CHOICE OF A DOMESTIC HEATING SYSTEM BY SEMI-GENERALIZED LOTTERIES OF I TYPE

IFAC WS ESC’06 ENERGY SAVING CONTROL IN PLANTS AND BUILDINGS, October 2-5, 2006 Bansko, Bulgaria Copyright © IFAC ENERGY SAVING CONTROL IN PLANTS AND BUILDINGS Bansko, Bulgaria, 2006

CHOICE OF A DOMESTIC HEATING SYSTEM BY SEMI-GENERALIZED LOTTERIES OF I TYPE N. Nikolova*, G. Ramos*, K. Tenekedjiev* * Technical University - Varna, Bulgaria, e-mail:{kiril,natalia}@dilogos.com ** Tokyo Institute of Technology, Interdisciplinary School of Science and Engineering, Hirota Lab, Japan, e-mail: [email protected]

Abstract: Discussed is the choice between five alternatives for domestic heating with ten-year decision horizon. Consequences are described as 8-D vectors. The main source of uncertainty is the only continuous feature – average annual NPV of year heating costs. The distributions of those costs are acquired by risk analysis, on the basis of subjective prognoses for the annual inflation and prices of energy resources. A novel choice modeling tool, called semi-generalized lottery of I type, is introduced, where the uncertainty of the attribute vector’s discrete and continuous parts is measured respectively with probabilities and conditional densities. Formulae for the expected utility of semi-generalized lotteries are justified for the case of additive independent preferences over the discrete and continuous attributes. The five alternatives are ranked in a special case of probability independence of the attributes. Copyright © 2006 IFAC Keywords: subjective probabilities, risk analysis, expected utility. following assumptions are made: a) the heating season lasts 150 days (November-march); b) 1/3 of the electricity consumed is paid on a night rate; c) each alternative provides normal temperature in all four rooms of the flat; d) alternatives а1, а2 and а4 assume the construction of a local heating system, which heats the service compartments, whereas the latter are not heated under the other alternatives; e) protracted boilers have 20% lower consumption than electrical boilers; f) the calculations of the alternatives start from the heating season 2006/2007. At alternative a1 a furnace for coal and wood shall be used (“ATMOS kombi”, power 12-26kW, efficiency 84%). For a heating season it needs 5m3 of wood, and 6t coal. In the summer period 10kW are consumer per day for hot water from an electrical boiler, since the furnace is not used. It is equally probable for the initial investment and the duration for installation to be 6100 BGN and 40 days, or 6300 BGN and 50 days. At this alternative, servicing the system is hard and unpleasant, with a long period for heating of the rooms and the consumption is changed slightly when heating only some of the rooms. In alternative a2 a furnace „DAKOM” with power 18kW and efficiency 98% shall be used. The system has a protracted boiler for hot water, and the electrical boiler may be removed. The daily consumption of the system is 125.6kWh during the winter and 8kWh in the summer. It is equally probable for the initial

1. INTRODUCTION The choice of a heating scheme from the common household in the country should reflect the specificity of the house, the financial means, the comfort and cleanness of the heating devices, and the price change tendencies of the energy resources. Several ways to heat an urban panel flat shall be analyzed, with area of 80m2, consisting of two bedrooms, a lining room, kitchen and service compartments (WC, bathroom, closet, corridor), with a standard height of 2.5m, outward isolation of the north wall and PVC framing. Till the spring of 2006 the flat was heated by electrical heaters with a total power of 9kW and there is no local heating installation. Hot water for household needs is provided by an electrical boiler. During the heating period a total of 118kWh are used per day, and in the rest of the time – 10kWh per day for hot water. 2. DESCRIPTION OF THE HEATING ALTERNATIVES It is necessary to rank according to preference the elements of the set L={a1,a2,a3,a4,a5}, containing the following heating alternatives: a1 – installation of a coal-wood furnace system; a2 – installation of electrical furnace system; a3 – installation of airconditioners; a4 – installation of a gas furnace system; a5 – maintaining the present heating system. The time horizon of the decision is 10 years (2006-2015). The

31

investment and the duration of installation to be 8100 BGN and 40 days or 8300 BGN and 50 days. There is a long heating period in this alternative, and the consumption is slightly changed when heating only some of the rooms. Heating with air conditioners (a3) requires two single devices for the kitchen and the living room („Panasonic 9 cs/cu-a” and „Panasonic 12 cs/cu-a”) and a double device for the two bedrooms („Panasonic 18 cs/cu-a”) with a total heating power of 4.3kW and investment of 4800 BGN. The ratio of used to acquired energy for heating is 3.2. Additionally, during the whole year an electrical boiler is used for hot water. The daiy consumption of the system is 43kWh during the winter and 10kWh during the summer. The duration of installation is only 3 days. At this alternative, there is a short period for heating of the rooms and the consumption is changed proportionally to the heated volume. For the gas furnace system (a4) a furnace “DACON duo” shall be used with power 13-30kW and efficiency of 93%. The system has a protracted boiler for hot water and the electrical boiler may be removed. The daily consumption of the system is 14.9m3 gas during the winter, and 0.89m3 gas during the summer. The initial investment and the duration of installation may be respectively 5500 BGN and 50 days with 60% probability, 6000 BGN and 75 days with 30% probability or 6500 BGN and 100 days with probability 10%. There is an average heating period at this alternative, and the consumption is slightly changed when heating only some of the rooms. Alternative a5 does not require investment and time for installation. Eight objectives Table 1. Description of the discrete may be defined in part of the consequences d pi,r ni consequence the choice, and ai xi,r the consequences d x (6.1, 40, 2, 1, 1, 1, 2) 0.5 may be presented a1 11, 2 d as 8-dimensional x1,2 (6.3, 50, 2, 1, 1, 1, 2) 0.5 vectors with one d coordinate for a2 x2 ,1 (8.1, 40, 3, 2, 2, 1, 2) 0.5 2 each objective: 1) x2d,2 (8.3, 40, 3, 2, 2, 1, 2) 0.5 coordinate for the d a3 x3,1 (4.8, 3, 1, 2, 3, 2, 1) 1 1 objective “minimize x4d,1 (5.5, 50, 3, 2, 2, 1, 2) 0.6 heating costs” – d (6, 75, 3, 2, 2, 1, 2) 0.3 3 NPV of the a4 x4 ,2 average annual d x4 ,3 (6.5, 100, 3, 2, 2, 1, 2) 0.1 heating costs; 2) coordinate for the a5 x5d,1 (0, 0, 1, 2, 3, 2, 1) 1 1 objective “minimize investment” – amount of the investment in thousand BGN; 3) coordinate for the objective “minimize installation time” – duration of installation in days; 4) coordinate for the objective “maximize accessibility to hot water during the year” – nominal criterion: 3-complete (during the whole year), 2partial (only in winter); 1-lack; 5) coordinate for the objective “minimize labor consumption” – nominal criterion: 2-low, 1-high; 6) coordinate for the objective “minimize inertia of the system” – nominal criterion: 3-low, 2-medium, 1-high; 7) coordinate for the objective “maximize space control” – nominal criterion: 2-good, 1-bad; 8) coordinate for the

objective “heating service compartments” – nominal criterion: 2-yes, 1-no. The second and third attributes, although naturally continuous, can be successfully approximated as discrete parameters, since the level of information the decision maker (DM) has for those is high enough, because uncertainty will be resolved in half an year. In table 1, a description is given of the uncertainty in the consequence discrete part. Let ni be the number of different discrete consequences from the i-th d alternative, and the r-th discrete consequence xi,r has probability pi,r for r=1,2,.., ni. It is necessary to obtain a probability description of the only continuous parameter x1c of the NPV of average annual heating costs in BGN. 3. CONSTRUCTING THE DISTRIBUTION LAWS OF THE ANNUAL DOSCOUNTED HEATING COSTS The construction of the density of the NPV of average annual costs is made by risk analysis (Hertz and Thomas, 1983). The simulation modeling includes four stages: 1) simulation of annual inflation from 2006 to 2016 for each pseudo-reality; 2) simulation of annual prices of energy resources for each pseudoreality; 3) calculation of NVP of average annual heating costs for each alternative in each pseudoreality; 4) construction of a frequentists density for the NPV of the average annual heating costs for each alternative using the numerical experiments in different pseudo-realities. Step 1. Simulation of inflation is made on the basis of an auto-regression model AR(1), taking into consideration the degree to which macroeconomic processes repeat itself: Inflt+1=Is × Inflt+(1-Is) × ∆ Inflt+1, for t=2006,…,2015 Infl2006= 100 1.04 × ( 1 + ∆ Infl1 / 100 ) − 100 (1) In (1), Inflt is the annual inflation in the t-th year, Is is a coefficient of economic stability, and ∆ Inflt is the inflation error in the t-th year. The latter is modeled by a subjective quantile-approximated distribution in the interval [– 0.005; 0.105], and is shown on fig. 1. In the 2006 inflation, 4% of the accumulated inflation from the beginning of the year is considered. Using an algorithm from (Tenekedjiev, et al., 2004), 11 values of ∆ Inflt are generated for each pseudoreality. The annual inflations in (1) are calculated at IS=0.4. Example inflation from 2006 to 2016 in a single pseudo-reality is shown in column 7 of table 4 (where the inflation for the 2006 is given from July onwards). Density of the Inflation Noise Inflt+1=0.40Inflt+0.60∆ Inflt+1 0.25

PDF(∆ Infl)

0.2

0.15

0.1

0.05

0 -2

0

2

4

6

8

10

12

∆Infl, %

Fig. 1. Density of the corrected inflation at Is=0.4

32

Step 2. Table 2 presents the subjective minimal and maximal prognoses for prices (at the end of the year) of the energy resources till the 2015. Their generation accounted for: a) the expected increase of the price of electricity by 30-40% after shutting down the atomic power plant in Kozloduj in 2007; 2) the expected reach of the present European price levels till the 2010; 3) the expected stabilization of growth by 4-8% after 2010; 4) the expected increase by 30-70% of the prices of natural gas till the 2010, when the supply contracts of the country with Russia expire.

3) The probability for correction of consumers’ prices is calculated (it is assumed that there is no correction for the 2005); 4) The prices are corrected with the calculated probability at the end of the year. Figure 2 shows the graphics of the minimal, maxima, rough (Pzrough), necessary pseudo-real (Pzpr) and end prices of coal for the period 2005-2015.

EC

Municipal elections

Pcorr (%)

2006

60

− ∆mun

50

+ mun

year

Electricity (BGN/kWh) min max

Natural gas (BGN/1000m3) min max

Coal (BGN/ton) min max

Wood (BGN/m3) min max

2005

0.15

0.15

650

650

160

160

40

40

2006

0.17

0.19

670

710

160

180

46

50

Parliament elections

Table 2. Prognoses for the prices of energy resources

∆

2007

0.22

0.27

700

780

170

190

53

63

2007

60

2008

0.23

0.31

730

860

170

210

61

78

2008

60+ ∆EC

− ∆parl

2009

0.24

0.34

750

940

180

230

70

98

2010

0.25

0.39

780

1000

180

260

80

120

2009

60+ ∆EC

+ ∆parl

2011

0.26

0.43

1000

1800

190

280

83

2012

0.28

0.46

1100

1900

200

310

85

150

2013

0.29

0.50

1100

2100

200

340

88

160

2014

0.30

0.54

1200

2200

210

380

91

180

2015

0.31

0.58

1200

2400

220

420

93

200

− + ∆pres , ∆pres ,

− ∆mun

and

+ ∆mun

=30%,

− ∆pres

− = ∆mun

=25%,

+ ∆pres

∆

− parl

∆

+ parl

75 30 100

∆

− pres

∆

+ pres

− mun

30

+ mun

100

∆ ∆

30

2013

60+ ∆EC

2014

60+ ∆EC

− ∆mun

55

2015

60+ ∆EC

+ mun

95

100

∆

Predicted and Pseudo-Real Prices for Coal 450 minimal maximal 400

pseudo-real rough cost necessary pseudo-real cost pseudo-real price

Price

350

300

250

200

150 2004

denote the

+ = ∆mun

60+ ∆EC

2012

2006

2008

2010 Year

2012

2014

2016

Fig. 2. Graphics of the minimal, maximal, rough (Pzrough), necessary pseudo-real (Pzpr) and end prices of coal for the period

probabilities for decrease/increase of the prices before/after parliament, presidential and municipal − elections. It is subjectively assessed that ∆parl =50%, + ∆parl

60+ ∆EC

2011

It is assumed that the necessary prices for effective work of the energetic enterprises will lightly be increased, representing a uniformly distributed random variable between the minimum and maximum. It is assumed that actual prices for population shall be corrected once a year at most and at correction, shall reach the necessary prices in each pseudo-reality. The base probability for correction at the end of the years for the analyzed period is given in table 3, where the main estimate of 60% is corrected due to the following factors: a) additional stabilizing influence over the prices from the side of the European commission with probability ∆EC =20%; b) the influence of elections in the country (presidential, parliament, municipal) over the probabilities for price correction on the national market, i.e. they decrease in the year preceding the − , elections, and then they increase again. Let ∆parl + ∆parl ,

60+ ∆EC

2010

130

Presidential elections

Year

Table 3. Base probabilities for increase of prices of energy resources

Step 3. The following procedure is executed: 1) Defined are the average annual prices of energy resources for the summer and winter period. The winter period comprises 2 and 3 months from two consecutive years, which is why the average winter price includes 2/5 of the price in the first and 3/5 of the price in the second year of the heating season; 2) Calculated are the expenditures for the summer and winter period of each year in the pseudo-reality; 3) The summer expenditures for each year are discounted to the beginning of the current year by

=15%. If

there has been price correction in a certain year, then the probability for correction in the next year is decreased by 10% from the base, and vice versa – at lack of price increase, an increase by 10% in the next year’ base probability is made. For each energy resource the following simulation procedure may take place: 1) For each year, rough values Pzrough of the price are generated, as a random part (rand) of the difference between the maximal and minimal price is added to the minimal price; 2) The values of Pzrough are sorted to form the vector of necessary pseudo-real prices Pzpr;

multiplication with pvyst=

1 (value of 1 1 + Inflt 100

BGN in the middle of the current year to its start);

33

4) The total annual discounted expenditures to the begging of the year are obtained; 5) The discounted annual expenditures for the i-th year are discounted to 30.06.2006 (i.e. to the moment of choice) by multiplication with the value of 1 BGN at the end of the t-th year to the middle of 2006 pvt= t     1 1 = ×   ;  1 + Infl2006 / 100  j =∏ 2007  1 + Infl j 100    6) The discounted sums from 5) for each year are summed to result in the total NPV of heating costs. The sum is then divided to the number of years, which gives a better description to the DM for the expected costs in each period. Table 4 describes the content of a single pseudoreality for the calculation of NPV of the average annual heating costs. Step 4. After repeating the six-stage procedure in step 3 M=1000 times, the PDF of the NPV of average annual costs is constructed according to an algorithm from (Tenekedjiev, et al., 2002). Figure 3 shows the densities of the NPV of average annual costs for a1, whereas table 5 contains the expectations, standard deviations and medians of the distributions, all measured in BGN/year. -3

6

x 10

Densities of Alternatives' Average Annual NPV

4

7

ni

j =1

r =1

β × Eid ( u / p ) = ∑ k dj ∑ pi ,r u dj ( xi,d j ,r ) ,

coal furnace electrical furnace air conditioners gas furnace electrical heaters

5

PDFi(NPV)

lotteries of I type, in particular, the elements of L are a finite set, whereas the prizes form a continuous n-D set. Sometimes in problems similar to the one here analyzed, there is a discrete and a continuous part of the consequences. It is proposed for the uncertainty of the attribute vector’s discrete and continuous parts to be measured respectively with probabilities and conditional densities. This modeling tool could be called semi-generalized lottery of I type. In appendix 1, formulae are justified for the expected utility of semi-generalized lotteries of I type, for the case of additive independent preferences (Keeney and Raiffa, 1993) over the continuous and the discrete attributes: u( x ) = α u( xc ) + β u( xd ) . (2) In the special case of probability independence of the attributes, the conditional probability distributions of the continuous part of the attributes transforms into unconditional, and the following holds for the expected utility of the of the i–th alternative (3) Ei(u/p)= α Eic ( u / p ) + β Eid ( u / p ) . In the case revised, the expected utilities of the discrete and the continuous part take the form respectively

Eic ( u / p ) =

NPVmax

c c c c c ∫ u1 (x1 ) f1 (x1 )dx1 ,

(4) (5)

NPVmin

where the values of the NPV of the average annual costs for all alternatives belong to the interval [NPVmin; NPVmax], and xid, j ,r is the r-th realization of

3

the j-th attribute at the i-th alternative. 2

5. CONSTRUCTING THE UTILITY FUNCTION 1

0 1000

2000

3000 4000 NPV,BGN/year

5000

A. Utility function over the continuous attribute The utility function u1c over the NPV of average annual costs is built in the interval [NPVmin; NPVmax] ≡ [1000; 7000]. Obviously, u1c (1000)=1, u1c (7000)=0. Nine values of utility are chosen with a step 0.1. The uncertain equivalence method (Tenekedjiev, et al., 2006) is used to assess the values of NPV that have those utilities, and the point estimates obtained are: u1c (5615.1)=0.1, u1c (4784.9)=0.2, u1c (4197.2)=0.3, u1c (3730.4)=0.4, u1c (3324.6)=0.5, u1c (2943.2)=0.6, u1c (2557.7)=0.7, u1c (21381)=0.8 and u1c (1643)=0.9. The utility of other NPV values may be found by linear interpolation.

6000

Fig. 3. Densities of the NPV of average annual heating costs at M=1000

4. SEMI-GENERALIZED LOTTERIES

Table 5. Mean, median and standard deviation of the average annual NPV of heating costs

Decision making under σj ( x0.5 ) j j Ej risk, according to utility 1 1922 80 1921 theory (von Neumann and 2 4871 271 4863 Morgenstern, 1947), 3 2066 116 2062 requires ranking 4 2081 118 2083 alternatives, modeled as 5 4705 262 4698 lotteries, i.e. a set of excluding prizes (consequences) and the probability to win each. A set of lotteries L and a set of prizes X are formed. If L and X are finite, then an ordinary lottery applies l=, where pi≥0

B. Utility function over the discrete attributes It is convenient for the utility function over the discrete attributes x1d and x2d to be built as continuous in the intervals [0;9000] (where the investments in each alternative fall) and [0;100] (where the installation time of each alternative falls). Obviously u1d (0)=1, u1d (9000)=0. Nine values of utility are chosen with a step 0.1. The uncertain equivalence method is used to assess the investments with those utilities, and the obtained point estimates are: u1d (7119.6)=0.1, u1d (6035.3)=0.2, u1d (5274.1)=

q

is the probability to receive the price xi and ∑ pi = 1 . i =1

X may take one of several continuous intervals in the n-D Euclidian space n and then transforms into a continuous n-dimensional set of prizes. In that case generalized lotteries of I, II or III type are formed (Tenekedjiev, 2004a). In the case of generalized

34

x1d,worst ,

=0.3, u1d (4663.5)=0.4, u1d (4118.8)=0.5, u1d (3585.1)= =0.6, u1d (3011.9)=0.7, u1d (2332.2)=0.8 and d u1 (1424.6)=0.9. The utility of the necessary investments may be found by linear interpolation. Obviously, u2d (0)=1, u2d (100)=0. Nine values of the utility are chosen with a step 0.1. The uncertain equivalence method is used to assess the installation periods with those utilities, and the obtained point estimates are: u2d (97.5)=0.1, u2d (94.5)=0.2, u2d (91)=0.3, u2d (86.6)=0.4, u2d (81.3)= =0.5, d d d u2 (74.3)=0.6, u2 (65)=0.7, u2 (52)=0.8 and u2d (32.5)=0.9. The utilities of the necessary installation periods may be found by linear interpolation. The most preferred value of the third attribute is 1, and the worst – 3, i.e. u3d (1)=1,

3, 3, 2, 3, 2, 2] and x worst =[7000, 9, 100, 1, 1, 1, 1, 1] becomes indifferent to the prize. Since there is only a single continuous attribute the corresponding scaling 7

constant equals to k1c = 1 − ∑ k dj , which follows from j =1

the additive independence preferences of the DM. Let k c = k1c = α be the 1-D scaling constant of the continuous attribute, whereas k d be the 7-D vector of scaling constants for the discrete attributes. 6. RANKING ALTERNATIVES The calculation of the expected utility integral (5) over the continuous attribute is performed by an original procedure in (Tenekedjiev, 2004b) with no additional integration error. According to (3), (4) and (5), in the case analyzed the expected utility over the whole prizes of the i-th alternative is calculated as

using the lottery equivalence method (McCord and De Neufville, 1986). For the fourth attribute, u4d (2)=1, u4d (1)=0. In the same fashion, u5d (3)=1, u5d (1)=0. The utility of the average inertia is

u5d (2)=0.6, assessed by the lottery equivalence

method. For the sixth attribute,

(2)=1,

u6d

x7d,worst ]. The

x dj,best , …,

assessment of scaling constants is commented in (Clemen, 1996). In this case, for each corner consequence the DM should find the probability pj= k dj at which the lottery between xbest =[1000, 0,

u3d (3)=0. The utility of 2 is u3d (2)=0.3, assessed

u6d

x2d,worst , …,

Ei ( u / p ) = k1c ×

(1)=0. In

NPVmax

c c c c c ∫ u1 (x1 ) f1 (x1 )dx1 +

NPVmin

the same fashion, u7d (2)=1, а u7d (1)=0.

7

+∑

j =1

C. Assessment of the scaling constants The constants in (3) and (4) are called scaling constants. At normalized 1-D utility functions u1c (.),

k dj

ni

∑

r =1

.

pi ,r u dj ( xi,dj,r

)

In lines 2 and 9 of table 6, two sets of scaling constants are given. The results for the expected utility of the alternatives as well as separately for the discrete and the continuous part of the attributes are given respectively in lines 3-7 and 9-13. It is obvious that estimates of the scaling constants, reflecting the DM’s value function, give different ranking of alternatives. The lines of the best alternative under both sets of constant are given shaded in the table.

u1d (.), u2d (.), …, u7d (.) the scaling constants equal the

utility of the so-called corner consequences, for which all attributes are fixed to their worst levels, except for the j-th attribute. For the discrete part of the attributes in the problem revised, the corner consequence for each attribute could be of the kind xdi,corner = [ x1c,worst ,

Tabl. 4. One pseudo-reality for the alternative “coal boiler” with annual NPV=1851 BGN Тime

Winter Costs (BGN)

Summer Costs (BGN)

PV of 1 BGN at year start

Annual costs, discounted to the beginning of the year (BGN)

PV of 1 BGN at 30.06.2006

1566

0.981

Annual Inflation (%)

Annual costs, discounted to 30.06.2006 (BGN)

31.12.2005

3.822 31.12.2006 31.12.2007 31.12.2008 31.12.2009 31.12.2010 31.12.2011 31.12.2012 31.12.2013 31.12.2014 31.12.2015

1188 387

0.980

502

0.975

552

0.979

613

0.976

632

0.972

690

0.973

747

0.970

913

0.967

954

0.965

1013

0.970

1330

0.817

1854

0.771

1907

0.730

1917

0.687

1949 6.910

2908

2634

1849

6.189 2836

1987

0.857

5.600 2625

1953

1748

5.929 2473

1901

0.895

4.989 2270

1802

1714

4.418 2157

1656

0.942 5.254

1953

1559

1537 4.152

1819

1413

(6)

0.643

1870 7.393

3616

0.599

2165 6.356

31.12.2016

35

Table 6. Expected utility and ranking of the alternatives

E cj ( u / p ) c

E dj ( u / p )

Ej(u/p)

d

of the continuous attribute in x , whereas x is the k–dimensional d

In that case, x is a t–dimensional vector of possible prizes (consequences), where t=m+k. Let Ω be the t-dimensional space

ℜt of prizes within the problem. If the space of continuous m attributes of the prizes is the m-dimensional non-singular ℜ

c

c

d

space Ω , then Ωi = ( Ω ; xi ) is the t–dimensional k–times

k =0.5; k =[0.24; 0.01; 0.06; 0.01; 0.01; 0.12; 0.05] 0.4218 0.1207 0.5425 3 a1 0.0965 0.1446 0.2411 5 a2 0.4069 0.2405 0.6475 1 a3 0.4054 0.1901 0.5954 2 a4 0.1086 0.3900 0.4986 4 a5

singular uni-connected space of all possible combinations of the continuous part of the consequences with the i-th combination of discrete attribute values. The union of all Ωi gives the t– dimensional space Ω

sing

= { Ω1 ∪ Ω 2 ∪ … ∪ Ω n } , which is the

subspace of Ω , where x may belong. Let’s denote Pi the probability of the discrete part of the attributes

7. CONCLUSIONS

d

d

taking the i–th combination: Pi = P( x = xi ) . The conditional densities of the continuous attribute provided the discrete attributes

The paper presented a detailed analysis of a task, where the consequence vector contained both continuous and discrete attributes. The problem is modeled as semi-generalized lottery of I type, where the uncertainty of the attribute vector’s discrete and continuous parts was measured respectively with probabilities and conditional densities. Formulae for the calculation of the expected utility for the case of additive independence of preferences over the continuous and the discrete attributes were presented in an appendix. Alternatives were ranked in a special case of probability independence of attributes. Results were acquired at two different sets of scaling constants, which led to different ranking.

d

takes any possible combination xi , are

f1 (x | x = x1 )   f 2 (xc | xd = xd2 )  c c c d d  fi (x | x = xi ) , for x ∈ Ω .  c d d  fn (x | x = xn )  Thus, the density of obtaining x is c

d

d

n

∑ P(xd = xdi ) fi (xc | xd = xdi ) =

f (x) = f (xc , xd ) =

i =1

n c d d sing ∑ Pi fi ( x | x = xi ), for x ∈ Ω . =  i =1  sing 0 for Ω x ∉ ,  Provided the utility function u(.) over the prizes is assessed, the expected utility may be calculated as follows:

ACKNOWLEDGEMENTS The authors would like to thank Mihail Mihajlov, Filip Filipov and Iliyana Nedeva for their precious support in the collection of information, used in this study.

∫

E(u/p)=

x∈Ω

n

∑ Pi ∫ i =1 x∈Ω

REFERENCES

n

sing

u( x )∑ Pi fi ( x | x = xi )dx = c

d

d

c

i =1

c

d

d

c

d

d

c

u( x , x = xi ) fi ( x | x = xi )dx . sing

If an additive form for the utility function may be constructed, of

Clemen, R. (1996). Making Hard Decisions: an Introduction to Decision Analysis, Second Edition, Duxbury Press, Wadsworth Publishing Company. Hertz, D. and H. Thomas, (1983). Risk Analysis and its Applications, New York, John Wiley. Keeney, R. L. and H. Raiffa, (1992). Decisions with Multiple Objectives: Preference and Value Tradeoffs, Cambridge University Press. McCord, M. and R. De Neufville, (1986). Lottery Equivalents’: Reduction of the Certainty Effect Problem in Utility Assessment, Management Science, 32, 56–60. Tenekedjiev, K. (2004a). Decision Making Problems and Their Place Among Operations Research, Automatics and Informatics, XXXVIII(1), 6-9. Tenekedjiev, K. (2004b). Quantitative Decision Analysis – Utility Theory and Subjective Statistics, Marin Drinov Academic Publishing House, Sofia, Bulgaria. Tenekedjiev, K., D. Dimitrakiev, and N.D. Nikolova (2002). Building Frequentist Distributions of Continuous Random Variables, Machine Mechanics, 47, 164-168. Tenekedjiev, K., N.D. Nikolova, and D. Dimitrakiev (2004). Quantile-Approximated Distribution Toolbox For Technical Diagnostics And Reliability Applications, Annual Proceeding of Technical University in Varna, Varna, Bulgaria, 233-244. Tenekedjiev, K., N.D. Nikolova and R. Pfliegl (2006). Utility Elicitation with the Uncertain Equivalence Method, Proceedings of the Bulgarian Academy of Sciences (Comptes Rendus), Book 3, 59, 283-288. Von Neumann, J. and O. Morgenstern (1947) Theory of Games and Economic Behavior, Second Edition, Princeton University Press

the kind u( x ) = α u( xc ) + β u( xd ) , then n

E(u/p)= ∑ Pi i =1

∫

x ∈Ω c

c

α u( xc ) + β u( xd ) f ( xc | xd = xd )dxc = i  i i 

c c d d c α u( x ) fi ( x | x = x i )dx +   c∫ c  x ∈Ω = = ∑ Pi  d c d d c fi ( x | x = xi )dx  i =1  + β u( x i ) ∫   c c x ∈Ω  

n

n

d d d c = ∑ Pi α E ( u / p,x = x i ) + β u( x i ) =   i =1 n

α ∑ Pi E c ( u / p,xd = xdi ) + β E d ( u / p ) i =1

n

n

= α ∑ Pi E ( u / p, x = xi ) + β ∑ Pu( xi ) . i d

c

d

i =1

d

i =1

If independence between the continuous and the discrete attributes exists, i.e.

c

d

d

c

fi ( x | x = x i ) = f ( x )

for i=1,2,…,n, then the

calculation of expected utility transforms into E(u/p) = n   = ∑ Pi α ∫ u( xc ) f ( xc )dxc + β u( xdi ) ∫ f ( xc )dxc  =  xc ∈Ω c  i =1 xc ∈Ω c n

∑ Pi α E c ( u / p ) + β u( xdi ) = i =1

n

n

i =1

i =1

d α E c ( u / p )∑ Pi + β ∑ Pu( xi ) . i

n

Since ∑ Pi = 1 => E(u/p)= α E c ( u / p ) + β E d ( u / p ) . i =1

APPENDIX 1 The prizes (consequences) within the problem may be represented d

d

d

d

c

d

where xi is the i-th combination of the discrete attributes’ values.

d

k =0.47, k =[0.11; 0.03; 0.1; 0.12; 0.05; 0.08; 0.04] 0.3965 0.1154 0.5119 3 a1 0.0908 0.3198 0.4105 5 a2 0.3825 0.3213 0.7038 2 a3 0.3810 0.3362 0.7173 1 a4 0.1021 0.3900 0.4921 4 a5 c

d

vector of the discrete attributes in x , and x ∈ {x1 ; x 2 ; …;xn } ,

Ranking

c

d

If n=1, then E(u/p)= α E ( u / p ) + β u( x1 ) .

c

as the vector x = (x ; x ) , where x is the m–dimensional vector

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