174 QR-96

Backward Qualitative Simulation of Structural Modelfor Strategy Planning

Takenao OHKAWA, Shinya RATA, and NorihisaDepartment of Information Systems Engineering

Faculty of Engineering, Osaka University2-1, Yamadaoka, Suita, Osaka 565, Japan

phone: +81-6-879-7826, fax : -1-81-6-879-7827e-mail : ohkawa@ise.eng .osaka-u .ac .j p

Abstract

In the process of estimating the effectiveness ofplans or policies, it is useful to construct a di-agrammatic causal model, named the structuralmodel, that represents causality between severalfactors in the target organization . We have al-ready proposed a method for qualitative simula-tion that can predict behaviors of a target systemmodeled with a structural model for the strategyplanning . The effectiveness of a supposed planis estimated by reviewing a predicted behavior ofthe target as a simulation result . However, trialsof the simulation have to be iterated many timesin order to find out a better plan, if the model islarge and complex .This paper proposes the backward simulation

method that can generate possible initial statesof the operable nodes from the desirable behav-ior of the utility nodes to cope with this problem .Through the comparison with the forward simu-lation, the efficiency of the method is clarified .

IntroductionThe process of estimating the effectiveness of plans orpolicies introduced into a new strategy in an organiza-tion plays an important role for strategy planning . Inthis process, it is useful to construct a diagrammaticcausal model, named the structural model, that rep-resents causality between several factors in the targetorganization in the form of a directed graph (Warfield1973) . Several effective methods for structural mod-eling have been proposed, so far, and the computersupport for construction and utilization of the struc-tural models has been recognized to be one of the mostsignificant works, as organization becomes large andcomplex (Axelrod 1976, Montazemi & Conrath 1986,Conrath, Montazemi & Higgins 1987, Zhang, Chen &Bezdek 1989, Zhang et al . 1992) .We have already proposed a method for qualitative

simulation that can predict behaviors of a target sys-tem modeled with a structural model (Ohkawa & Ko-moda 1993) . In this method, the notion of time scalethat is determined based on the time lag for propagat-ing influence from one factor to another factor in the

KOMODA

model, is introduced to cope with ambiguities causedby lack of quantitative information . The time scale canhelp to divide a whole model into several sub-models .In addition, the redundant state transitions in the sim-ulation process can be reduced by using several typicalpatterns, which specify primitive behaviors in equilib-rium for each time scale .By using the simulation, we can estimate the effec-

tiveness of a supposed plan, by which initial conditionsof the simulation are specified, by reviewing a predictedbehavior of the target as a simulation result . However,trials of the simulation, in which various initial condi-tions, namely, various plans are considered, have to beiterated many times in order to find out a better plan,if the model is large and complex .

This paper presents a method of backward simula-tion of the structural model, which can derive onlypossible initial plans or policies backwardly from thedesirable behavior of the target . In this method, ba-sically, an influence of a utility node of the model ispropagated to the other nodes along the directed arcsinversely. The notion of the one-step (forward) simu-lation is introduced in order to select only meaningfulbehaviors from possible behaviors derived through theprocess of backward influence propagations .The following two sections describe briefly the def-

initions of the structural model treated in this paperand the overview of the (forward) simulation methodrespectively. Then the procedure of the backward sim-ulation is presented . Finally, the effectiveness of thebackward simulation is verified through some experi-mental results .

Structural ModelThe structural model represents causal relationsamong several factors in an organization or a system .It takes the form of a directed graph that consists ofsome nodes and arcs . Each node corresponds to thefactor in the system . Each arc expresses the causal re-lation with the arrow indicating the direction from acause to an effect . Figure 1 illustrates an example ofthe structural model . The figure suggests that the ex-istence of reliable executives in a corporation improves

Figure 1 : An example of the structural-model .

the bank's confidence, and it brings the funds raising,and so on .We have defined the following four types of qualita-

tive parameters that specify a state of the structuralmodel . For further details of the definitions, see refer-ence (Ohkawa & Komoda 1993) .Definition 1 (Qualitative status value)

Let x bea node and t be a time . [x(t)], the qualitative statusvalue of x at t, is defined as follows :

H,

M,

L,

Definition 2 (Change tendency)

Let x be a nodeand t be a time . [8x(t)], the change tendency of x att, is defined as follows :

[8x(t)] =

if status value of x cannot increaseto the higher value than the cur-rent value,if status value of xcrease and decrease,if status value of x cannot decreaseto the lower value than the currentvalue .

if statuscreasingif statusat t,if statust,if statusat t,if status valuecreasing at t .

can both in-

value of x is greatly in-at t,value of x is increasing

value of x is steady at

value of x is decreasing

of x is greatly de-

Definition 3 (Direction of influence) Let x bea cause node and y be an effect node . D(x, y), thedirection of influence from x to y, is defined as follows :

if the status value of y increasesin proportion as the status valueof x increases,if the status value of y decreasesin proportion as the status valueof x increases .

Definition 4 (Propagation speed) Let x be acause node and y be an effect node . V(x, y), the prop-agation speed from x to y, is defined as follows :

Vo,

if the change of the status valueof x causes instantaneously thechange of the status value of y,

V,1 , if the time order (e .g . hours,days, weeks, months, years, etc)for propagating influence from xto y is longer than the time ordersuch that V(x, y) = V� _1 .

Qualitative Simulation Using TypicalPatterns

The goal of qualitative simulation of a structural modelis to derive model's behaviors that are triggered bychanging states of several nodes in the model . Thismethod is based on the following two assumptions .

Behaviors of slower sub-models can be ignored whenfocusing on a faster sub-model .A faster sub-model is regarded as being in equilib-rium, whenever evaluating other slower sub-models .The simulation of each sub-model is performed in or-

der of time scale . The result of each simulation, whichis obtained in equilibrium, is propagated to the slowersystem, so that the behaviors of the whole system arederived .The propagation of influence between nodes in a sub-

model is determined based on the change tendencyof the cause node that is the source of the influenceand the qualitative status value of the effect nodethat is influenced by the cause node according to thepropagation rules summarized in Table 1 in case ofD(x, y) = -h . If D(x, y) = -, the table in which `I'and `D' in the column of [8x(t)] are exchanged is usedinstead of Table 1 . If more than one influences arepropagated to a node, the sum of the change tenden-cies of the cause nodes is treated as the influence to theeffect node . The sum of the change tendencies is calcu-lated according to Table 2 . The symbol `?' in the tableindicates an unknown value, which means the changetendency cannot be determined uniquely. If the un-known value is obtained, all possible values must beconsidered . The behavior of a node is specified with asequence of the change tendencies in the node.

If the state transitions according to the propagationrules are in equilibrium, in other words, if the repeti-tion of the same state transitions is observed, the rep-etition part of the state transition is evaluated qualita-tively with a typical pattern for each node using Table3 .

Let TS2 be a sub-model in a certain time scale, andif each sequence of change tendencies that specifies thebehavior of each node in TS, is terminated by a typ-ical pattern, the sub-model TSi is regarded as beenin equilibrium . In this case, the influence is propa-gated from the sub-model TSz to TS,+i, which is a

Ohkawa 175

Table 1 : Propagation from x to y ([y(t)], [ay(t)D.

Table 2 : Summation of the change tendency .

176 QR-96

sub-model with slower time scale than TSi . This typeof influence propagation is achieved based on the sumof change tendencies and the initial value of the qual-itative status values according to the rules shown inTable 4, where [x(TSj )], [ax(TSj )] indicate the initialvalue of qualitative status value of node x in time-scaleTSi and the final state of node x in TSi respectively .Each element of the table represents the influence toTSi+1, namely, a pair of the initial value of qualitativestatus value of node x in TSi+1 and the initial value ofchange tendencies for node x in TSi+1 . The final stateof a node is estimated from the combination of the sumof a series of change tendencies and the typical patternof the node in equilibrium using Table 5 .

After propagating influences from a fast sub-modelTSi to a slow sub-model TSi+1, the propagation of in-fluence in TSi+1 is performed similarly . As a resultof simulation, the sequence of change tendencies of ev-ery nodes that is terminated by a typical pattern for

Table 3 : Identification of the typical pattern .

Overview

Table 4 : Propagation from TSi to TSi+I .

Table 5 : Calculation of the final state .

every sub-model is obtained . For detailed argumentson the simulation procedure, see reference (Ohkawa &Komoda 1993) .

Backward Simulation

The backward simulation is considered as a processof deriving possible initial states of the target struc-tural model from a behavior of a utility node (Axelrod1976), in which we have interested particularly . Theinitial state of the model is specified with the quali-tative status value and the change tendency of everynode at t = 0 and the change tendencies of operablenodes at t = 1, which correspond to actions of a plan .In other words, the backward simulation aims at enu-merating possible change tendencies of operable nodesat t = 1 based on the states of all node at t = 0 andsequence of change tendencies of utility nodes given asinput .

In the backward simulation method, firstly, changetendencies of all nodes in a sub-model TSi are calcu-lated by the influence propagation with utility nodesas starting points . If the states of all nodes are clar-ified in TSi, the results are propagated to the nextslower model TSi+i using the manner similar to thenormal (forward) simulation methods . The state tran-sitions that cannot follow given behaviors of the utilitynodes are discarded in this process, and the survivedstate transitions give the possible solution, namely thechange tendencies of operable nodes at t = 1 .

Backward influence propagation in asub-modelBasic propagation rule The backward propaga-tion rules in a sub-model can be defined on the basis

y(t[8x(t)1

- 1)lM H L

S (M, S) (H, S)_(L, S)

I (M, I) (H, S) (M, I)D (M, D) (M, D) (L, S)I+ (M, I+) (II, S) (M, I+)D+ (M, D+) (M, D+) (L, S)

x(TS,)l[ax (TS, M H L

S (M, S) (H, S) (L, S)I (M, I) (H, S) (M, I)D (M, D) (M, D) (L, S)I+ (H, I+) (I1, S) (H, I+)D+ (L, D+) (L, D+) (L, S)

+ S I D I+ D+S

I D I+ D+

I I I ? I+ DD D ? D I D+I+ I+ I+ I I+ ?D+ D+ D D+ ? D+

Change tendency of repetition part Typicalpattern

S only appear S*I or I+ appear, neither D nor D+ I*D or D+ appear, neither I nor I+ D*I and D appear, neither I + nor D+ (ID)*, (Dl)*I+ and D+ appear (ID)*, (DI)"I+ and D appear, D+ not appear I*D+ and I appear, I+ not appear D*

S* I* D* (ID)*, (DI)*S S I+ D+ S

I, I+ I I+ D+ ID, D+ D I+ D+ D

Table 6 : Backward propagation from y to x[ax(t)1 / [y(t)] .

Table

7:

Backward

propagation

from

z

to

x, y([ax(t)], [ay(t)]) / [z(t)] (no special order in parenthe-ses) .

of the normal propagation rules shown in Table 1 . Ta-ble 6 summarizes the backward propagation rules forD(x, y) = -}- . In case of D(x, y) _ -, `I' and `D' for[ax(t)] are swapped .

If more than one influences are propagated to anodes, Table 7, which is generated by interpreting therule of summation (Table 2) inversely, is applied .Propagation considering operable node Thebackward propagation rules shown in Table 6 and 7can be applied to neither the nodes connected with anoperable node nor the operable node itself, because thechange tendency of the operable node is determined bythe plan as well as the influence propagation . For ex-ample, the change tendency of an operable node att = 1 can be regarded as an action of a plan, and if theoperable node is not influenced from other nodes, thechange tendency of the node at t > 1 keeps `S' . Table 8shows the rules for backward propagation consideringthe operable node for each position of them as shownin Figure 2 . The initial value of flag(x) is 0 . If theresult of propagation is given as `?', all possible valuesare assumed .One-step simulation If behaviors derived by thebackward propagation rules cannot appear in the nor-mal simulation process because of feedback loops in themodel, these inconsistent behaviors must be removed .We introduce a mechanism of one-step (forward) sim-ulation in order to find out inconsistent behaviors .The results of backward propagation at time t are

verified by the one-step simulation using the result ofbackward propagation at t - 1 and the change ten-dency of operable node derived at t . If a result of the

operablenode

Figure 2 : Positions of the operable node .

backward propagation is consistent with the one-stepsimulation, it is accepted .

Typical pattern identificationThe result of (forward) simulation is described in theform of a sequence of the change tendencies includinga typical pattern for each node (e .g . ISI(ID)* for nodexi , SSSID* for x2, etc) . The description is dividedinto two parts, the transitional part (such as ISI forxl and SSSI for x2) and the equilibrium part (such as(ID)* for xl and D* for x2) . The typical pattern inthe equilibrium part is identified easily by the forwardsimulation based on the final state of the transitionalpart . By comparing the typical pattern derived bythe forward simulation with the given behavior, theconsistency of the state transition is verified .

Procedure of backward simulationThe direction of influence, the propagation speed, thequalitative status value at t = 0 and the change ten-dency at t = 0 are given as initial states of the targetmodel . In addition, the change tendencies for someutility nodes are given . Under these preparations, theprocedure ofthe backward qualitative simulation of thestructural model is shown as follows .Step 1 : Divide a given structural model into several

sub-models TSo . . . TSI based on the time scale .Suppose current sub-model TS = TSo and currenttime t = 1 .

Step 2: Propagate influences backwardly from thechange tendencies of the utility node at t in sub-model TS according to the rule shown in Table 6and 7, until the change tendency at t = 1 for ev-ery node in TS has been determined . If the changetendency cannot be determined uniquely, enumer-ate all possible behaviors and store a condition tobe required for each behavior .

Step 3 : Execute the one-step simulation based on theresult of backward propagation at t - 1 and thechange tendencies of operable nodes at t . If the re-sult of the one-step simulation matches the behaviorderived in Step 2, save them .

Step 4: If t = t,ax where t,a2 indicates the last timeof the transitional part, identify the typical pattern

Ohkawa 177

z(t - 1)][8z(t)1

M H L

(S, S)/M (S, S)/H (S, S)/L(1, D)/M (1, ?)/H (D, ?)/L

I (I, ?)/M - (I, ?)/MD (D, ?)/M (D, ?)/M -

(t - 1)][ay(t)]

M H L

S S/M S/H, I/H S/L, D/LI I/M - I/MD D/M D/M -

~oad toervice sec .

Table 8 : Backward propagation from y to x [ax(t)] / [y(t)] with operation node .

sales

[(+, VO I

price

share

I( ,,7V-) =8(- 1V.):onsumer'smage(+, Vo)

b

d

cost

CM

Figure 3 : An example of simulation model (model 1) .

of the equilibrium part for each node using the for-ward simulation . If derived patterns are consistentwith given behaviors, consider them as possible statetransitions . If t < trnax, increase t and go Step 2 .

Step 5: If TS = TSi, output the sequences of thechange tendencies of all nodes for each sub-model asthe result, and terminate the simulation .

Step 6: Propagate the last state in the current sub-model TS(= TS;) to the succeeding sub-modelTS;+I according to the rules defined in Table 4 . Sup-pose TS = TS1+1, go Step 2 .

An example of backward simulationThe procedure of the backward qualitative simula-tion has been implemented on the workstation (SUNSPARCstation2) using the C language . We show anexample of application of the method to a structuralmodel about consumer's image shown in Figure 3 . Inthis model, the nodes 'price' and `Cibl' are operablenodes and the node `sales' is a utility node .

178 QR-96

*1 : In case that no influence is propagated to node x .*2 : In case that node x is influenced from other node .

Table 9 : Simulation results .

The backward simulation was executed under thecondition that the initial values of the change tendencyand the qualitative status value of every node are `S'and `M' respectively and the behavior of `sales' are (be-havior of TSo, behavior of TSI)= (SII(DI)*,II*) . Asa result, seven kinds of initial states of the operablenodes were calculated in about 0.5 second . The simu-lation results are summarized in Table 9 . In the table,E(x, y) means the degree of influence from node x to y,and `-' indicates the "don't care" state . For example,the result of No.3 suggests that the strategy of keepingthe price and improving the CM produces soaring salesunder any condition .

DiscussionThe efficiency of the backward simulation is evaluatedby comparing the number of trials and the total exe-cution time to derive the initial states of the operablenodes with the forward simulation, where all possiblecombinations of the states of operable nodes are con-sidered .

Operable node Condition Result of propagation Post processx* I t = 1 [ax(t)] = [ay(t)]

t > 1, flag (x) = 0, [ax(t-1)] = [ay(t)] [ax(t)] = [ay(t)]t > 1, flag(x) = 0, [ax(t-1)] 0 [ay(t)] [ax(t)] = S flag(x) = 1t > 1, flag(x) = 1 [ax(t)] = S

x*2 t = 1 [ax(t)] = [ay(t)]t > 1, flag (x) = 0, [ax(t-1)] = [ay(t)] [ax(t)] = [ay(t)]t > 1, flag (x) = 0, [ax(t -1)] i4 [ay(t)] [ax(t)] = [ay(t)] flag(x) = 1t > 1, flag(x) = 1 [ax(t)] = [ay(t)]

y flag(y) = 0 [ax(t)] =flag(y) = I [ax(t)1 = [ay(t)]

N°'

Initial stateConditionprice CM

1 D 12 1 I E(d, g) < E(e, g)3 S I4 D D E(d,g) > E(e,g)5 D -6 - I E(d, g) < E(e, g)7 - D E(d, g)

E(a, b)> E(e, g) and< E(e, b)

We evaluated the performance of the methods usingtwo structural models, the same model as shown in Fig-ure 3 consisting of 7 nodes and 9 arcs (model 1) andthe more large model about the same theme consistingof 14 nodes and 20 arcs where factors such as 'capi-tal investment' and 'labor cost' had been additionallyconsidered (model 2 shown in Figure 4) . The operablenodes of the model 1 are `price' and `CM' . The model 2has two additional operable nodes (`extending branch'and `office automation') including them . The utilitynodes is `sales' for both models .The result of comparison is shown in Table 10 . `#

of state' indicates the number of initial states derivedby the backward simulation, and '# of exec .' meansthe number of trials to obtain all possible solutions .`Time' means the total execution time .This result tells the following effectiveness of the

backward simulation .

Performance of the backward simulation for the rel-atively large scale model is more than 10 times ashigh as the one of the forward simulation from theview point of the total execution time .

Difference of the efficiency between both of methodswidens as the target model becomes large .

ConclusionThis paper reported the backward simulation methodthat can generate possible initial states of the operablenodes from the desirable behavior of the utility nodes .Through the comparison with the forward simulation,the efficiency of the method was clarified .On the other hand, the backward simulation never

supersedes the forward simulation completely . Thecombination of both of them is more useful for thestrategy planning . For example, after clarifying an es-sential factor that is inferred from the results of thebackward simulation, we can evaluate the effect of theessential factor in detail using the forward simulation .We have developed the GUI based system SPLEQS(Strategy Plan Evaluation system base on Qualita-tive Simulation) that integrates both of the methodand other features, such as the edit of the structuralmodels, the visualization of the simulation results withgraphs (Rata, Ohkawa & Komoda 1994), the auto-matic scenario generating (Hiramatsu et al . 1995), etc .

Table 10 : Comparison with forward simulation .

References

Figure 4 : Simulation model (model 2) .

Warfield, J . N . 1973 . Binary Matrices in System Mod-eling . IEEE Transactions on Systems, Man and Cy-bernetics SMC-3(5) : 441-449 .Axelrod, R. 1976 . Structure of Decision : PrincetonUniversity Press .Montazemi, A. R., and Conrath, D. W. 1986. The Useof Cognitive Mapping for Information RequirementsAnalysis . MIS Quarterly 10 : 45-56.Conrath, D. W ., Montazemi, A. R., and Higgins, C .A . 1987 . Evaluating Information in Ill-Structured De-cision Environments . Journal of the Operational Re-search Society 38(5) : 375-385 .Zhang, W. R., Chen, S . S ., and Bezdek, J . C . 1989 .Pool2 : A Generic System for Cognitive Map Devel-opment and Decision Analysis . IEEE Transactions onSystems, Man and Cybernetics SMC-19(1) : 31-39 .

Ohkawa 179

Model of Backward Forward# state # of exec . Time (ms) # of exec . Time (ms)model 1 7 1 531 10 2,380model 2 43 1 97,412 131 1,183,884

Zhang, W. R ., Chen, S . S ., Whenhua, W ., and King,R. S . 1992 . A Cognitive-Map-Based Approach tothe Coordination of Distributed Cooperative Agents .IEEE Transactions on Systems, Man and CyberneticsSMC-22(1) : 103-114 .Ohkawa, T ., and Komoda, N . 1993 . Multiple TimeScaling Qualitative Simulation Using Typical Pat-terns . In Proceedings of the IEEE International Work-shop on Emerging Technologies & Factory Automa-tion, 129-136 . Cairns, Australia : IEEE IndustrialElectronics Society.Hata, A ., Ohkawa, T., and Komoda, N . 1994 . Devel-opment of Qualitative Simulation System Using Typ-ical Patterns . In Proceedings of the IEEE Interna-tional Conference on Systems, Man and Cybernetics,2749-2754 . San Antonio, Texas : IEEE Systems, Manand Cybernetics Society .Hiramatsu, A., Hata, S., Ohkawa, T., and KomodaN . 1995 . Scenario Generator for Qualitative Simula-tion System . In Proceedings of the IEEE InternationalConference on Systems, Man and Cybernetics, 143-148 . Vancouver, Canada : IEEE Systems, Man andCybernetics Society.