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Decision-making in chemical engineering and expert systems: application of the analytic hierarchy process to reactor selection
009%1354/92
Computersthem. Engng,Vol. 16, No. 9, pp. 849-860, 1992 Printedin Great Britain.All rightsreserved
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Copyright Q 1992 PergamonPressLtd
DECISION-MAKING IN CHEMICAL ENGINEERING AND EXPERT SYSTEMS: APPLICATION OF THE ANALYTIC HIERARCHY PROCESS TO REACTOR SELECTION P. J. HANRATTY and B. JOSEPH Washington University, St Louis, MO
63130, U.S.A.
(Received 9 May 1991;fina/ version received 17 February 1992; received for publication 5 March 1992) Abstract-Expert systems have been used to model and automate decision making in engineering. Often the problem is cast as a decision tree which is then translated into a set of rules. However, all decision-making problems cannot be cast into this mold. Decisions are usually based on a number of criteria (and possibly subcriteria) which are accepted or established as important to the particular decision. The experience, expertise and accuracy with which the criteria are rated and combined into an overall raring determine the success of the decision. Experts often find it difficult to express their decisions in clear-cut terms. Two experts may not completely agree on the answers to a given problem. The question then is how to capture and represent this uncertainty in decision-making? Current implementations of expert systems employ probability theory, certainty factors and fuzzy set theory. These methods are often found inadequate or unsuitable for the problem leading to less than satisfactory implementation and performance. Within the social science and business fields, a method called the analytical hierarchy process (AHP) has been utilized on a number of different problems. AHP offers a mathematical methodology based on pairwise comparisons which is well-suited for a variety of different problems in both the chemical engineering domain and the expert system domain and it is easily understood by the experts. The AHP structure is applicable to the chemical engineering domain and it is easily implemented in an expert system. This paper will present the AHP methodology and illustrate its use for the engineeringproblem of
chemicallaboratory reactor selection.Also, the resultsof the implementationof AHP into an expert system for laboratory reactor selectionwill be discussed. INTRODUCTION
Decison-making
is part of everyday
life. Exactly how
one makes decisions is difficult to quantify and tranlsate into mathematical algorithms. A variety of tangible and intangible costs and benefits are evaluated by the brain in arriving at a choice. In building expert systems that imitate the decision-making process of experts, it is necessary to somehow capture the essential ingredients of the experts’ decision-making process. The objective of this paper is to address this problem from the perspective of a process engineer faced with multiple choices in a design environment. For example, consider the problem faced by a chemical engineer in deciding what type of laboratory reactor he should use for a given application. The engineer needs to chose the laboratory reactor which will get the best possible data, in the least possible time, at a reasonable cost and with the easiest possible experiments and analysis. Usually the best reactor choice for a given situation is a compromise of all these considerations. We shall use the above paradigm in this paper to investigate the decision-making process and its automation into an expert system. One common method used in decision-making is to establish a number of criteria (and possibly subcri-
teria) upon which to base the decision_ Each criterion (and subcriterion) is rated and based upon these ratings the final decision is made. Making decisions based on established sets of criteria has a number of advantages; it forces one to establish what is important and what is not; it allows one to focus on smaller more tractable problem domains (one criterion at a time); and it seems to model well how decisions are actually made. If one proceeds to utilize the above structure to establish a model for the decision-making process, a few problems are quickly encountered. First, how are the different criteria rated? After each of the criteria are rated, how can we merge all the different criteria into one rating? How is consistency among the many choices maintained? etc. Three methods commonly used by expert systems in the engineering domain to atack these problems are probability theory (and variations thereof), fuzzy logic and utility functions. These methods have enjoyed some success in implementation on different problems. However, many problems are not well modeled by any of these methodologies. One method that has been overlooked in both the engineering field and in expert systems is the Analytic Hierarchy Process, AHP (Saaty, 1980). This method has been
P. J. HANFU-I-IY and B. Jos~~ir
850
used widely in the social science and business fields. AHP is a methodology based on relative instead of absolute ratings. Utilizing pairwise comparisons of the many options, AHP gives a structure and a mathematical base upon which many problem domains can be modeled. The methodology is applicable to a large range of engineering problems and it is easily implemented into an expert system. The objective of this article is to present the AHP methodology, illustrate its use for the engineering problem of chemical reactor selection, and discuss its implementation into an expert system for laboratory reactor selection. AHP
METHODOLOGY
AHP is a technique developed by Saaty (1980, 1987) which aids in the decision-making analysis.
AHP has been applied to a wide variety of areas from the ranking of countries according to social and political factors (Peniwati and Hsiao, 1987) to the selecting of a microcomputer (Arbel and Seidmann, 1984). Zahedi (1986) gives a listing of the various applications of AHP. In the following section, the AHP methodology will be outlined. For more details on the method see Saaty (1980). The AHP technique involves a four-step procedure (Johnson, 1980; Zahedi, 1986): Set up a decision hierarchy by breaking down the decision problem into a hierarchy of interrelated elements. Collect the input data of pairwise comparisons of the decision elements. Use the eigenvalue method to estimate the relative weights of the decision elements.
Level 1
Level 2
DdSiill AttrIbute 1
Level k
Fig. 1. General decision hierarchy for AHP.
Decision-makingin chemicalengineering Table Relative
importance
: 5 7 9 2,4 6, g Reciprocals of above numbers
1. Scak
of relative
851
importance
D&n&ion Equal importancc Moderate importance of one over another Ekential or strong impoltancc Demonstrated importance Extreme importance Intermediate values between two adjacent judgements If an activity has one of the above numbers assigned to it when compared to a second then the second activity has the reciprocal when compared to the first
4. Aggregate the relative weights of the decision elements to arrive at a set of ratings for the decision alternatives (or outcomes). In Step 1, the decision hierarchy for the process is developed. Here the decision problem is broken down into a hierarchy of related decision elements. The decision hierarchy can be viewed as a type of tree diagram with each node of the tree representing a decision element and each connection representing the corresponding weight. At the top of the hierarchy is the objective of the decison-making process. The level below the top level contains the criteria (attributes) upon which the top level decision is based. The levels below this add details to the decision elements in level above it. The final level contains the decision alternatives. Figure 1 illustrates a general hierarchy format. Establishing a hierarchy which represents the decision-making process well is crucial to accurate decision-making. The complexity of the problem domain and the detail of the problem modeling will dictate the number of levels in the hierarchy and the number of decision elements per level. The next section of the paper illustrates the decision hierarchy for laboratory reactor selection. In Step 2 of the AHP process, the pairwise comparisons of the decision elements are established in a matrix A. Here each decision element is pairwise compared against every decision element in its level for its relative importance to achieving the objective(s) of the level above it. For instance for level two of Fig. 1, attribute 1 is compared against attribute 2 then attribute 3, then . . . attribute n for achieving the objective listed in the level above it. If an attribute is considered to be more important than the other it is given a value of l-9 depending upon how much more important it is than the other element. Table 1 gives a listing of the suggested definition scale for entering the judgcment values for the pairwise comparisons. Take al to represent the pair-wise comparison of element i to element j, and A to represent the nWbiX of the pairwise comparisons. For AHP it can be argued by reasons of consistency that if element i is 5 times more important than element j that element j must be l/5 as important as element i. Putting this
in more general terms, the reciprocal axiom can be written (Saaty, 1986): aii= T/a/,.
(1)
Also, it is obvious if element i is compared against itself that each element has equal importance so that it can be written that: a,, = 1.
(2)
So to create the pairwise comparison matrix for n decision elements n(n - 1)/n pairwise comparisons need to be made among the elements. In Step 3 of AHP process, the relative weights of the decision element are estimated from the pairwise comparison matrices developed in Step 2. To estimate the weights of each of the decision elements, it should he remembered that each entry in the decision matrix can be thought of as the estimate of the relative importance or weight wi of one element to another. So that if the input values which are entered are exact relative comparisons of the decision elements the decision (pairwise comparison) matrix could he written: attributes *
1 1
Wl /WI
2 WI/W2
..
n WI IWPI
(3)
A=:
n
W”/W”
W”l%
From the above matrix, the weights could easily be determined from the equation by solving for W: A-W=n.W,
(4)
where W = (wi, w,, . . . , w.)= is the vector of the weights and n is the number of decision elements.
P. J. HANRATIYand B. JOSEPH
852
However, in actual implementations, the values for the relative weights are not known exactly so that the created decision matrix a contains inconsistencies and this is the basis of AHP. It is proposed that the weights be estimated as follows (Saaty, 1980): B.W=&,,-ti,
(5)
where A^ is the created matrix of pairwise comparisons, &,_ is the principal eigenvalue of a and J@ is the vector of the estimated values of the relative weights. For the above proposed method, L,_ may be considered to be an estimate of n from equation (4). It has been shown that &_ is always greater than or equal to n. For a completely consistent matrix &,,_ = n, and the closer &_ is to n, the more consistent are the pairwise comparisons of matrix d (Saaty, 1980). A matrix is said to be consistent if the following is true: aii.aik=aik.
(6)
Saaty (1980) suggested that L_ could be used as a measure of the departure from consistency. Utilizing this argument a consistency index CI and consistency ratio CR have been defined: CI=S,
1
(7)
CR = CI/RI,
(8)
where RI is an average random consistency index of randomly generated matrices of size n x n. The values for RI have been calculated to be as follows (Saaty, 1986):
n
Random consistency index (RI)
12 0 0
3 0.58
A general rule-of-thumb is that CR < 0.1 is considered to be acceptable. For CR > 0.1, it is suggested that the matrix of pairwise comparisons be updated to eliminate inconsistencies. The eigenvalue method is one method used within AHP to estimate the relative weights of the decision elements. Other methods do exist, but the eigenvalue method is the most widely accepted and used. In the final step (Step 4) of the AHP process the weight vectors determined in Step 3 are used to produce a vector of composite weights which serve as the ratings of the decision alternatives. The composite relative weight vector elements at the k th level with
respect to that of the tist level may be computed as follows (Zahedl, 1985, 1986): wc = fi Bi,
(9)
i-2
where WC is a k-dimensional vector of composite weights of elements at level k with respect to level 1, and Bi is the ni_, x ni matrix with rows consisting of estimated I@ vectors and ni represents the number of decision elements at level i. The composite weight vector WC from equation (9) represents the final decision vector (or ratings) of the possible choices. From this vector of weights the decision is made-the largest weight represents the best choice. Alternative
methoa!s
Other methods are available to aid the decisionmaking process-probability theory, certainty factors, fuzzy logic and utility functions. Each of these methods offers a way to combine and account for the different criteria which are essential to the decisionmaking process. Probability theory measures the extent to which one set of propositions, out of logical necessity and apart from human opinion, confirms the truth of another. Proponents generally regard this view as an extension of logic (Ng and Abramson, 1990). The appeal and advantage of probability theory is that it is solidly based in the mathematics of logic. For probability theory P(E) is the probability that event E will occur or that it is true. There is extensive literature on the interpretation and manipulation of the probability of the occurrence of different events and interrelationships among events. Bayes’ rule is commonly used
4 0.90
5 1.12
6 1.24
7 1.32
8 1.41
9 1.45
10 1.49
for implementing probabilistic inference in expert systems (Rolston, 1988). Probability theory (and variations thereof) rcquires that probabilities of truth be assigned to each fact, criteria and the relationships among them. The determination of these probabilities is often quite difficult and sometimes artificial. For instance, for the reactor selection problem, what is the probability that the best reactor for a given situation has a low heat of reaction or average operating cost? The values for these probabilities are not obvious nor is there a methodology available for determining these probabilities. A major assumption in many implemen-
Decision-makingin chemicalengineering tations is that all of the probabilities are conditionally independent (they do not alfect each other). However, it is often not possible to construct a set of conditionally independent relation+for example, the probability of how operating cost affects the reactor choice is not independent of the heat of reaction. Therefore, probability theory suffers from the problem that the probabilities required for the application of Bayes’ rule are hard to acquire (Buchanan, 1984), even for experts. However, there have been some successful applications of probability theory to expert systems, most notably PROSPECTOR (Duda, 1980). AHP offers an easier method for extracting and determining the importance of a fact or an attribute to the decision-making process especially if the decision process is not easily quantified. Another variance of the use of probability theory in expert systems is through certainty factors. Certainty factors give an informal mechanism to quantify the degree to which, based on a set of evidence, a conclusion is to be believed or disbelieved. Certainty factors are widely used for systems that operate with incrementally acquired evidence (Rolston, 1988). See Adams (1984) for a comparison between certainty factors and probability theory. The concept of certainty factors was first developed and used in MYCIN (Buchanan, 1984). Certainty factors is an ad hoc method developed because of the difficulty of acquiring a priori and conditional probabilities required for application of Bayes’ rule. However, it has been shown to be another interpretation of Bayes’ inference rule (Adams, 1984). Fuzzy logic was first proposed by Zadeh (1965) and since it has been implemented and modified by many others, see Maiers and Sherif (1985) for an extensive listing of over 500 applications of fuzzy logic in the literature. Fuzzy logic is an implementation of fuzzy set theory, which is an analogy to set theory where the “edge” of the set is fuzzy (Rolston, 1988). Within the theory, values are assigned a membership from 0 to 1, where 0 means no membership in the set, 1 means membership in the set and 0.5 means that it is equally likely to be in the set or out of the set. For example, the value for the heat of reaction could be defined as 0.7 low, 0.2 medium and 0.1 high. Fuzzy set theory lays out the means by which to relate fuzzy sets (assertions) and to manipulate fuzzy relations. The practical application of fuzzy set theory can be difficult because the coding of relationships and quantification of membership sets and fuzzy relationships becomes a much more intricate and ad hoc process. Utilizing fuzzy mathematics, these values (or sets) can be propagated through the system and assist the
853
decision-making process. However, the addition of fuzzy logic greatly increases the difficulty of implementation and the complexity of the system. It also demands that the expert expresses his choices in precise quantitative terms that most experts are not prepared to do. Utility finctions are based on derived (theoretical, experimentally or ad hoc) equations which represent the utility of given property. Quite often the utility function is based on strictly economic considerations, i.e. cost or return on investment. If a mathematical function which represents the worth (utility) of the desired choices is available, it is perhaps the best method on which to base decisions. Unfortunately, for many problem domains, there does not exist a theoretical function by which to rate the utility of the various choices. Therefore, ad hoc utility functions must be developed. Two possibile utility functions are shown below: rating = 5 b,(r,), i-1
(10)
rating = fi (r,)4, i=L
(11)
where ri is the rating (O-l) of the attribute determined to be important to the selection process (such as the attributes listed in the next section), n is the number of attributes and b, is the weight or the importance of an attribute. The major problems with these functions are that there usually is no theoretical or experimental proof that they are accurate, the weight, bi given to each attribute is very difficult to assign and determine and sometimes the decision-making is based on subjective criteria which cannot be quantified in the utility function. Utility functions require an empirical and/or theoretical equation that rates the performance of each choice. Sometimes other subjective considerations which cannot be quantified easily (i.e. environmental concerns, company relations, etc.) need to be considered in the decision-making process. If an empirical and/or theoretical utility function is not available then ad hoc utility functions can and have been formulated, but this is often unrealistic. Another possible approach to this decision-making process would be to collect enough examples of the decisions made covering all possible choices. Then one could try to construct a decision tree using inductive methods or build a functional approximation using neural networks. Unfortunately, this will work only on very small problems where the number of choices and attributes of the decisionmaking problem are small. Most engineering design problems do not fall into this category. In the
854
P. J. HANRATTY and B. JOSEPH
example problem considered in this paper, the size of the decision tree generated is extremely large. Thus we did not consider using such a scheme. An application study: selection of laboratory
reactors
Laboratory reactors are used to collect the data used for design of processes, improving and diagnosing existing processes, etc. Basically, the data from laboratory reactors is used to improve the performance of existing and future processes. Therefore, the accuracy and reliability of the data (and therefore of the reactor itself) directly affects the performance of the process. Unfortunately, one reactor type will not give the needed data for all reacting systems and applications. Each reactor type has its own set of advantages and disadvantages which change as the reacting system and application change. So for each reacting system and application, the advantages and disadvantages of all the reactor types need to be reevaluated so that the best reactor type can be chosen. The selection of laboratory chemical reactors was explored to illustrate the utilization of AHP for a chemical engineering domain problem. Weekman (1974) suggested that to select a laboratory reactor one first should decide which attributes are important to the reactor performance, rate these attributes and then select the best reactor based upon the attribute ratings. Weekman did not suggest a method on how to combine these attribute ratings into an overall rating with which to compare the reactors. AHP is well-suited for this problem. The example of Weekman (1974) will be used to demonstrate AHP applied to the laboratory selection problem. The example utilized a three-phase system with a highly active catalyst, rapid catalyst decay, highly endothermic reaction and a complex reaction system. For kinetic studies, he suggested that selection of the reactor should be based upon the following attributes: 1. Ease of sampling composition. 2. Isothennality.
and analysis of
3. Accuracy of residence contact time measurement. 4. Selectivity time averaging disguise. 5. Construction difficulty and cost.
product
Weekman rated each of the above attributes for all the reactors as shown in Table 2. Based on these ratings, the best reactor is chosen. To implement AHP for the reactor selection process, a decision hierarchy first must be developed. From the example of Weekman, a decision hierarchy is quite obvious and is illustrated in Fig. 2. After establishing the decision hierarchy (Fig. 2) the pairwise comparisons of the decision elements need to be determined. The relative weights of the decision elements are then estimated by the normalized eigenvalue method [equation (S)] from the matrix of pairwise comparisons. The pairwise comparisons of the second level and its associated weights were determined and calculated first. The painvise wmparisons between the attributes used to choose the best reactor could not be derived from the paper of Weekman, so the comparisons were elicited from three different experts in chemical reaction engineering-two experts in chemical reaction engineering from Washington University and one expert from industry. Table 3 shows the pairwise comparisons of one of the experts. This table represents the matrix of the pairwise comparisons A^with each of the entries classified as av. The weights for each attribute is determined from the normalized principal eigenvector of matrix A, equation (5). The weights of each of the attributes for each of the three experts are shown in Table 4 (note that expert 1 corresponds to the values given in Table 3). The consistency ratio [defined by equation (8)] for the pairwise comparison matrices of the three experts is less than 0.04 for each case, which is well within the suggested guideline for consitency and therefore the pairwise comparison matrices are sufficiently consistent. As shown in Table 4, the weights derived for each of the experts are somewhat different. To determine a compromised (averaged value) value for the attribute weights, two different methods are available:
Table 2. Reactor ratings for a given uample Sampling and analysis
Reactor type Differential Fixed bed Stird batch Stirred contained solids Continuous stirred tank straight-through transpon Recirculating transuort Pulse G-good,
F=fair,
P-poor.
P-F G F G F F-G F-G G
Isothennality F-G P-F z G P-F G F-0
Residencecontact time F z F-G F-G F-G G P
Selectivity disguise-decay P P : G G G FCr
ConstNction problems z G F-G P-F F-G P-F - _ G
Decision-makingin chemical engineering
855
Choose Reactor
h2rllnne 1 ISOthWldity
hmlbme 2
hnrkne
MaesTmrBtef
D_
httdbUe5
4
Renaor
Lbni?&ons
COSI
Reaaor 2:
Rmcaor 3
-18 .
S-0-W
.
.
.
.
TV
-1:
-:
awry:
Batch
BaIdl
DiltarenlU
Fig. 2. Laboratory reactor selectionhierarchy.
one, average the values of the weights determined for each of the experts; or two, take the pairwise comparisons of the experts ag and determine a new pairwise comparison matrix based on a combination of all the experts values utilizing the equation below: h’ - No. a &Slew =
(
of experta
l-l
t-1
LIff
.
au,,urpak
(12)
>
new weight values are then determined from the new matrix of averaged pairwise comparisons. The compromised weight values in Table 4 correspond to the second method discussed above. After the weight vector for level 2 has been determined, the weight vectors for level 3 need to be calculated. The pairwise comparison matrices for each of the reactor attributes were derived directly from Table 2 (Weekman’s rating for each of the reactor attributes). From the pairwise comparison The
Table 3. Attribute Sampling Sampling and analysis Isothemmlity Residence-contact time s&ctivity time average disguise Construction cost and difikulty
I % 3 l/3
matrices, the weight vectors for level 3 were calculated using equation (5). The values for the weights are shown in Table 5. Note that the ratings in Table 5 represent the weights specific to this problem example. A change in the chemical system will change the weights. The consistency ratio RI for the pairwise comparison matrices is less than 0.025 for all five cases which is well within the suggested guideline for consistency indicating that the pairwise comparison matrices are consistent. Now the overall reactor ratings can be calculated using the results in Tables 4 and 5. The overall ratings were calculated using equation (9). The weight vectors and therefore B, from equation (9), are taken from the results shown in Table 4 (level 2) and Table 5 (level 3). The overall reactor ratings (or the composite weight vector Wc) are listed in Table 6.
painvisc comparisons Isathermality 117 1 1 l/5 w
(expert
Contact 119 1 :,3 117
time
1) s&ctivity l/3 5 3 :,5
Construction 3 9 7 5 1
P.
856
J. HANRATIY Table
Expert Sampling Isotbermality Contact time Selectivity Construction
1
0.0579 0.4094 0.3690 0.130 0.034
4.
and
Attribute weights
Expert
2
OF
EXPERT
AHP
INTO
AN
SYSTEM
The AHP methodology has been implemented into an expert system for laboratory reactor selection. The expert system was designed to acquire information about the reacting system from the user and then select the best reactors for this system. AHP was utilized to discriminate between the many different laboratory reactor configurations. As illustrated before, a laboratory reactor is best chosen based on its expected performance on many different critical attributes. In the following the implementation of AHP into an expert system for chemical laboratory reactor selection is discussed. To implement AHP into an expert for laboratory reactor selection a couple of issues must be handled first: (i) what programming environment should be used? and (ii) how should the system and system knowledge be structured and organized? Commercially available “shells” are commonly used to build today’s expert systems. However, for the implementation of AHP, not all of the available shells are suitable. The requirements of the shell are that it has considerable flexibility to allow liberal structuring of the knowledge, and that it allows for linking/incorporating of external programs which conduct the necessary numerical calculations. A commercial shell CxPERT was utilized for this system. The shell is frame and rule-based system which allows for incorporation of C programming directly into the Table Reactor
type
Differential Fixed bed
Stirred batch
Stirred contained solids Continuous stirred tank Straight-through transport
Rccirculatinn
Pulse
tmnstmrt
-
-
Expert
0.0681 0.2805 0.4106 0.1906 0.0501
From the ratings for the reactors shown in Table 6, it can be seen that the recirculating transport reactor has the highest rating for this application followed by the stirred contained solids reactor and the continuous stirred tank. These rankings of the reactors agree with experts’ choices and rankings.
IMPLEMENTATION
5. Level
3
Compromised
0.0664 0.4390 0.1221 0.2220 0.149
weight
0.0642 0.377 0.301 0.181 0.078
code. Therefore, CxPERT gives the needed flexibility for this implementation. To structure the knowledge for the AHP implementation the four-step methodology (structure) of AHP mentioned before is kept in mind. The first step is to establish the decision hierarchy. The decision hierarchy establishes the structure by which the decision will be based, also it sets the structure for the organization of knowledge in the expert system. For the laboratory reactor selection problem, the hierarchy illustrated in Fig. 2 was utilized. The attributes and reactor types utilized are similar to those listed in the example. After the hierarchy is set, the structure of the system needs to be designed and incorporated around the AHP hierarchy while implementing the necessary comparisons. A multilevel multimodule approach was used to organize and structure the system, as illustrated in Fig. 3. The top level of the system maintains the overall system control. The levels below this correspond to the levels of the AHP hierarchy. Each level of the hierarchy is broken down into modules, one module per decision element, i.e. level 3 of Fig. 3 has five modules corresponding to each of the attributes. The AHP decision hierarchy utilized in the system is similar to that illustrated in Fig. 2 using different attributes and reactors. The attributes and reactors utilized for the AHP hierarchy of the system are dependent upon the application for which the reactor will be used. For each different reactor application the system will set up a different hierarchy (attributes and reactors) for the selection process. The goal of the top level of the system is to direct and overview the selection process. The structure of the system is set by the top level when the reactor application (i.e. kinetic studies or catalyst screening) is chosen by the user. Once the structure is set, the
3 weights
for
Sampling
Isothermality
0.0232 0.230 0.0442 0.230 0.0442 0.0988
o.og13 0.0237 0.198 0.198 0.198 0.0237
0.0988 0.230
B. JOSEPH
0.198 0.0813
reactor
selection
Contact time
Selectivity
0.0497 0.0497 0.270 0.115 0.115
0.0277 0.0277 0.0277 0.0277 0.268
0.115 0.270 0.0178
0.268 0.268 0.0851
Construction problems 0.198 0.198 0.198 0.08 13 0.0237
0.0813 0.0237 0.198
Decision-makingin chemicalengineering Table 6. Overall laboratory reactor ratings Reactor
Rating
Differential Fixed bed Stirred batch Stirred contained solids Continuous stirred tank Straight-through transport Recirculating transport PUISC
0.0674 0.0590 0.1789 0.1356 0.1623 0.1047 0.2124’ 0.0815
‘Highest rated reactor
selection process can begin. The lower levels are directed to conduct the pairwise comparisons and calculations needed to implement the AHP process. The lower levels determine the matrices of the estimated weight vectors Bi with which the final reactor ratings can be calculated using equation (9). Based on the reactor ratings, the final reactor recommendations are given (the highest rated reactor is the
Overall
first choice). Note that in addition to overviewing the selection process, Level 1 also overviews the explanation of the selection process. The levels below the top level correspond to the AHP decision hierarchy. Each level determines the matrix of the estimated weight vectors B,. The estimated weight vectors J@ are determined from the pairwise comparison matrix A, utilizing the eigenvalue method [equation (5)]. The pairwise comparison matrix is created based on the knowledge and expertise stored in the system. One weight vector is determined per decision element. Each level is divided into modules with each module corresponding to a decision element. The goal of the module is to determine the estimated weight vector ti associated with decision element. The weight vectors determined by each of the
System Control:
-set structure -0vWview systemoperation -oaldete finalralings(eqn. B) give finalreac(w recommendationa -ovWiewe~
Choose Reactor
Fig.
3.
Schematicof
857
expert
system structure.
P.
858
J.
HABIRA-I-TY and
Knowledge Base:
B.
JOSEPH
Weight Vector.W
Generate Pairwise Comparison Matrix -RUlf?S
-FEUWS
Eigenvalue
-Numerics
method (eqn. 6.5)
System user Fig. 4. Schematicof module in expert system.
the modules in a level make up the matrix of estimated weight vectors B,. The structure of each module is illustrated in Fig. 4. The module has a knowledge base which interacts with the user to determine the pairwise comparison matrix. The pairwise comparison matrix is sent to subroutine which calculates the estimated weight vector I&’by the eigenvalue method. The knowledge base creates the pairwise comparison matrix using a rule-base, a frame structure and some simple calculations. The frame structure is a convenient manner to store, share and pass knowledge about the reactors and the chemical system. The rule base for this implementation is different than traditional expert systems. Insteady of concentrating on one element at a time, each module determines ratings for one element relative to another. Therefore, the rules within the modules are designed to look at the element compared to another so that the pairwise comparisons can be assigned. Calculations are used to support the rule base. This type of implementation requires a good understanding of the strengths and weaknesses of each reactor under a variety of different situations. The knowledge about the reactors is obtained from two sources-literature (textbooks, published papers, etc.) and three experts in multiphase reactors. The final laboratory reactor ratings are calculated from equation (9) using the matrices of estimated weight vectors from each of the levels of the AHP
structure. The ratings determined for the reactors, via AHP, are used to rank the reactors and to suggest which is the probable best reactor for the given chemical reacting system. For this expert system-18 different reactor types are possible options. Note that for this implementation that the system has been limited to multiphase catalytic reaction systems and two application areas (catalyst screening and kinetic studies). The expert system for laboratory reactor selection is currently functioning with the AHP implementation. The system contains approx. 700 rules. The logic and knowledge of the system has been covered and tested in cooperation with a number of experts in multiphase reactors. The recommend& reactor selections of the system have also been shown to agree with the selections of the experts. This indicates that the implementation of AHP to expert systems, the chemical engineering domain and, in particular, to the reactor selection problem is feasible.
DISCUSSION
Capturing the decision-making process of experts with expert systems is usually a difficult task. The decomposition and modularization of the problem is usually an essential first step. Modularization of the problem makes the problem easier to understand, model, code, maintain and update. The difficulty with
Decision-makingin chemicalengineering decomposition of the problem-solving approach is that somehow all of the components need to be combined in the final decision process. Different approaches to this problem are available. Unfortunately, no one approach is applicable to all problem domains. AHP offers a useful approach which appears to be particularly well-suited for modeling the decision making in choosing among design alternatives. The determination of the probabilities (weights) that a given fact or attribute contributes to (or proves) the final decision is often quite difficult. For instance, even though the reaction system can be established to be isothermal for a given reactor it is difficult to quantify the probability that this reactor is the correct reactor choice Experts have a very difficult time giving answers to such questions. However, the relative importance of isothermality as opposed to reactor cost to the selection of the best laboratory reactor can be estimated by the experts. In these cases where it is easier and more natural to deduce relative ratings than absolute ratings, AHP offers a preferable option. Probability theory methods offer some advantages over AHP. Probability methods are widely used within expert systems; therefore, there is a more established acceptance and coding methodology. Encoding the knowledge for AHP requires rules which compare and contrast differences instead of looking For strengths and weaknesses of a single object at a time. Probability methods allow For encoding of a flexible decision structure: a structure which can change rapidly as the system evolves. AHP offers a more rigid structure-a structure which does not change rapidly over time. Therefore, for constantly evolving decision structures probability methods offer an easier implementation strategy (note for rapidly changing structures it is possible to implement AHP, but it is more difficult). For rigid decision structures, AHP provides a good decision-making expert system approach in implementation, Utility Functions are also used to Facilitate the decision-making process. IF a reasonable, soundly based (theoretically or empirically) Function is available to rate the different options available, then a utility Function approach is probably the best. However, if an appropriate utility function is not available, the utility function approach is risky. Developing ad hoc Functions for a specific domain is difficult. Often the ad hoc functions are usually reasonable for a small range of conditions but not over the entire domain. Unfortunately, it is hard to establish where the utility functions are valid and where they are not.
859
The reactor selection problem was also implemented utilizing the utility function expressed by equation (11) (the product of the attribute ratings). The diliIculty of this implementation was the assignment of the constants. There was no basis on which to assign the constants, yet the values of these constants is crucial to the results given by this Function. The results are also very sensitive to the values of the constants. After considerable experimentation a set of constants was established. The results of this implementation gave good discrimination between the different reactor choices and the rankings of the reactors were, in general, good. However, the accuracy and reliability of the utility function still remained a concern since it is not possible to establish its results for all the alternatives. These concerns prompted the utilization of AHP. Another difficulty with the utility function approach is in knowledge acquisition from experts. The experts were uncomfortable with the assignment of arbitrary numerical weights to various attributes. On the other hand, AHP offered a much better altemative to extracting relative weights via pair-wise comparisons. Note that while AHP offers a good structure and methodology For aiding the decision-making process, it does not readily handle uncertainty in the input information which is manipulated to make the pairwise comparisons. Other methods can and should be used in cooperation with AHP to handle the input uncertainty. For this system, fuzzy logic was found to be a good method for handling input uncertainty. A merging of Fuzzy logic in the AHP structure was found to give good results For this expert system implementation.
CONCLUSION
This paper has introduced the analytical hierarchy process, AHP, illustrated its implementation, discussed its implementation into an expert system and discussed its application to a chemical engineering domain problem. AHP offers a method to assist, guide and structure the decision-making process. This paper has shown its successful application to the engineering and expert system domains. Many of the decisions made in the engineering domain are well-suited to the hierarchical structure, and conflict resolution techniques of AHP. It represents a quantitative method to guide the decisions which are currently made based on qualitative arguments. For expert systems, AHP gives the structure For which expert systems can easily be designed. It guides
a60
P. J. HANRATTY and B. JOSEPH
the hierarchical order of the decision process and it gives the struqture on which to base the modularization of the problem (one module per decision element). AHP is best applied to stable problems whose decision structures do not change much with time. Finally AHP represents a useful alternative to the currently used methods in engineering and expert systems for decision-making. It is especially useful when relative judgements are easier and more natural than absolute judgements. AHP should be taken as another useful tool for both engineering and experts systems with which to model the decision process.
NOMENCLATURE A = oij = Bi = bj = CI = CR = CF(c, e) = n= P(B,) = r, = RI = W = @ = W, = w, = 5, =
Matrix of pairwise comparisons Pairwise comparison of element i to element i Matrix of estimated weight vectors Weight or importance of attribute i Consistency index Consistency ratio Certainty that conclusion c is true given evidence e Numer of decision elements Probability that Bi is true Rating of attribute i Random consistency index Weight vector Estimated weight vector Composite weight vector (final decision vector) Importance (weight) of element i Principal eigenvalue of pairwise comparison matrix
REFERENCES
Adams J., Probabilistic reasoning and certainty factors. In Rule Based Expert Systems (Buchanan B. et al.. Eds). Addison-Wesley, Menlo Park, CA (1984). Arbel A. and A. Seidmann, Selecting a microcomputer for process control and data acquisition. IE Trans. 14,73-80 (1984). Buchanan B., R. Duda and E. Shortcliffe, Rule Based Expert Addison-Wesley, Menlo Park, CA (1984). Systems. Duda R. O., The PROSPECTOR system for mineral exploration. Final Report, SRI Project 8172, SRI International, Artificial Intelligence Center, Menlo Park, CA (1980). Johnson C. R., Constructive critique of a hierarchical prioritization scheme employing pairwise comparisons. Proc. Int. Conf. on Cybernetics and Society of the IEEE, Institute of Electrical Engineering, Cambridge, MA, pp. 373-378 (1980). Maiers J. and Y. S. Sherif, Application of fuzzy set theory. IEEE Trans Sysr. Cybemet. 15, 175-189 (1985). Ng K. and B. Abramson, Uncertainty management in expert systems. IEEE Expert. 29-47 (1990). Peniwati K. and T. Hsiao, Ranking countries according to economic, social and political indicators. Math1 Model. 9, 203-209 (1987). Rolston D. W., Principles of Artificial Intelligence and Expert Systems Development. McGraw-Hill, New York (1988). Saaty R. W., The analytic hierarchy process-what it is and how it is used. Math1 Model. 9, 161-176 (1987). Process. McGraw-Hill, Saaty T. L., The Analytic Hierarchy New York (1980). Saaty T. L., Axiomatic foundation of the analytic hierarchy process. Mgmt Sci. 32, 841-855 (1986). Weekman V. W., Laboratory reactors and their limitations. AIChE Jl20, 833-840 (1974). Zadeh L., Fuzzy sets. Inform. Control 8, 358-363 (1965). Zahedi F., Database management system evaluation and selection decision. Deck Sci. 16, 91-l 16 (1985). Zahedi F., The analytic hierarchy process-a survey of the method and its applications. Interfaces 16,96-108 (1986).
Computersthem. Engng,Vol. 16, No. 9, pp. 849-860, 1992 Printedin Great Britain.All rightsreserved
$5.00 + 0.00
Copyright Q 1992 PergamonPressLtd
DECISION-MAKING IN CHEMICAL ENGINEERING AND EXPERT SYSTEMS: APPLICATION OF THE ANALYTIC HIERARCHY PROCESS TO REACTOR SELECTION P. J. HANRATTY and B. JOSEPH Washington University, St Louis, MO
63130, U.S.A.
(Received 9 May 1991;fina/ version received 17 February 1992; received for publication 5 March 1992) Abstract-Expert systems have been used to model and automate decision making in engineering. Often the problem is cast as a decision tree which is then translated into a set of rules. However, all decision-making problems cannot be cast into this mold. Decisions are usually based on a number of criteria (and possibly subcriteria) which are accepted or established as important to the particular decision. The experience, expertise and accuracy with which the criteria are rated and combined into an overall raring determine the success of the decision. Experts often find it difficult to express their decisions in clear-cut terms. Two experts may not completely agree on the answers to a given problem. The question then is how to capture and represent this uncertainty in decision-making? Current implementations of expert systems employ probability theory, certainty factors and fuzzy set theory. These methods are often found inadequate or unsuitable for the problem leading to less than satisfactory implementation and performance. Within the social science and business fields, a method called the analytical hierarchy process (AHP) has been utilized on a number of different problems. AHP offers a mathematical methodology based on pairwise comparisons which is well-suited for a variety of different problems in both the chemical engineering domain and the expert system domain and it is easily understood by the experts. The AHP structure is applicable to the chemical engineering domain and it is easily implemented in an expert system. This paper will present the AHP methodology and illustrate its use for the engineeringproblem of
chemicallaboratory reactor selection.Also, the resultsof the implementationof AHP into an expert system for laboratory reactor selectionwill be discussed. INTRODUCTION
Decison-making
is part of everyday
life. Exactly how
one makes decisions is difficult to quantify and tranlsate into mathematical algorithms. A variety of tangible and intangible costs and benefits are evaluated by the brain in arriving at a choice. In building expert systems that imitate the decision-making process of experts, it is necessary to somehow capture the essential ingredients of the experts’ decision-making process. The objective of this paper is to address this problem from the perspective of a process engineer faced with multiple choices in a design environment. For example, consider the problem faced by a chemical engineer in deciding what type of laboratory reactor he should use for a given application. The engineer needs to chose the laboratory reactor which will get the best possible data, in the least possible time, at a reasonable cost and with the easiest possible experiments and analysis. Usually the best reactor choice for a given situation is a compromise of all these considerations. We shall use the above paradigm in this paper to investigate the decision-making process and its automation into an expert system. One common method used in decision-making is to establish a number of criteria (and possibly subcri-
teria) upon which to base the decision_ Each criterion (and subcriterion) is rated and based upon these ratings the final decision is made. Making decisions based on established sets of criteria has a number of advantages; it forces one to establish what is important and what is not; it allows one to focus on smaller more tractable problem domains (one criterion at a time); and it seems to model well how decisions are actually made. If one proceeds to utilize the above structure to establish a model for the decision-making process, a few problems are quickly encountered. First, how are the different criteria rated? After each of the criteria are rated, how can we merge all the different criteria into one rating? How is consistency among the many choices maintained? etc. Three methods commonly used by expert systems in the engineering domain to atack these problems are probability theory (and variations thereof), fuzzy logic and utility functions. These methods have enjoyed some success in implementation on different problems. However, many problems are not well modeled by any of these methodologies. One method that has been overlooked in both the engineering field and in expert systems is the Analytic Hierarchy Process, AHP (Saaty, 1980). This method has been
P. J. HANFU-I-IY and B. Jos~~ir
850
used widely in the social science and business fields. AHP is a methodology based on relative instead of absolute ratings. Utilizing pairwise comparisons of the many options, AHP gives a structure and a mathematical base upon which many problem domains can be modeled. The methodology is applicable to a large range of engineering problems and it is easily implemented into an expert system. The objective of this article is to present the AHP methodology, illustrate its use for the engineering problem of chemical reactor selection, and discuss its implementation into an expert system for laboratory reactor selection. AHP
METHODOLOGY
AHP is a technique developed by Saaty (1980, 1987) which aids in the decision-making analysis.
AHP has been applied to a wide variety of areas from the ranking of countries according to social and political factors (Peniwati and Hsiao, 1987) to the selecting of a microcomputer (Arbel and Seidmann, 1984). Zahedi (1986) gives a listing of the various applications of AHP. In the following section, the AHP methodology will be outlined. For more details on the method see Saaty (1980). The AHP technique involves a four-step procedure (Johnson, 1980; Zahedi, 1986): Set up a decision hierarchy by breaking down the decision problem into a hierarchy of interrelated elements. Collect the input data of pairwise comparisons of the decision elements. Use the eigenvalue method to estimate the relative weights of the decision elements.
Level 1
Level 2
DdSiill AttrIbute 1
Level k
Fig. 1. General decision hierarchy for AHP.
Decision-makingin chemicalengineering Table Relative
importance
: 5 7 9 2,4 6, g Reciprocals of above numbers
1. Scak
of relative
851
importance
D&n&ion Equal importancc Moderate importance of one over another Ekential or strong impoltancc Demonstrated importance Extreme importance Intermediate values between two adjacent judgements If an activity has one of the above numbers assigned to it when compared to a second then the second activity has the reciprocal when compared to the first
4. Aggregate the relative weights of the decision elements to arrive at a set of ratings for the decision alternatives (or outcomes). In Step 1, the decision hierarchy for the process is developed. Here the decision problem is broken down into a hierarchy of related decision elements. The decision hierarchy can be viewed as a type of tree diagram with each node of the tree representing a decision element and each connection representing the corresponding weight. At the top of the hierarchy is the objective of the decison-making process. The level below the top level contains the criteria (attributes) upon which the top level decision is based. The levels below this add details to the decision elements in level above it. The final level contains the decision alternatives. Figure 1 illustrates a general hierarchy format. Establishing a hierarchy which represents the decision-making process well is crucial to accurate decision-making. The complexity of the problem domain and the detail of the problem modeling will dictate the number of levels in the hierarchy and the number of decision elements per level. The next section of the paper illustrates the decision hierarchy for laboratory reactor selection. In Step 2 of the AHP process, the pairwise comparisons of the decision elements are established in a matrix A. Here each decision element is pairwise compared against every decision element in its level for its relative importance to achieving the objective(s) of the level above it. For instance for level two of Fig. 1, attribute 1 is compared against attribute 2 then attribute 3, then . . . attribute n for achieving the objective listed in the level above it. If an attribute is considered to be more important than the other it is given a value of l-9 depending upon how much more important it is than the other element. Table 1 gives a listing of the suggested definition scale for entering the judgcment values for the pairwise comparisons. Take al to represent the pair-wise comparison of element i to element j, and A to represent the nWbiX of the pairwise comparisons. For AHP it can be argued by reasons of consistency that if element i is 5 times more important than element j that element j must be l/5 as important as element i. Putting this
in more general terms, the reciprocal axiom can be written (Saaty, 1986): aii= T/a/,.
(1)
Also, it is obvious if element i is compared against itself that each element has equal importance so that it can be written that: a,, = 1.
(2)
So to create the pairwise comparison matrix for n decision elements n(n - 1)/n pairwise comparisons need to be made among the elements. In Step 3 of AHP process, the relative weights of the decision element are estimated from the pairwise comparison matrices developed in Step 2. To estimate the weights of each of the decision elements, it should he remembered that each entry in the decision matrix can be thought of as the estimate of the relative importance or weight wi of one element to another. So that if the input values which are entered are exact relative comparisons of the decision elements the decision (pairwise comparison) matrix could he written: attributes *
1 1
Wl /WI
2 WI/W2
..
n WI IWPI
(3)
A=:
n
W”/W”
W”l%
From the above matrix, the weights could easily be determined from the equation by solving for W: A-W=n.W,
(4)
where W = (wi, w,, . . . , w.)= is the vector of the weights and n is the number of decision elements.
P. J. HANRATIYand B. JOSEPH
852
However, in actual implementations, the values for the relative weights are not known exactly so that the created decision matrix a contains inconsistencies and this is the basis of AHP. It is proposed that the weights be estimated as follows (Saaty, 1980): B.W=&,,-ti,
(5)
where A^ is the created matrix of pairwise comparisons, &,_ is the principal eigenvalue of a and J@ is the vector of the estimated values of the relative weights. For the above proposed method, L,_ may be considered to be an estimate of n from equation (4). It has been shown that &_ is always greater than or equal to n. For a completely consistent matrix &,,_ = n, and the closer &_ is to n, the more consistent are the pairwise comparisons of matrix d (Saaty, 1980). A matrix is said to be consistent if the following is true: aii.aik=aik.
(6)
Saaty (1980) suggested that L_ could be used as a measure of the departure from consistency. Utilizing this argument a consistency index CI and consistency ratio CR have been defined: CI=S,
1
(7)
CR = CI/RI,
(8)
where RI is an average random consistency index of randomly generated matrices of size n x n. The values for RI have been calculated to be as follows (Saaty, 1986):
n
Random consistency index (RI)
12 0 0
3 0.58
A general rule-of-thumb is that CR < 0.1 is considered to be acceptable. For CR > 0.1, it is suggested that the matrix of pairwise comparisons be updated to eliminate inconsistencies. The eigenvalue method is one method used within AHP to estimate the relative weights of the decision elements. Other methods do exist, but the eigenvalue method is the most widely accepted and used. In the final step (Step 4) of the AHP process the weight vectors determined in Step 3 are used to produce a vector of composite weights which serve as the ratings of the decision alternatives. The composite relative weight vector elements at the k th level with
respect to that of the tist level may be computed as follows (Zahedl, 1985, 1986): wc = fi Bi,
(9)
i-2
where WC is a k-dimensional vector of composite weights of elements at level k with respect to level 1, and Bi is the ni_, x ni matrix with rows consisting of estimated I@ vectors and ni represents the number of decision elements at level i. The composite weight vector WC from equation (9) represents the final decision vector (or ratings) of the possible choices. From this vector of weights the decision is made-the largest weight represents the best choice. Alternative
methoa!s
Other methods are available to aid the decisionmaking process-probability theory, certainty factors, fuzzy logic and utility functions. Each of these methods offers a way to combine and account for the different criteria which are essential to the decisionmaking process. Probability theory measures the extent to which one set of propositions, out of logical necessity and apart from human opinion, confirms the truth of another. Proponents generally regard this view as an extension of logic (Ng and Abramson, 1990). The appeal and advantage of probability theory is that it is solidly based in the mathematics of logic. For probability theory P(E) is the probability that event E will occur or that it is true. There is extensive literature on the interpretation and manipulation of the probability of the occurrence of different events and interrelationships among events. Bayes’ rule is commonly used
4 0.90
5 1.12
6 1.24
7 1.32
8 1.41
9 1.45
10 1.49
for implementing probabilistic inference in expert systems (Rolston, 1988). Probability theory (and variations thereof) rcquires that probabilities of truth be assigned to each fact, criteria and the relationships among them. The determination of these probabilities is often quite difficult and sometimes artificial. For instance, for the reactor selection problem, what is the probability that the best reactor for a given situation has a low heat of reaction or average operating cost? The values for these probabilities are not obvious nor is there a methodology available for determining these probabilities. A major assumption in many implemen-
Decision-makingin chemicalengineering tations is that all of the probabilities are conditionally independent (they do not alfect each other). However, it is often not possible to construct a set of conditionally independent relation+for example, the probability of how operating cost affects the reactor choice is not independent of the heat of reaction. Therefore, probability theory suffers from the problem that the probabilities required for the application of Bayes’ rule are hard to acquire (Buchanan, 1984), even for experts. However, there have been some successful applications of probability theory to expert systems, most notably PROSPECTOR (Duda, 1980). AHP offers an easier method for extracting and determining the importance of a fact or an attribute to the decision-making process especially if the decision process is not easily quantified. Another variance of the use of probability theory in expert systems is through certainty factors. Certainty factors give an informal mechanism to quantify the degree to which, based on a set of evidence, a conclusion is to be believed or disbelieved. Certainty factors are widely used for systems that operate with incrementally acquired evidence (Rolston, 1988). See Adams (1984) for a comparison between certainty factors and probability theory. The concept of certainty factors was first developed and used in MYCIN (Buchanan, 1984). Certainty factors is an ad hoc method developed because of the difficulty of acquiring a priori and conditional probabilities required for application of Bayes’ rule. However, it has been shown to be another interpretation of Bayes’ inference rule (Adams, 1984). Fuzzy logic was first proposed by Zadeh (1965) and since it has been implemented and modified by many others, see Maiers and Sherif (1985) for an extensive listing of over 500 applications of fuzzy logic in the literature. Fuzzy logic is an implementation of fuzzy set theory, which is an analogy to set theory where the “edge” of the set is fuzzy (Rolston, 1988). Within the theory, values are assigned a membership from 0 to 1, where 0 means no membership in the set, 1 means membership in the set and 0.5 means that it is equally likely to be in the set or out of the set. For example, the value for the heat of reaction could be defined as 0.7 low, 0.2 medium and 0.1 high. Fuzzy set theory lays out the means by which to relate fuzzy sets (assertions) and to manipulate fuzzy relations. The practical application of fuzzy set theory can be difficult because the coding of relationships and quantification of membership sets and fuzzy relationships becomes a much more intricate and ad hoc process. Utilizing fuzzy mathematics, these values (or sets) can be propagated through the system and assist the
853
decision-making process. However, the addition of fuzzy logic greatly increases the difficulty of implementation and the complexity of the system. It also demands that the expert expresses his choices in precise quantitative terms that most experts are not prepared to do. Utility finctions are based on derived (theoretical, experimentally or ad hoc) equations which represent the utility of given property. Quite often the utility function is based on strictly economic considerations, i.e. cost or return on investment. If a mathematical function which represents the worth (utility) of the desired choices is available, it is perhaps the best method on which to base decisions. Unfortunately, for many problem domains, there does not exist a theoretical function by which to rate the utility of the various choices. Therefore, ad hoc utility functions must be developed. Two possibile utility functions are shown below: rating = 5 b,(r,), i-1
(10)
rating = fi (r,)4, i=L
(11)
where ri is the rating (O-l) of the attribute determined to be important to the selection process (such as the attributes listed in the next section), n is the number of attributes and b, is the weight or the importance of an attribute. The major problems with these functions are that there usually is no theoretical or experimental proof that they are accurate, the weight, bi given to each attribute is very difficult to assign and determine and sometimes the decision-making is based on subjective criteria which cannot be quantified in the utility function. Utility functions require an empirical and/or theoretical equation that rates the performance of each choice. Sometimes other subjective considerations which cannot be quantified easily (i.e. environmental concerns, company relations, etc.) need to be considered in the decision-making process. If an empirical and/or theoretical utility function is not available then ad hoc utility functions can and have been formulated, but this is often unrealistic. Another possible approach to this decision-making process would be to collect enough examples of the decisions made covering all possible choices. Then one could try to construct a decision tree using inductive methods or build a functional approximation using neural networks. Unfortunately, this will work only on very small problems where the number of choices and attributes of the decisionmaking problem are small. Most engineering design problems do not fall into this category. In the
854
P. J. HANRATTY and B. JOSEPH
example problem considered in this paper, the size of the decision tree generated is extremely large. Thus we did not consider using such a scheme. An application study: selection of laboratory
reactors
Laboratory reactors are used to collect the data used for design of processes, improving and diagnosing existing processes, etc. Basically, the data from laboratory reactors is used to improve the performance of existing and future processes. Therefore, the accuracy and reliability of the data (and therefore of the reactor itself) directly affects the performance of the process. Unfortunately, one reactor type will not give the needed data for all reacting systems and applications. Each reactor type has its own set of advantages and disadvantages which change as the reacting system and application change. So for each reacting system and application, the advantages and disadvantages of all the reactor types need to be reevaluated so that the best reactor type can be chosen. The selection of laboratory chemical reactors was explored to illustrate the utilization of AHP for a chemical engineering domain problem. Weekman (1974) suggested that to select a laboratory reactor one first should decide which attributes are important to the reactor performance, rate these attributes and then select the best reactor based upon the attribute ratings. Weekman did not suggest a method on how to combine these attribute ratings into an overall rating with which to compare the reactors. AHP is well-suited for this problem. The example of Weekman (1974) will be used to demonstrate AHP applied to the laboratory selection problem. The example utilized a three-phase system with a highly active catalyst, rapid catalyst decay, highly endothermic reaction and a complex reaction system. For kinetic studies, he suggested that selection of the reactor should be based upon the following attributes: 1. Ease of sampling composition. 2. Isothennality.
and analysis of
3. Accuracy of residence contact time measurement. 4. Selectivity time averaging disguise. 5. Construction difficulty and cost.
product
Weekman rated each of the above attributes for all the reactors as shown in Table 2. Based on these ratings, the best reactor is chosen. To implement AHP for the reactor selection process, a decision hierarchy first must be developed. From the example of Weekman, a decision hierarchy is quite obvious and is illustrated in Fig. 2. After establishing the decision hierarchy (Fig. 2) the pairwise comparisons of the decision elements need to be determined. The relative weights of the decision elements are then estimated by the normalized eigenvalue method [equation (S)] from the matrix of pairwise comparisons. The pairwise comparisons of the second level and its associated weights were determined and calculated first. The painvise wmparisons between the attributes used to choose the best reactor could not be derived from the paper of Weekman, so the comparisons were elicited from three different experts in chemical reaction engineering-two experts in chemical reaction engineering from Washington University and one expert from industry. Table 3 shows the pairwise comparisons of one of the experts. This table represents the matrix of the pairwise comparisons A^with each of the entries classified as av. The weights for each attribute is determined from the normalized principal eigenvector of matrix A, equation (5). The weights of each of the attributes for each of the three experts are shown in Table 4 (note that expert 1 corresponds to the values given in Table 3). The consistency ratio [defined by equation (8)] for the pairwise comparison matrices of the three experts is less than 0.04 for each case, which is well within the suggested guideline for consitency and therefore the pairwise comparison matrices are sufficiently consistent. As shown in Table 4, the weights derived for each of the experts are somewhat different. To determine a compromised (averaged value) value for the attribute weights, two different methods are available:
Table 2. Reactor ratings for a given uample Sampling and analysis
Reactor type Differential Fixed bed Stird batch Stirred contained solids Continuous stirred tank straight-through transpon Recirculating transuort Pulse G-good,
F=fair,
P-poor.
P-F G F G F F-G F-G G
Isothennality F-G P-F z G P-F G F-0
Residencecontact time F z F-G F-G F-G G P
Selectivity disguise-decay P P : G G G FCr
ConstNction problems z G F-G P-F F-G P-F - _ G
Decision-makingin chemical engineering
855
Choose Reactor
h2rllnne 1 ISOthWldity
hmlbme 2
hnrkne
MaesTmrBtef
D_
httdbUe5
4
Renaor
Lbni?&ons
COSI
Reaaor 2:
Rmcaor 3
-18 .
S-0-W
.
.
.
.
TV
-1:
-:
awry:
Batch
BaIdl
DiltarenlU
Fig. 2. Laboratory reactor selectionhierarchy.
one, average the values of the weights determined for each of the experts; or two, take the pairwise comparisons of the experts ag and determine a new pairwise comparison matrix based on a combination of all the experts values utilizing the equation below: h’ - No. a &Slew =
(
of experta
l-l
t-1
LIff
.
au,,urpak
(12)
>
new weight values are then determined from the new matrix of averaged pairwise comparisons. The compromised weight values in Table 4 correspond to the second method discussed above. After the weight vector for level 2 has been determined, the weight vectors for level 3 need to be calculated. The pairwise comparison matrices for each of the reactor attributes were derived directly from Table 2 (Weekman’s rating for each of the reactor attributes). From the pairwise comparison The
Table 3. Attribute Sampling Sampling and analysis Isothemmlity Residence-contact time s&ctivity time average disguise Construction cost and difikulty
I % 3 l/3
matrices, the weight vectors for level 3 were calculated using equation (5). The values for the weights are shown in Table 5. Note that the ratings in Table 5 represent the weights specific to this problem example. A change in the chemical system will change the weights. The consistency ratio RI for the pairwise comparison matrices is less than 0.025 for all five cases which is well within the suggested guideline for consistency indicating that the pairwise comparison matrices are consistent. Now the overall reactor ratings can be calculated using the results in Tables 4 and 5. The overall ratings were calculated using equation (9). The weight vectors and therefore B, from equation (9), are taken from the results shown in Table 4 (level 2) and Table 5 (level 3). The overall reactor ratings (or the composite weight vector Wc) are listed in Table 6.
painvisc comparisons Isathermality 117 1 1 l/5 w
(expert
Contact 119 1 :,3 117
time
1) s&ctivity l/3 5 3 :,5
Construction 3 9 7 5 1
P.
856
J. HANRATIY Table
Expert Sampling Isotbermality Contact time Selectivity Construction
1
0.0579 0.4094 0.3690 0.130 0.034
4.
and
Attribute weights
Expert
2
OF
EXPERT
AHP
INTO
AN
SYSTEM
The AHP methodology has been implemented into an expert system for laboratory reactor selection. The expert system was designed to acquire information about the reacting system from the user and then select the best reactors for this system. AHP was utilized to discriminate between the many different laboratory reactor configurations. As illustrated before, a laboratory reactor is best chosen based on its expected performance on many different critical attributes. In the following the implementation of AHP into an expert system for chemical laboratory reactor selection is discussed. To implement AHP into an expert for laboratory reactor selection a couple of issues must be handled first: (i) what programming environment should be used? and (ii) how should the system and system knowledge be structured and organized? Commercially available “shells” are commonly used to build today’s expert systems. However, for the implementation of AHP, not all of the available shells are suitable. The requirements of the shell are that it has considerable flexibility to allow liberal structuring of the knowledge, and that it allows for linking/incorporating of external programs which conduct the necessary numerical calculations. A commercial shell CxPERT was utilized for this system. The shell is frame and rule-based system which allows for incorporation of C programming directly into the Table Reactor
type
Differential Fixed bed
Stirred batch
Stirred contained solids Continuous stirred tank Straight-through transport
Rccirculatinn
Pulse
tmnstmrt
-
-
Expert
0.0681 0.2805 0.4106 0.1906 0.0501
From the ratings for the reactors shown in Table 6, it can be seen that the recirculating transport reactor has the highest rating for this application followed by the stirred contained solids reactor and the continuous stirred tank. These rankings of the reactors agree with experts’ choices and rankings.
IMPLEMENTATION
5. Level
3
Compromised
0.0664 0.4390 0.1221 0.2220 0.149
weight
0.0642 0.377 0.301 0.181 0.078
code. Therefore, CxPERT gives the needed flexibility for this implementation. To structure the knowledge for the AHP implementation the four-step methodology (structure) of AHP mentioned before is kept in mind. The first step is to establish the decision hierarchy. The decision hierarchy establishes the structure by which the decision will be based, also it sets the structure for the organization of knowledge in the expert system. For the laboratory reactor selection problem, the hierarchy illustrated in Fig. 2 was utilized. The attributes and reactor types utilized are similar to those listed in the example. After the hierarchy is set, the structure of the system needs to be designed and incorporated around the AHP hierarchy while implementing the necessary comparisons. A multilevel multimodule approach was used to organize and structure the system, as illustrated in Fig. 3. The top level of the system maintains the overall system control. The levels below this correspond to the levels of the AHP hierarchy. Each level of the hierarchy is broken down into modules, one module per decision element, i.e. level 3 of Fig. 3 has five modules corresponding to each of the attributes. The AHP decision hierarchy utilized in the system is similar to that illustrated in Fig. 2 using different attributes and reactors. The attributes and reactors utilized for the AHP hierarchy of the system are dependent upon the application for which the reactor will be used. For each different reactor application the system will set up a different hierarchy (attributes and reactors) for the selection process. The goal of the top level of the system is to direct and overview the selection process. The structure of the system is set by the top level when the reactor application (i.e. kinetic studies or catalyst screening) is chosen by the user. Once the structure is set, the
3 weights
for
Sampling
Isothermality
0.0232 0.230 0.0442 0.230 0.0442 0.0988
o.og13 0.0237 0.198 0.198 0.198 0.0237
0.0988 0.230
B. JOSEPH
0.198 0.0813
reactor
selection
Contact time
Selectivity
0.0497 0.0497 0.270 0.115 0.115
0.0277 0.0277 0.0277 0.0277 0.268
0.115 0.270 0.0178
0.268 0.268 0.0851
Construction problems 0.198 0.198 0.198 0.08 13 0.0237
0.0813 0.0237 0.198
Decision-makingin chemicalengineering Table 6. Overall laboratory reactor ratings Reactor
Rating
Differential Fixed bed Stirred batch Stirred contained solids Continuous stirred tank Straight-through transport Recirculating transport PUISC
0.0674 0.0590 0.1789 0.1356 0.1623 0.1047 0.2124’ 0.0815
‘Highest rated reactor
selection process can begin. The lower levels are directed to conduct the pairwise comparisons and calculations needed to implement the AHP process. The lower levels determine the matrices of the estimated weight vectors Bi with which the final reactor ratings can be calculated using equation (9). Based on the reactor ratings, the final reactor recommendations are given (the highest rated reactor is the
Overall
first choice). Note that in addition to overviewing the selection process, Level 1 also overviews the explanation of the selection process. The levels below the top level correspond to the AHP decision hierarchy. Each level determines the matrix of the estimated weight vectors B,. The estimated weight vectors J@ are determined from the pairwise comparison matrix A, utilizing the eigenvalue method [equation (5)]. The pairwise comparison matrix is created based on the knowledge and expertise stored in the system. One weight vector is determined per decision element. Each level is divided into modules with each module corresponding to a decision element. The goal of the module is to determine the estimated weight vector ti associated with decision element. The weight vectors determined by each of the
System Control:
-set structure -0vWview systemoperation -oaldete finalralings(eqn. B) give finalreac(w recommendationa -ovWiewe~
Choose Reactor
Fig.
3.
Schematicof
857
expert
system structure.
P.
858
J.
HABIRA-I-TY and
Knowledge Base:
B.
JOSEPH
Weight Vector.W
Generate Pairwise Comparison Matrix -RUlf?S
-FEUWS
Eigenvalue
-Numerics
method (eqn. 6.5)
System user Fig. 4. Schematicof module in expert system.
the modules in a level make up the matrix of estimated weight vectors B,. The structure of each module is illustrated in Fig. 4. The module has a knowledge base which interacts with the user to determine the pairwise comparison matrix. The pairwise comparison matrix is sent to subroutine which calculates the estimated weight vector I&’by the eigenvalue method. The knowledge base creates the pairwise comparison matrix using a rule-base, a frame structure and some simple calculations. The frame structure is a convenient manner to store, share and pass knowledge about the reactors and the chemical system. The rule base for this implementation is different than traditional expert systems. Insteady of concentrating on one element at a time, each module determines ratings for one element relative to another. Therefore, the rules within the modules are designed to look at the element compared to another so that the pairwise comparisons can be assigned. Calculations are used to support the rule base. This type of implementation requires a good understanding of the strengths and weaknesses of each reactor under a variety of different situations. The knowledge about the reactors is obtained from two sources-literature (textbooks, published papers, etc.) and three experts in multiphase reactors. The final laboratory reactor ratings are calculated from equation (9) using the matrices of estimated weight vectors from each of the levels of the AHP
structure. The ratings determined for the reactors, via AHP, are used to rank the reactors and to suggest which is the probable best reactor for the given chemical reacting system. For this expert system-18 different reactor types are possible options. Note that for this implementation that the system has been limited to multiphase catalytic reaction systems and two application areas (catalyst screening and kinetic studies). The expert system for laboratory reactor selection is currently functioning with the AHP implementation. The system contains approx. 700 rules. The logic and knowledge of the system has been covered and tested in cooperation with a number of experts in multiphase reactors. The recommend& reactor selections of the system have also been shown to agree with the selections of the experts. This indicates that the implementation of AHP to expert systems, the chemical engineering domain and, in particular, to the reactor selection problem is feasible.
DISCUSSION
Capturing the decision-making process of experts with expert systems is usually a difficult task. The decomposition and modularization of the problem is usually an essential first step. Modularization of the problem makes the problem easier to understand, model, code, maintain and update. The difficulty with
Decision-makingin chemicalengineering decomposition of the problem-solving approach is that somehow all of the components need to be combined in the final decision process. Different approaches to this problem are available. Unfortunately, no one approach is applicable to all problem domains. AHP offers a useful approach which appears to be particularly well-suited for modeling the decision making in choosing among design alternatives. The determination of the probabilities (weights) that a given fact or attribute contributes to (or proves) the final decision is often quite difficult. For instance, even though the reaction system can be established to be isothermal for a given reactor it is difficult to quantify the probability that this reactor is the correct reactor choice Experts have a very difficult time giving answers to such questions. However, the relative importance of isothermality as opposed to reactor cost to the selection of the best laboratory reactor can be estimated by the experts. In these cases where it is easier and more natural to deduce relative ratings than absolute ratings, AHP offers a preferable option. Probability theory methods offer some advantages over AHP. Probability methods are widely used within expert systems; therefore, there is a more established acceptance and coding methodology. Encoding the knowledge for AHP requires rules which compare and contrast differences instead of looking For strengths and weaknesses of a single object at a time. Probability methods allow For encoding of a flexible decision structure: a structure which can change rapidly as the system evolves. AHP offers a more rigid structure-a structure which does not change rapidly over time. Therefore, for constantly evolving decision structures probability methods offer an easier implementation strategy (note for rapidly changing structures it is possible to implement AHP, but it is more difficult). For rigid decision structures, AHP provides a good decision-making expert system approach in implementation, Utility Functions are also used to Facilitate the decision-making process. IF a reasonable, soundly based (theoretically or empirically) Function is available to rate the different options available, then a utility Function approach is probably the best. However, if an appropriate utility function is not available, the utility function approach is risky. Developing ad hoc Functions for a specific domain is difficult. Often the ad hoc functions are usually reasonable for a small range of conditions but not over the entire domain. Unfortunately, it is hard to establish where the utility functions are valid and where they are not.
859
The reactor selection problem was also implemented utilizing the utility function expressed by equation (11) (the product of the attribute ratings). The diliIculty of this implementation was the assignment of the constants. There was no basis on which to assign the constants, yet the values of these constants is crucial to the results given by this Function. The results are also very sensitive to the values of the constants. After considerable experimentation a set of constants was established. The results of this implementation gave good discrimination between the different reactor choices and the rankings of the reactors were, in general, good. However, the accuracy and reliability of the utility function still remained a concern since it is not possible to establish its results for all the alternatives. These concerns prompted the utilization of AHP. Another difficulty with the utility function approach is in knowledge acquisition from experts. The experts were uncomfortable with the assignment of arbitrary numerical weights to various attributes. On the other hand, AHP offered a much better altemative to extracting relative weights via pair-wise comparisons. Note that while AHP offers a good structure and methodology For aiding the decision-making process, it does not readily handle uncertainty in the input information which is manipulated to make the pairwise comparisons. Other methods can and should be used in cooperation with AHP to handle the input uncertainty. For this system, fuzzy logic was found to be a good method for handling input uncertainty. A merging of Fuzzy logic in the AHP structure was found to give good results For this expert system implementation.
CONCLUSION
This paper has introduced the analytical hierarchy process, AHP, illustrated its implementation, discussed its implementation into an expert system and discussed its application to a chemical engineering domain problem. AHP offers a method to assist, guide and structure the decision-making process. This paper has shown its successful application to the engineering and expert system domains. Many of the decisions made in the engineering domain are well-suited to the hierarchical structure, and conflict resolution techniques of AHP. It represents a quantitative method to guide the decisions which are currently made based on qualitative arguments. For expert systems, AHP gives the structure For which expert systems can easily be designed. It guides
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P. J. HANRATTY and B. JOSEPH
the hierarchical order of the decision process and it gives the struqture on which to base the modularization of the problem (one module per decision element). AHP is best applied to stable problems whose decision structures do not change much with time. Finally AHP represents a useful alternative to the currently used methods in engineering and expert systems for decision-making. It is especially useful when relative judgements are easier and more natural than absolute judgements. AHP should be taken as another useful tool for both engineering and experts systems with which to model the decision process.
NOMENCLATURE A = oij = Bi = bj = CI = CR = CF(c, e) = n= P(B,) = r, = RI = W = @ = W, = w, = 5, =
Matrix of pairwise comparisons Pairwise comparison of element i to element i Matrix of estimated weight vectors Weight or importance of attribute i Consistency index Consistency ratio Certainty that conclusion c is true given evidence e Numer of decision elements Probability that Bi is true Rating of attribute i Random consistency index Weight vector Estimated weight vector Composite weight vector (final decision vector) Importance (weight) of element i Principal eigenvalue of pairwise comparison matrix
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