INVERSE SIMULATION FOR RECOMMENDATION OF BUSINESS

SCENARIO WITH QUALITATIVE AND QUANTITATIVE

HYBRID MODEL

Keisuke Negoro, Takeshi Nakazaki, Susumu Takeuchi and Masanori Akiyoshi

Graduate School of Information Science and Technology, Osaka University

2-1 Yamadaoka Suita, Osaka,565-0871, Japan

Keywords:

Qualitative and Quantitative Simulation, Monte Carlo Method, Inverse Propagation, Contradiction.

Abstract:

In order to decide an effective management plan, managers often draw up and evaluate business scenarios.

To make the evaluation, a simulation method on the qualitative and quantitative hybrid model represented as

causal graph has been proposed. There is a strong need to get optimal input values for the target outputs in

the simulation, but exhaustive search can not be realistically applied to it from considering the processing

time. Therefore, we propose a quick search method for optimal input values cencerning the qualitative and

quantitative hybrid simulation. Our approach is to get optimal values of input nodes by inverse propagation of

effects from the value of target output nodes on the simulation model. However, it generates the contradiction

that the value of a separated node in the causal graph decided from one of destination nodes is different from

the value of the other destination nodes. Therefore, we re-execute the inverse propagation repeatedly from

the nearest qualitative node connecting to a quantitative node for solving the contradiction. By experimental

results about the proposed method, time could be reduced for reaching the solution. We also could conﬁrm a

certain level of accuracy about the solution.

1 INTRODUCTION

In order to decide an effective management plan,

managers often draw up and evaluate business sce-

narios that indicate a series of changes about busi-

ness factors. For evaluating the business scenarios,

the modeling and simulation is often used(Forrester,

1961). However, it was considered to be difﬁcult to

execute simulation on the models including both qual-

itative and quantitative information which are general

factors of business systems(Kuipers, 1986).

Based on this situation, a simulation method on

qualitative and quantitative hybrid model has been

proposed(Samejima et al., 2007a). In the method,

both qualitative and quantitative factors can be han-

dled in a model. The model consists of nodes which

show business factors and arcs which show cause-

effect relations between the business factors. All

nodes have values, and in the simulation, they are

changed by propagation of effects through arcs. The

value ofinput nodes are decided according to the busi-

ness plan, and the value of output nodes are checked

for evaluation of the plan. In the method, the propaga-

tion of the effect between quantitative and qualitative

nodes is achieved using random numbers by monte

carlo method. So, the outputs are represented as a

form of probability distribution.

On the hybrid simulation, it is required to decide

optimal input values that bring target output values at

the highest probability. Of course such an input can

be decided by exhaustive search, but it takes much

time. Therefore, we propose the quick search method

for optimal input in the hybrid simulation.

In order to decide the optimal input for the target

output, our approach is simply utilizing the simulation

model proposed in the work(Samejima et al., 2007a),

which is made by users for cut-and-try solution. In the

model, propagation of effects between nodes is de-

ﬁned as a rule. It is considerd to be possible to decide

the value of source node from the value of destination

node by using the rule inversely. Therefore, values

of input nodes are obtained by deciding the value of

source nodes repeatedly. The values of input nodes

obtained from that way is considered nearly as the op-

timal input value, so we repeat the inverse simulation

to get probability distribution on input values.

333

Negoro K., Nakazaki T., Takeuchi S. and Akiyoshi M. (2008).

INVERSE SIMULATION FOR RECOMMENDATION OF BUSINESS SCENARIO WITH QUALITATIVE AND QUANTITATIVE HYBRID MODEL.

In Proceedings of the Third International Conference on Software and Data Technologies - PL/DPS/KE, pages 333-338

DOI: 10.5220/0001900203330338

 SciTePress

2 INPUT VALUE SEARCH ON

HYBRID SIMULATIONL

MODEL

2.1 Qualitative and Quantitative

Hybrid Simulation

In qualitative and quantitative hybrid simulation, both

factors are handled together in a model. The model

consists of qualitativeand quantitativenodes, and arcs

between nodes. Qualitative nodes were given with

the value range. In this method, because the propaga-

tion of an effect between quantitative and qualitative

factors is achieved using random numbers by monte

carlo method, output is distribution of probability as

shown in Fig.1(Samejima et al., 2007b).

The number of

quality manager

[10, 20]

Production time

[2.0, 5.0]

Quality level

Lead time

Volume

of sales

:Quantitative node

:Qualitative node

Output

Probability

Evaluate

as probability

distribution

Input 2

Input 1

Deciding values using Monte Carlo method

Volume

of sales

Figure 1: Qualitative and quantitative hybrid simulation.

Propagationbetween quantitativenodes is decided

by mathematical expression given by the user. Be-

tween qualitative nodes that are given with state val-

ues (ﬁve kinds of values), the value of destination

node is decided as the pair of the value of source node

in consideration of cause-effect relation (ex. mono-

tonically increasing or decreasing).

We describe about the propagation between qual-

itative and quantitative nodes as follows. A quantita-

tive node connecting to a qualitative node is divided

into ﬁve ranges by landmarks given by the user, and

the particular range is paired with the state value of

the qualitative node on the basis of cause-effect rela-

tion, as shown in Fig.2

• Propagation from a qualitative node to a quantita-

tive node

In the corresponding pair of range with the state

value of the qualitativenode, a unique value of the

quantitativenode is decided by generating random

numbers.

• Propagation from a quantitative node to a qualita-

tive node

The state value of the qualitative node is decided

as the corresponding pair of range made by land-

Sum of propagated

values in that

reflected the weights

M1 C

M2 C

z = C

Weights from magnitude

Correlation at random

= 1

M(normal)

H(high)

dH(slightly high)

Qualitative value is decided at

random in the corresponding

pair of range

Ranges are

divided by

landmarks

State value is decided

as the corresponding

pair of range

Magnitude

correlation

Source node

Destination node

Figure 2: Propagation of effect between qualitative and

quantitative nodes.

marks in which the value of quantitative node ex-

ists.

• Propagation from multiple nodes to a node

Propagation from each source node to the desti-

nation node is executed by the above-mentioned

manner. On the basis of magnitude correlation

about connecting arcs to the destination node,

weighting coefﬁcient about each arc is decided

using random number. As a result, sum of each

propagatedvalue multiplied by the coefﬁcient was

decided as the value of destination node.

In propagation of effect from a qualitative node to

a quantitative node, the value of destination node is

not decided in a unique manner. So by repeating a

chain of propagation in a model, distributions of the

values about output nodes are generated.

2.2 Problems about Searching Optimal

Input Values

As we mentioned in the introduction, it is required to

decide optimal input values that bring target output

values at the highest probability. On the hybrid simu-

lation, the optimal input can be decided by exhaustive

search. However, it takes much time because a certain

amount of time is needed to get an output of distribu-

tion in the simulation. The brute-force search is not

considered to be realistically available. Therefore, we

propose the quick search method for optimal inputs in

the hybrid simulation.

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334

3 MODEL-BASED INVERSE

PROPAGATION

As mentioned in section.2.2, it is difﬁcult to decide

optimal input values by exhaustive search in the qual-

itative and quantitative hybrid simulation. Therefore,

we propose the method to obtain optimal input values

for the target outputs quickly.

3.1 Problem on Inverse Propagation

In order to decide the optimal input for the target out-

put, our approach is to use original simulation models

deﬁned by users. On the model, propagation of ef-

fects between a source node and a destination node is

deﬁned as a rule. So, it is possible to decide the value

of source node from the value of destination node by

using the rule inversely. We deﬁne the inverse propa-

gation as shown in Fig.3.

V1, V2 are decided at random

（Condition：C1V1+C2V2 = z）

State value is decided as the

corresponding pair of V1or V2

M1 C

M2 C

State value is decided

as the corresponding

pair of range

Qualitative value is decided at

random in the corresponding

pair of range

Source node

Destination node

Weights from magnitude

Correlation at random

= 1

Source

node

Destination

node

Inverse propagation

Figure 3: Inverse propagation between qualitative and quan-

titative nodes on a model.

• Inverse propagation from a quantitative node to a

qualitative node

The state value of the qualitative node (source

node) is decided as the corresponding pair of

range made by landmarks in which the value of

quantitative node (destination node) exists.

• Inverse propagation from a qualitative node to a

quantitative node

In the corresponding pair of range with the state

value of the qualitative node (destination node),

a unique value of the quantitative node (source

node) is decided by generating random numbers.

• Inverse propagation from a node to multiple nodes

On the basis of magnitude correlation about con-

necting arcs to the destination node, weighting co-

efﬁcient value about each arc is decided at ran-

dom. Moreover, a random number about each arc

is generated so that sum of the products about the

random numbers multiplied by the weighting co-

eeﬁcients is corresponding to the value of destina-

tion node. Finally, the value of each source node

is decided, by inverse propagation of each arc’s

random number in the above-mentioned manner.

A set of values about input nodes that will gener-

ate the target output can be obtained by deciding the

value of source nodes repeatedly. The set of values

about input nodes is considered as candidates of the

optimal values of input nodes, so we repeat the in-

verse simulation to get probability distribution on in-

put values. The sets of values got at high possibility

are deﬁned as the optimal inputs.

However, if a source node has more than two des-

tination nodes, called “branch node”, there is a possi-

bility that the value of source node from one of desti-

nation nodes is different from the value of other ones

as shown in Fig.4. It contradicts the rules of propaga-

tion about normal direction that the value of each des-

tination node is decided from the same source node.

It is possible to adjust the different values to the aver-

age, but that is not an appropriate way for executing

inverse propagation correctly.

Output

Input

Values of destination

nodes are decided

from identical node

= f(x)

= g(x)

Contradiction!

Figure 4: Contradiction about inverse propagation at a

branch node.

In case of the contradiction, our approach is to ad-

just the values by executing the inverse propagation

of effect between qualitative and quantitative nodes

again. The factors of the contradiction can be prop-

agated from the cause-effect relation between quali-

tative and quantitative nodes, because values of the

qualitative nodes are decided as a random number in

the corresponding pair of range to the value of qual-

itative node. Therefore, if the contradiction has oc-

curred at a branch node, we re-execute inverse prop-

agation from the nearest qualitative node connecting

to a quantitative node. Re-execution of propagation

from the qualitative node is repeated and stopped in

case of getting the corresponding values at branch

node. If the contradiction can not be solved in the

pre-deﬁned number of times, all the inverse propaga-

tion is considered to be unavailable and canceled.

INVERSE SIMULATION FOR RECOMMENDATION OF BUSINESS SCENARIO WITH QUALITATIVE AND

QUANTITATIVE HYBRID MODEL

335

3.2 Input Value Search Method by

Inverse Propagation

It is shown in Fig.5 the outline of input value search

method by inverse propagation with re-propagation.

At the beginning of the method, the values of target

output are set. And, inverse propagation is executed

from the output nodes to the input nodes.

The judgment is necessary whether contradiction

is occurred or not. There are realistically few cases

inversely propagated values to a branch node are en-

tirely the same value. So, inconsistency needs to be

allowed to some extent and we introduced a function

for the judgment of contradiction. The difference be-

tween the maximum value and the minimum value of

the inverselypropagated values to the separate node is

deﬁned as “d”, and the value range about the branch

node is deﬁned as “R”. And, the judgment of contra-

diction is done with the following expression.

ε =

If ε is equal to or more than the threshold k, contra-

diction is considered to be occurred. In case that con-

tradiction is not occurred, the value of branch node is

decided as the random number between the maximum

value and the minimum value of the inversely propa-

gated values. Effects are propagated inversely in this

manner, and ﬁnally values of input nodes are decided.

Re-propagation

Judgment of

contradiction

Contradicted

Re-execute inverse propagation

from the nearest qualitative node

connection to a qualitative node

Yes

Not contradicted

The value of node

is decided from

the range [V

, V

]

at random

Figure 5: Input value search method by inverse propagation.

The process for getting a set of values about input

nodes from the target outputs is shown as follows.

step 1 A chain of inverse propagation is executed

from the output nodes to the input nodes.

step 2 If contradiction is occurred at a branch node,

it is re-executed inverse propagation from the

nearest qualitative node connecting to a quan-

titative node.

(a) In case that the contradiction is solved, in-

verse propagation is continued.

(b) If the contradiction can not be solved in

the pre-deﬁned number of times, all the in-

verse propagation is considered to be un-

available and canceled, and the process goes

back to the step 1.

step 3 A set of values about input nodes is decided by

inverse propagation.

Repeating the above process for the pre-deﬁned

number of times, the distribution of probability is ob-

tained.

4 EVALUATION EXPERIMENT

4.1 Model and Output of Our

Experiment

We execute experiment of the proposed method. Us-

ing the model described in Fig.6, we evaluated the

processing time for getting the distribution of inputs

and checked whether the sets of optimal values about

inputs can be obtained or not.

Volume

of sales

Probability

The number of

quality manager

[15, 25]

Step interval =1

Production time

[3.0, 5.0]

Step interval =0.1

Frequency of tests

Amount of production

in a day

Lead time

Opportunity

loss rate

Volume of sales

[0, 1600]

Target output

Input 1

Input 2

output

:Quantitative node

:Qualitative node

Quality level

Figure 6: The model for the experiment.

“The number of managers in charge of quality”

and “Production time” are set as input nodes. The

value of “The number of managers in charge of qual-

ity” is given in range from 15 to 25 on step intervals

1, and “Production time” is given in range from 3 to 5

on step interval 0.1.

In qualitative and quantitative hybrid simulation

in normal propagation, 10000 values of output are

got from an input value. And, also in the proposed

method, 10000 sets of values about input nodes are

got from a target output. The limit of re-propagation

from qualitative nodes is set 1000 times. The thresh-

old “k” for the judgment of contradiction is set 0.01.

In this model, some target outputs produce input

distributions with multiple peaks. It is shown in the

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336

Fig.7 the probability distribution about sets of input

nodes for the target output “700”. According to the

distribution, the input sets of high probability for the

output “700” are scattered about input space.

Input 1

Input 2

Figure 7: Probability distribution for the target output

“700”.

For ﬁfteen target outputs values between 100 to

1500 on step intervals 100 using the model, we evalu-

ated the processing time for getting the distribution of

inputs and checked whether the sets of optimal values

about inputs can be obtained or not.

4.2 Processing Time

We compared the processing time for getting the dis-

tribution of inputs by the proposed method to the

time by exhaustive search. The program is described

with Java. Windows XP and Pentium M 1.6GHz and

512MB memory are used as the execution environ-

ment of the program.

It takes 5 minutes on an average by the proposed

method, which is at least 3 minutes and at most 11

minutes. On the other hand, it takes 2 hours 10 min-

utes to search all sets of inputs values by exhaustive

search. It is concluded that much time is reduced to

get the distribution of inputs.

The reason for scattering time about proposed

method is that the number of input sets that can pro-

duce target outputs is different between the target val-

ues. If the number of inputs sets that can bring about

the target outputs is quite few to the input space, it is

occurred re-propagation or cancellation at high prob-

ability. And it takes more time to get a distribution of

inputs.

4.3 Precision of Probability Distribution

about Inputs

On the probability distribution about input got by the

proposed method, users probably regard the input sets

at the highest probability and around them as optimal

inputs. So, we deﬁned the set of inputs at the high-

est probability as “suggested inputs (x, y)”(x is the

value of input1, y is the value of input2), and counted

the number of right optimal input sets in the range of

suggested inputs and around the point as ”hit num-

ber”. Optimal sets of input values are deﬁned as sets

of top 30% at probability about all candidates. The

counted hit numbers are shown in Fig.8.

Probability distributions

from exhaustive search

Probability distributions

from the proposed method

In case of target output “400”

Input 2

Input 1

suggested input

and around it

Input 2

Input 1

Figure 8: Suggested inputs sets and around the point.

By the result about hit numbers in Fig.9, opti-

mal sets of inputs are included in suggested inputs or

around the point for the most case of target outputs.

The reason why optimal sets are not included at the

case the target out put is 300 or 1500 is that the num-

ber of optimal sets is very small.

Additionally, we checked degree of coincidence

between probability distributions from the proposed

method and distributions from exhaustive search. We

Hit number：　　

The number of

optimal inputs

：

Figure 9: Hit numbers and optimal sets of inputs.

INVERSE SIMULATION FOR RECOMMENDATION OF BUSINESS SCENARIO WITH QUALITATIVE AND

QUANTITATIVE HYBRID MODEL

337

calculated degrees of precision and recall for each dis-

tribution about inputs sets of top 50% at probability.

The result is shown in Fig.10, for example, the ten-

dencies of distributions are mostly corresponding to

each other at the target output “400” in which both the

precision and the recall are large. In the other target

outputs, precisions are considered to be large. By that

result, we can say probability distributions from the

proposed method have coincidence with the distribu-

tions from exhaustivesearch. Therefore, the proposed

method has a possibility to be applied to the problem

with multiple peaks.

Recall

Precision

Figure 10: Precision and recall for the distributions.

At the target output “1500”, however, the preci-

sion value is small while the recall value is large. It

is considered that the number of inputs sets of top

50% at probability is large in the distribution from the

proposed method, and they include inputs sets bot-

tom 50% at probability in the distribution from the

exhaustive search. On the contrary, the recall value

is small while the precision value is large at the tar-

get output “800”. It is considered that the number of

inputs sets of top 50% at probability is large in the

distribution from the proposed method is small, and

most of inputs sets top 50% at probability in the dis-

tribution from the exhaustive search are not included

in them.

5 CONCLUSIONS

In this paper, we proposed inverse simulation for rec-

ommendation of business scenario with qualitative

and quantitative hybrid model. As an approach, we

used the cause-effect simulation model that was made

by the user and executed propagation of effects in-

versely in order to decide a value of a source node

from a value of a destination node. Executing the in-

verse propagation in a chain of cause-effect relations

from output nodes, we decided the values of input

nodes for the target outputs. Because the value of

source node is decided at random between a qualita-

tive destination node and a quantitative source node,

contradiction is occurred concerning the value of a

branch node where one of destination nodes is dif-

ferent from the value of the other destination nodes.

Therefore, in the case of the contradiction, we re-

executed the inverse propagation repeatedly from the

nearest qualitative node connecting to a quantitative

node for solving the contradiction. According to the

repeating the whole re-propagation, probability distri-

bution of input nodes were obtained.

Evaluation was done by the processing time and

checked whether optimal inputs could be obtained or

not about ﬁfteen values of target output. It took 5

minutes for the processing and much time could be

reduced to get the distribution of inputs. We could get

optimal inputs about thirteen target outputs out of ﬁf-

teen, so it is conﬁrmed a certain level of coincidence

between probability distributions from the proposed

method and distributions from exhaustive search.

REFERENCES

Forrester, J. W. (1961). Industrial Dynamics. MIT Press,

Cambridge, USA.

Kuipers, B. (1986). Qualitative simulation. Artiﬁcial Intel-

ligence, 29(3):289–338.

Samejima, M., Akiyoshi, M., Mitsukuni, K., and Ko-

moda, N. (2007a). Business scenario design sup-

port by qualitative-quantitative hybrid simulation. In

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Technology, e-Commerce, and e-Services (EEE’07)

and the IEEE International Conference on Electronic

Commerce (CEC’07), pages 401–408.

Samejima, M., Negoro, K., Akiyoshi, M., Norihisa, K., and

Mitsukuni, K. (2007b). Hybrid simulation on quali-

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