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Procedia Computer Science 102 (2016) 168 - 175

12th International Conference on Application of Fuzzy Systems and Soft Computing, ICAFS

2016, 29-30 August 2016, Vienna, Austria

Decision making on oil extraction under z-information

Lala M.Zeinalovaa, M.A.Mammadovab

aDepartment of Computer Engineering, Azerbaijan State Oil and Industry University,Baku, Azadlyg ave. 20, AZ1010, Azerbaijan , bDepartment of Engineering and Computer Graphics, Azerbaijan State Oil and Industry University,Baku, Azadlyg ave. 20, AZ1010, Azerbaijan

Abstract

In modern conditions, the refining process is complicated and ambiguous, requiring a precise knowledge of all the internal and external factors. However, in many cases, it is impossible to get complete information. Therefore, the process of oil production takes place in conditions of uncertainty accompanying the various situations. A partial absence of beliefs and fuzziness are some of the aspects of uncertainty. In this paper we consider a somewhat different framework for representing our knowledge. Zadeh suggested a Z-number notion, based on a reliability of the given information. In this study we apply Z- information to decision making on oil extraction problem and suggest the framework for decision making on a base of Z-numbers. The method associates with the construction of a non-additive measure as a lower prevision and uses this capacity in Choquet integral for constructing a utility function.

© 2016Publishedby ElsevierB.V. Thisisanopenaccess article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Organizing Committee of ICAFS 2016

Keywords: Behavioral modeling, decision making, combined states, low prevision, Choquet inteqral

1. Introduction

Decision making almost always takes place under uncertain information. The uncertainty may be expressed as incomplete information on the alternatives, utility value and states of nature, and in the lack of confidence in the knowledge of experts. As it is known the basics of decision theory were developed by von Neumann and Morgenstern1. Later Savage introduced the concept of subjective probability and subjective expected utility2.

Subjective probability is determined by a survey of expert or group of experts. Expected Utility Theory states that the decision maker chooses between risky or uncertain prospects by comparing their expected utility values, i.e. the weighted sums obtained by adding the utility values of outcomes multiplied by their respective probabilities3. In most of real-world cases it becomes impossible to determine the values of objective probabilities4. It is more plausible to determine the values of subjective probabilities, reflecting the beliefs of a decision maker. Imprecision and uncertainty may be in the aspects of measurement, probability, or descriptions5. The theory of fuzzy sets by

1877-0509 © 2016 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Organizing Committee of ICAFS 2016 doi:10.1016/j.procs.2016.09.385

Zadeh is very effective for the mathematical formalization of uncertainty, allowing to make the necessary decision in a given situation on the basis of expert assessments of subjective probability values. Imprecision in description is the ambiguity, vagueness, qualitativeness, or subjectivity in natural language5. In all existing classical decision making theories the probability measures are regarded to be described in a precise manner, which, in many real-world cases could be impossible to achieve. There are a lot of approaches for describing imprecision of probability relevant information. One of the approaches is the use of hierarchical imprecise models. These models capture the second-order uncertainty inherent in real problems. According this approach an expert opinion on probability assessments is usually imprecise6,7,8. The method, proposed in7 uses a Choquet integral for determination the values of utility functions for further comparing the preferences among acts. The major advantage of the Choquet integral is the use of fuzzy measure9 for an estimation of relation between the different states of nature. In7 an imprecise hierarchical decision-making model has the first and the second levels described by interval probabilities. In8, where a hierarchical uncertainty model which exhibits imprecision at its second level in sense of the use of lower probabilities at this level is represented. However one should mention that this model doesn't deal with probability distribution (multiple priors) which are more general description of incomplete probability relevant information. In10 Zadeh introduced a concept of Z-numbers to describe information which is uncertain, incomplete or partially truth.

A Z-number is a pair of fuzzy numbers (A, R) . Here A is a fuzzy value of some variable and R is a fuzzy

reliability or a fuzzy probability for this value11. In12 author shows how to use Z-number based information for decision making. In this case Z-information is given in terms in of a Dempster-Shafer belief structure and in terms of type-2 fuzzy sets. In13 authors considered multi-criteria decision making problem under Z-information. They don't use any operations over the Z-numbers using extension principle but suggest easier method, converting the Z-numbers to classical fuzzy numbers and determining a weight for each alternative. Thus, two approaches are used for decision making with Z-information. The first approach is based on reducing of Z-numbers to classical fuzzy numbers, and generalization of expected utility approach and use of Choquet integral with an integrant represented by Z-numbers. The second approach is based on direct computation with Z-numbers14. To illustrate a validity of suggested approaches to decision making with Z-information an example of decision making on oil extraction at a potentially oil-bearing region is used. In this paper we suggest a generalization of Choquet integral for environment described by Z-valuation of the uncertain information.

2. Preliminaries

Definition 1. A Z-number10. A Z-number is an ordered pair of fuzzy numbers, (A, R) . A -is a fuzzy restriction

on the values which a real-valued uncertain variable is allowed to take. R is a measure of reliability of the first component.

Definition 2. Choquet integral16'21. Let (j> :Q.^ R be a measurable real-valued function on Q and r] : F —> [0,1] be a non-additive measure defined over F .The Choquet integral of ^ with respect to r/ is defined as

= ± L (Bw) - V (B{i_1)) • ) (la)

where index (i)implies that elements a>i eO, i =1,...,n are permuted such that <j>[co^j ^ ^¿y^j , = 0

and B^ = j^,...,®^! c Q.

A value of fuzzy utility function for an action is determined as a fuzzy number-valued Choquet integral

(fi (B, (B, fa,)

(i) means that utilities are ranked such that ¡¡> ^¿o^ j > ... >(j> (®(n1) j , <j> (®(n+1)) ~ 0 .

Definition 3. Fuzzy measure16'22. Let En be a space of all fuzzy subsets of Rn. These subsets satisfy the conditions of normality, convexity, and are upper semicontinuous with compact support. Let V,W e En. A fuzzy number-valued fuzzy measure on F is a fuzzy number-valued fuzzy set function ij: F with the properties: (1) fj (0) = 0; (2) if V c. WW then fj {v)<ij {WW); (3) if V c V2 c ... ,Vn c ... e F V c V2 c ..., Vn c ... eF, then

* (U If. ) = n&i V);

(4) if V 3 V2 3 ..., Vn c ..., Vn e F , and there exists n0 such that fj(W^ j # co , then fj^^JVn j = lim77{Wn^ .

Definition 5. Lower prevision23'28. A coherent lower prevision is defined as a lower expectation functional from the set of gambles to the real numbers that satisfies some rationality criteria. This function is conjugate to another that is called a coherent upper pre vision. When a coherent lower prevision coincides with its conjugate coherent upper prevision we call it a linear prevision. An unconditional lower prevision P (X) is coherent if and only if it is the lower envelope of dominating linear previsions. If the lower prevision P is represented as the lower envelope of a closed convex set P of linear previsions then

P = min {P (X): X c S} (2)

Lower prevision P is characterized by probability density function of each linear prevision in extreme points29. In particular case, when linear prevision is a probability measure the lower prevision is the lower envelope of multiple priors. In this work we use lower prevision as non-additive measure. So we can define P as rj.

3. Problem statement

Below we formulate in general a problem of decision making under Z-information. Let we have a set of states of the nature s1,s2,...,sm e S. As decision maker usually is uncertain about first-order imprecise probabilities, we

describe the prior probabilities of states of nature Zp^ |Zp^ j = |Aj,R jj ,withanumber of possible utilities

Z ^ Z ^ ^ = ^ f. (sj R1 j foraset of acts f1, f2,..., fn as Z-numbers. R1 is a confidence degree for the

value of probability of the state of nature. We have the following events: s1,s2,...,sm e S and b1,b2,...,bn e B . Now we can revise the prior probabilities of the states of nature on the base of given Z-valued conditional

probabilities Z p^b |ZP(b /s^ = |Ap^b /sj,R jj ofpossible combination of these events. According to Bayesian theorem we recount the posterior probabilities of events. The total conditional probabilities are determined as

ZP(b) = ZP(i1) x ZP(bJ s) + ZP (s2 )x ZP(bJ s2) + ••• ZP (Sj ) x ZP(bJ Sj ) + + ZP(s. ) X ZP(bJ sm ) (3)

The conditional posterior probabilities for these events combinations are determined as

ZP(s )x ZPlk /. ) / ZP(b, ) (4)

Z posterior _

p U, b) _

Formally the problem is formulated as follows. Decision-making under uncertainty can be considered as 4-tuple (S,ZX,A,y) ,where S - {s1,s2,...,sm} - a space of mutually exclusive and exhaustive states of nature, ZX - a set

of outcomes, described by Z-valuation. A is the set of actions that are functions f : S ^ ZX , y is the non-additive preference relation on the set of actions. In decision-making under uncertainty, a probability over S is imprecise. FS is a (7 - algebra of subsets B of S . Denote by A0 the set of all FS - measurable step-valued functions from S to X and denote A the constant actions in A0 . Let A be a convex subset of Xs which includes A . X can be

considered as a subset of some linear space, and X S can then be considered as a subspace of the linear space of all functions from S to the first linear space. The problem is to determine preferences among alternatives by means of

a utility function. The Choquet expected utility function used here is as follows: ^ =|^dZ'n . The

decision making problem in this case consists in the determination of an optimal action f * e A such that

Z= max<iZssdZ'> . u(f) IJ "UW) nJ

4. A solution to the problem

To determinate an unknown probability of state s. - Z , > on a base of given probabilities

Zp(s),ZFjs),...,Zj,...,Zr^s ^ or unknown conditional probabilities Zp^ /s^, we use the method suggested in16.

According to Bayesian theorem we have to recount the posterior probabilities of events and determine the total conditional probabilities Z'^ ^ and conditional posterior probabilities for the given events combinations Zp^'™)

according to (3) and (4). We apply an approach based on direct computation with Z-numbers14. Let consider the several aspects of arithmetic operations on discrete Z-numbers30. As it was mentioned above we consider the values of outcomes and probabilities represented as Z-numbers Z (A, R), where A and R are fuzzy numbers with

trapezoidal and triangle membership functions. Let we have two Z-numbers: Z (Ax,Rx) and Z (Ay,Ry). The first

part A is computed as Az = Ax * Ay , where * is any arithmetic operation. Using the convolutions where pR and

pR are the distributions with trapezoidal membership functions we determine the membership function for the

reliability part B:

For sum of Z-numbers (Ax, Bx) and (A„, B„): pR^ +Rjr (v) = J pRx (u (v - u )du,

Vz (pz ) = suP[Vbx [\Vax (u)Px (u)du(JX (u)Py (u)du^ Vbz (w) = sup(MrZ (Pz )) s.t.w.(u)pZ (u)du . For subtraction of Z-numbers (Ax,Bx) and (Av,B„):

Prx-rt (v) = JPrx (u)Prt (v + u)du, Vpz (pz ) = sup[Vbx [\VAx (u) Fx (u)du)^Mby [\Vay (u) Py (u)du^ ju^ (w) = sup(^p (pZ)) s.t.w. (u)pZ (u}du . For multiplication of Z-numbers (Ax,Bx) and

(av,B ) : Prx-ry (v) = JPrx (u)Prt (v/uMpz (Pz ) = sup[vbx [\Vax (u)Px (u)du)aVby [\^ay (u)Py (u

(w) =sup(^p (pZ)) s.t.w. = |(u)pZ (u}du . For division of Z-numbers (Ax,Bx) and

(•Ay , By ) : Prx / ry (v) = J Prx (u )Pry (v •u )du, Vpz ( Pz ) = sup [vbx [\Vax (u ) Px (u )du )^Mby [\Vay (u ) Py (u )du)],

Mbz ( w )= sup (MrZ ( Pz )) Stw=\VAz (u ) Pz (u )du

Given the payoff table and the complete probability distribution we can evaluate the values of Choquet integral on base of (1a,1b) and construct lower prevision, that is defined16 as

Z'n~(H) - U«'[Z;~ a(H),z;~ a(H)], H c S - {sp...,sm} (5)

a e (0,1]

P --mn Pl'ft P right

where ) = mf P {sj ),-, P {sm )e PaJ , Pa = j(p (*),..., p (s„)) e P«x...x Pam ¿p (Sj ) = lj Here

Pla,...,P.m are a-cuts of fuzzy probabilities P1;...,Pm , p(s1 ),...,p(sm) are basic probabilities for P^...,Pm , x

denotes the Cartesian product. Now we can construct a fuzzy measure with trapezoidal membership function from fuzzy set of possible probability distributions as its lower probability function (lower prevision) taking into consideration (5) and the method used in15. Now we obtain the fuzzy values of utility function U(f (s)) for each

alternative by (1b), where an optimal action f * e A is obtained in accordance with (6):

Zt>u )=i ^uw)^ 6

5. An application to a problem of decision making on oil extraction at a potentially oil-bearing region

We consider a problem of decision making on oil extraction at a potentially oil-bearing region where the outcomes of the decisions depend on two systems of statistically depended events. Assume that a manager of an oil-extracting company needs to make a decision on oil extraction at a potentially oil-bearing region. The manager formulates his/her knowledge about oil occurrence in natural language by using of Z-information. The geologic investigations show that a prior probability of oil occurrence is estimated as "less than medium" with a confidence "likely". The manager can make a decision using this information or using the seismic investigation results. The seismic investigations confirm an oil occurrence with probability (high, quite sure) and an oil absence with probability (below than high, quite sure). Thus the manager has the set of alternative actions to choose from. The goal is to find the optimal action. Let us now give a general formal description of the problem. The set of states of the nature is S = {s1,s2} , where s1 denotes "oil's occurrence" and s2 denotes "oil's absence". The seismic investigation results give an opportunity to estimate the probabilities of the states of nature s1 and s2 , but these results are not accurate and the manager beliefs them with some confidence basing on his own experience. Thus, the probabilities of the states of nature are estimated by the manager as Z-number values ZP^s ^ = RR j . Let the

first and the second components of the probabilities ZP^ ^ AR j of the states of nature si e S where

Ri1 "j^,(x)):x e[0,1]j be represented by trapezoidal and triangle fuzzy numbers: ZP^^ = "lower than

medium, likely"=((0.2, 0.3, 0.4, 0.5;1), (0.7, 0.8, 0.9;1)). The probability of the state S2 is unknown. So in our problem we have two types of events: geological events (states of the nature) - "oil's occurrence" (s1) and "oil's

absence" (s2) and two seismic events (results of seismic investigation) - "seismic investigation shows oil

occurrence" (b1) and "seismic investigation shows oil absence" (b2). The event b1 corresponds to real confirmation

that seismic investigation shows oil occurrence and to wrong conclusion about oil occurrence while its factual absence. By analogy, the event b2 corresponds to real confirmation that seismic investigation shows oil absence.

Below we list possible combinations of geological and seismic events: 1 1 - there is indeed oil and seismic investigation confirms its occurrence, ZP^/s ^ - "high, quite sure" = ((0.7,0.75,0.8, 0.85;1),(0.8,0.9, 1.0;1)); b2 / s1 -

there is indeed oil but seismic investigation shows its absence, Z^ /s ^ is unknown; b1 / s2 - there is indeed no oil

but seismic investigation shows its occurrence, ZP^,s ^ = is unknown; b2 / s2- there is indeed no oil and seismic

investigation confirms its absence, ZP^b ts ^ - "below than high, quite sure"= ((0.6,0.65, 0.7, 0.75; 1), (0.8, 0.9,

1.0; 1)) . We obtain unknown prior probability Zp(s2) = ((0.5, 0.6, 0.7, 0.8;1),(0.7, 0.8, 0.9;1)),conditional

probabilities Zp{h ^} = ((0.15,0.2,0.25,0.3;1), (0.8,0.9,1.0;1)) and Zp{h} = ((0.25,0.3,0.35,0.4;1),) ((0.8,0.9,1.0;1)). The use of seismic investigation will allow the manager to update the prior knowledge about

actual state of the nature with the purpose to obtain more credible information. This means that given a result of seismic investigation, manager can revise prior probabilities of the states of the nature on the base of linguistic

probabilities Z , >,i -1,2, j = 1,2 of possible combinations of geological and seismic events. According to

F(bj/-v

Bayesian theorem we can recount the posterior probabilities of events. The total conditional probability of the seismic event "Oil occurrence" is determined as Z'F(b]} = ZF( } x ZF{^/ ) + ZF( ) x ZF(^/s ) = ((0.28, 0.40, 0.56, 0.72;1), (0.72, 0.81, 0.9;1)). By analogy, the total conditional probability of the seismic event "Oil absence" is determined as Z^} = Z^) x Zp(hi^) + ^ x ^ = ((0.34,0.45,0.57,0.72;1),(0.72,0.81,0.9;1)).The conditional posterior probabilities for these events combinations are determined as

7 posterior _I 7 \y 7 I / 7f 7 Posteri°r _I 7 n/7 \ / 7' 7 posterior _I 7 w 7 I / 7f

ZF(Vb, ^LZP(-1 )X ZF(\/sx )J ' ZF(bx)' ZP(-1/b2 ^LZP(-1 )X Z F^/sx )J 1 Z P(b2) ; ^/b, ) " ^(-2 ) ^/-2) J ' Z F(\)'

ZP^T™) ~ [ZF(s) x ZF^b /s ^ / Z p(-b y Assume that the manager evaluate utilities for various actions taken at various

states of the nature from some scale. Because of incomplete and uncertain information about possible values of profit from oil sale and possible costs for seismic investigation and drilling of a well, the manager would linguistically evaluate utilities for various actions taken at various states of the nature. For outcomes estimation we consider the following Z-number values for profit and loss: -the seismic investigation of an area costs Zu -

"low, sure" =$((917911.8, 927911.8, 936911.8; 1),(0.7,0.8,0.9;1)); -well-drilling costs Z^ = "medium, sure" =$((3701647.4, 3711647.4, 3721647.4;1),(0.7, 0.8, 0.9;1)); -in case of indeed oil occurrence the profit from its selling excluding the expenses will be equal to Z - "high, sure"

=$((18548237.1,18578237.1,18598237.1;1),(0.7,0.8,0.9;1)). Now we can construct the decision tree for oil extraction taking into account the possible outcomes. A decision about seismic investigation of the area must be done before the decision about oil extraction. Thus a node of seismic investigation will be on left and must have the two branches "Yes" and "No", corresponding to the decision about prior seismic investigation or refuse from it. If we will have the decision about refuse from it then we have to make a decision about oil extraction. Thus we have 11 possible outcomes. Now taking into account opportunities of the manager, we will consider the following set of the manager's possible actions: A ={f1, f2, f3, f4, f5, f6} , where the description of the manager actions fk, k - 1,6 is the following: for f1 - conduct seismic investigation and extract oil if seismic investigation shows its occurrence (outcomes (1) and (2)), for f2 - conduct seismic investigation and do not extract oil if seismic investigation shows its occurrence (outcomes (3) and (4)), for f - conduct seismic investigation and extract oil if seismic investigation shows its absence (outcomes (5) and (6)), for f4 - conduct seismic investigation and do not extract oil if seismic

investigation shows its absence (outcomes (7) and (8)), for f5 - extract oil without seismic investigation (outcomes f

(9) and (10)), for j6 - abandon seismic investigation and oil extraction (outcome (11)). Assume that the manager evaluate utilities for various actions taken at various states of the nature. We consider the next expenditures and profits for outcomes estimation: =((9179.86,9279.86, 9369.86;1),(0.7,0.8,0.9;^),

Z = ((37016.44,37116.44,37216.44;1), (0.7,0.8,0.9;1)), Z = ((18548237.19,

uwell—drilling uprofit

18578237.19,18598237.19;1),(0.7,0.8,0.9;1)). Thus the outcomes values for alternatives are the following: (Outcome (1)): the profit from oil selling - the expenditures on seismic investigation - the expenditures on welldrilling Z= Z - Z - Z = ((13928677.89, 13938677.89, 13939677.89), (0.7,0.8,0.9;1)); Outcome

° u1 uprofir useismic uwell-drillmg 7 V "

(2)): the profit from oil selling - the expenditures on seismic investigation - the expenditures on well-drilling Z = 0 - Z - Z = - ((4619559.3,4639559.3, 4658559.3), (0.7,0.8,0.9;1)); (Outcome (3)): the loss from

u2 useismic uwell -drilling

oil non-selling - the expenditures on seismic investigation Z^ = —Zu -Zu =- ((194661.05,195061.05, 195351.05), (0.7,0.8,0.9;1)); (Outcome (4)): the expenditures on seismic investigation Zu^ = ~ZUi_ =-((91791.8, 92791.8, 93691.8;1),(0.7,0.8,0.9;1)); (Outcome (5)): the profit from oil selling - the expenditures on seismic investigation - the expenditures on well-drilling Z - Z - Z - Z -

^ 1 ^ u5 uprofir useismic uwell -drilling

((13928677.8,13938677.8,13939677.8),(0.7,0.8,0.9;1)); (Outcome (6)): the profit from oil selling - the expenditures on seismic investigation - the expenditures on well-drilling Z = 0 - Z - Z = - ((46195.3,46395.3,

° ^ ° u6 useismic uwell-drilling VV ' '

46585.3), (0.7, 0.8, 0.9;1)); (Outcome (7)): the loss from oil non-selling - the expenditures on seismic investigation Z^ =-Z -Z-us_ = - ((19466149.05, 19506149.05, 19535149.05), (0.7, 0.8, 0.9;1)); (Outcome (8)): the

expenditures on seismic investigation Z^ = ~Zu^mc = - ((917911.86, 927911.86, 937911.86;1),(0.7, 0.8, 0.9;1)); (Outcome (9)): the profit from oil selling - the expenditures on well-drilling

Z = Z - Z = ((14846589.7,14866589.7, 14876589.7), (0.7,0.8,0.9;1)); (Outcome (10)): - the

u9 uprofir uwell -drilling VV ' ' V V//

expenditures on well-drilling Z^ =-((3701647.4,3711647.4, 3721647.4;1), (0.7,0.8,0.9;1)); (Outcome

(11)): Z - 0 . Below we give the representation of utilities for actions made at the states by Z-numbers defined on the scale [0,1]: Z^ = ((0.9724,0.9727,0.9727;1),(0.7,0.8,0.9;1)); Z^ = ((0.4323,0.4328,0.4334;1),(0.7,0.8,0.9;1)); Zun = ((0.0,0.0008,0.002,0.003;1),(0.7,0.8,0.9;1)); Z.^ = ((0.5404,0.5407,0.541;1),(0.7,0.8,0.9;1));

Z^ = ((0.9724,0.9727,0.9727;1),(0.7,0.8,0.9;1)); Z.^ = ((0.4323,0.4328, 0.4334;1),(0.7,0.8,0.9;1));

Z^ = ((0.0,0.0008,0.002,0.003;1),(0.7,0.8,0.9;1)); Z^ = ((0.5404,0.5407,0.541;1),(0.7, 0.8,0.9;1));

Z^ = ((0.9991,0.9997,1.0;1),(0.7,0.8,0.9;1)); Z^ = ((0.4595,0.4598,0.4601;1),(0.7,0.8,0.9;1)); Z.^ = 0. To find the optimal action on the base of the methodology we propose at first it is needed to calculate for each action fi its Z-valued utility function as a Choquet integral ZUf) - J Zu(/(sj)dZn , where Zn is a Z-valued measure. Let us mention that depending on actions a Z-valued measure will be constructed on the base of either prior or revised probabilities. For actions f1, f2 the Z-valued measure will be constructed on the base of Z^^''T) and Zp^"™) and

for actions f3, f4 the Z-valued measure will be constructed on the base of Zp^'b") and Zp™'"™ (because seismic

investigation is used). For action f5 the Z-valued measure will be constructed on the base of prior probability

ZPS j. For action f6 its utility, i.e. Choquet integral, is obviously equal to zero. The Z-valued measures Z^ ^ and

Z^B) defined on the base of Zp, Zp^'J0 and Zp^J™, Zp°s^, respectively, and also the Z-valued measure

Z^B) defined on the base of prior probability Zp^s ^ are the triangle Z-numbers. As outcomes Zu are Z-numbers,

the corresponding values of Choquet integrals will be also Z-numbers. We will calculate a Z-number-valued utility function for every action fk. The form of a Choquet integral for action f will be:

As ZUn =((139286.8,139386.8,139396.8),(0.7,0.8,0.9;1)), Z.^ =-((4619559.3,4639559.3,4658559.3),(0.7,0.8,0.9; 1)), using ranking we find that Z < Z . Then Z , . ,■= Z , Z , , ^ = Z and s,,, = s., s/n, = s, . So,

u12 u11 u( f (sW)) ' uf (s(2))) u11 W 2' (2) 1

Zu f J = Zu^ + {ZUn - ZUn )z, ({sj). We determined the action with the highest Z-number utility value applying Jaccard compatibility-based ranking method. The best action is f2 as one with the highest utility value.

6. Conclusion

In this study, we have considered a problem of decision making under Z-valued information represented by Z-number which induces a possibility distribution over probability distributions associated with decision variables. We developed method of decision making which associates with the construction of a non-additive measure as a lower prevision and uses this capacity in Choquet integral for constructing a utility function in Z-valuation environment. Computation with Z-information is based on direct arithmetic over Z-numbers. We applied the suggested theory and methodology to solving a real world problem of oil extraction at a potentially oil-bearing region.

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