 and λ3>0, getting

F(x1,x2)=λ1f1(x1,x2)+λ2f2(x1,x2) + λ3f3(x1,x2). With this function, problem (PE) turns into the following problem (P):

F(x1,x2) = λ1(- pEN x1 - pN x2) + λ2(pED x1 + pD x2) + λ3

−

E

N1αα → min,

when x1 + x2 = 1, (x1, x2) ∈ {0,1} × {0,1}. By elementary calculus one gets

F(0,1) = - λ1 pN + λ2 pD + λ3 E

Nαα

,

F(1,0) = - λ1 pEN + λ2 pED + λ3

and, as consequence,

F(1,0) - F(0,1) = λ1(pN � pEN) + λ2(pED - pD) + λ3

−

E

N1αα .

If F(1,0) - F(0,1) ≤ 0 then one can decide that the environmental policy is profitable. Also, an environmental policy is better than another one if its F(1,0) - F(0,1) is the lowest one (i. e. its absolute value is the greatest one). This result is the reason of using the difference F(1,0) - F(0,1) as a method of making the decision, when the choice of an environmental policy is under debate.

3.4. The ecologic-economical effectiveness index In this subsection we investigate the properties of the difference F(1,0) - F(0,1) and the manner in which it is able to turn into a decision making tool in the process of choosing an environmental policy. Definition 3.3. The ecologic-economical effectiveness index of an environmental policy is the number

− +) − (+ )−(= 21

E

N3DD 1ppppEEf

ααλλλ ΕΕΝΝ .

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As one can remark, this index provides an inner characterization of the effect of an environmental policy on humans living or working in the environment under treatment, since it does not depend only on costs, taking into account the effects of the policy on human beings.

The monotony properties of this index are: � EEf decreases when the environmental policy increases the healthy persons percentage; � EEf decreases when the percentage of persons completely recovered is greater under the

environmental policy than otherwise; � EEf decreases when the percentage of persons dying under the environmental policy is

lower than otherwise; � EEf decreases when the persons with complications and side effects under the

environmental policy produces lower costs than the persons with complications in absence of an environmental policy.

As consequence, one can say that an environmental policy is more efficient than another one if it has a lower negative EEf. On another hand, it depends on the social or moral system of values of the decision makers: the pounds λ1, λ2 and λ3 are chosen according to the importance given to each criterion f1, f2 and f3 within the company.

Example 3.4. This index was tested as a decision method in choosing the policy of reducing the temperature within a company in Arad County during the summer time. The increase of the temperature at more than 35o C had severe consequences not only by drastically decreasing the work efficiency of the employees but also on their momentary health condition. All the effects of the increase of the temperature on employees will be referred in what follows as disease. Therefore, a policy of reducing the temperature within the company was urgently applied, choosing it by some rules of thumbs. The unit of the company, we have been allowed to study, has 2540 employees, having no healthy issues in normal conditions (average temperature of 24o C). The total cost of the complete recovery per one season is 6000 �, the total cost of treating the complications is 10000 � and the total cost of treating the side effects of the environmental policy is 15000 � for the entire personnel per season. Before taking action for reducing the temperature, the behaviour of the company employees working within the unit under investigation was recorded as follows: pD = 0, pN = 15%, pC = 80%, pR = 5%. Therefore, αN = 433.1. Three solutions to reduce the temperature were presented to the company. The estimated consequences of each policy are presented in the following table.

PROJECT REDUCED TEMPERATURE

TOTAL COST

(�)

PEN

% PER

% PED

% PEA %

ΑE

1 28o C 45600 80 10 0 5 1861.1

2 30o C 250000 70 15 0 7 9952.75

3 22o C 490000 100 0 0 0 19291.34

Taking the pounds λ1 = λ2 = 2 and λ3 = 1 (according to the opinion of company�s staff) and computing the ecologic-economical effectiveness index of each environmental policy, we got

EEf(project1) = - 129.24,

EEf(project2) = - 109.04, EEf(project3) = - 169.02.

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It shows that Project 3, which reduces the temperature at 22o C, is the most efficient from the point of view of the employees� behaviour at their workplace. According to our opinion, the use of an index of this kind instead of the known ones, referring mainly to costs and benefits, when an environment strategy is discussed, is an ethical option. As one can see, in this method of approaching the problem, the costs may be present as auxiliary parameters, with secondary impact on the decision making process.

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Representation of the UK Energy System, UK Energy Research Centre, on line [http://www.ukerc.ac.uk];

2. G. Cristescu, Liana Lupşa, Non-Connected Convexities and Applications, Kluwer Academic Publishers, Dordrecht / Boston / London, 2002;

3. E. A. Galperin, Nonscalarized Multiobjective Global Optimization, J. O. T. A. 75, 1(1992), 69-85;

4. Luciana Lupşa, Construction of a Medico-Economic Effectiveness Index which Characterizes a Vaccine, Séminaire de la Théorie de la Meilleure Approximation, Convexité et Optimisation (editor E. Popoviciu), Srima, Cluj-Napoca, 2001, 61-65;

5. L. Neamţiu, Applications of the Numerical Analysis, Optimisation and Informatics in Medecine (in Romanian), Ph.D. Thesis, �Babeş-Bolyai� University of Cluj-Napoca, Faculty of Mathematics and Computer Sciences, Cluj-Napoca, 2007;

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7. H-P. Siderius, R. Harrison, An Energy Efficiency Index for TVs, In 'Energy Efficiency in Household Appliences and Lighting' (editors Bertoldi, Ricci, de Almeida), Springer Verlag, Berlin, 2001, 267-277;

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