THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 389
A Case Study in System Dynamics Optimisation
R. Keloharju
(Helsinki School of Economics)
E. F. Wolstenholme
(University of Bradford Management Centre
Abstract
This paper presents the use of optirnisation as a tool for policy analysis
and design in system dynamics models and presents a demonstration of its
use on the ‘project model’ developed by G. P. Richardson and A. L. Pugh IIt
in their book “Introduction to System Dynamics Modelling with DYNAMO”.
The use of optimisation to design parameters, table functions and new
model structure is shown to produce a very significantly improved
performance for this mode] compered to conventional approaches.
INTRODUCTION
The purpose of this paper is to present a case study to demonstrate the
merits of using optimisation for the purpose of policy design in system
dynamics models.
Traditionally, system dynamics has relied very extensively on the use of
intuition and experience by system owners and analysts to help design
policies for improving system behaviour over time. This situation is now
changing and much effort is being expounded in the development of policy
design methods. Basically, two schools of thought are emerging. The first
of these concerns the epplicetion of control thearetic methods such as
eigenvalue analysis, linear control theory, Routh stability criterion, model
control theory and optional control theory. (Sharma (1985), Mohapatra and
Sharma (1985). These approaches can be powerful but do required a level
of assumption and analytical ability out of keeping with the original aims
of system dynamics, which was to facilitate the exploration of systems by
as wide @ range of practitioners as possible. Intensive use of these
methods is not anticipated until computer software is developed to
improve their ease of application.
390 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOUIEIY. SEVILLA, ULTUBER, 1950.
The second major aproach te policy design which has emerged in recent
years is that of sirmulation by optimisation (Keloharju (1983), Gustafson
and Wiechowski (1986)). This approach also relies fundementally on
computer softwere but is not inhibiting in it's dependence on sophisticated
analytical techniques. The software to be described and applied for
optimisation in this paper was originally developed in the late 1970's as
an appendage to DYSMAP (Dynamic Simulation Model Application
Programme) (Covana and Coyle 1982) and known os DYSMOD (Dynamic
Simulation Model Gptimiser and Developer). This software is currently
under further develapment by Bradford end Salford Universities in the
United Kingdorn and will be released shortly ss a specific version of the
redeveloped and restructured DYSMAP2.
The DYSMOD Optimiser
Although the concept of optimisation is not new in system dynamics, the
DYSMOD approsch to model development snd analysis provides 6 new
dimension to system dynamics. The softwere uses ¢ hill climbing routine
to heuristically determine the optimum values for: any nurnber of model
perameters relative to predefined objective functions or performance
measures. Essentially, the method assumes a system dynamics model as a
starting point. However, experience uses might formulate their models
somewhat unconventionally to give the software maximum scope to assist
with the task of model development.
Optimisation in parameter space is achieved by interleafing simulation
and optimisation. One iteration of the procedure consists, firstly, of a
DYSMAP simuletion run, in which the value of the objective function is
recorded, and secondly in a run of the optimiser to choose parameter
values which might improve the objective function. Subsequent iterations
consist of rerunning DYSMAP to test out the resultant improvement in the
objective function under the new parameters and further refinement of
them by optimisation. Any one experiment with the software might take @
100 or more iterations. This procedure presents few problems, however,
given efficient software and the current downward trend in computer
hardware costs.
One of the more basic and but perhaps more trivial uses of such an
optimisation technique (which is often wrongly considered as it's only
use) is in fitting models to past data. Whilst the method hes an important
role in the area of validation, it's much more influential role is in
parameter and table function policy design and, less obviously, in
structural policy design. The letter is achieved by combining alternative
Policy equations using pseudo parameters to achieve ‘mixed’ rather than
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 391
‘pure’ policy analysis. Further, by combining real and pseudo parameters in
an experiment, parameter and structural policy design can be carrried out
simultaneously and in @ process which subsumes sensitivity analysis. In
conventional system dynamics practice these activities must be carried
out separately and sequentislly and ina very limited way. In optimisation
the model structure can be considered as 6 continuum and the process
considered as one of choosing or synthesising a final model from en
infinite number of possible models, comprised of all the parameter and
structural permutations offered (a variety set).
in large scale models the policy design process is further facilitated in
DYSMOD by the use of a base vector convention similar to those used
elsewhere; for example in the simplex algorithms of mathematical
programming. That is, only a limited number of parameters are considered
‘free’ or in the base at any time; but that the candidate parameters for the
bese can be changed as optimisation proceeds. Other sophistications
available, but outside the scope of this writing are the concepts of model
simplification (thet is, the driving of psuedo perameter variables to zero
to eliminate model structure) and optimisation over variable time
horizons within the simulation.
Figure 1 gives an overview of the structure of, and interactive imputs to
DYSMOD. During optimisation/simulation only the finsl values of
perameters and objective functions are printed. The rerun facility allows
‘the final ‘model’ ta be run conventionally to examine other model variables
and their behaviour over time. Subsequently, further optimisation can be
undertaken with revised peramters, objective functions, numbers of
iterstions, ete.
A Case Study
|. A Specimen Model, Optimisation as a policy design tool is best
demonstrated by using 6 system dynamics model as its starting point. In
order to avoid the devotion of time and spece here to introducing a new
model it was decided to choose @ well known model as a candidate for
optimisation. The model chosen for this purpose was the excellent
‘project model’ (Richerdson and Pugh (1961)), originally developed to
explain the process of applying system dynamics and to dernonstrate the
Power af the method for exploring the merits of alternative system
operating policies. It should be stressed that this choice is not in any way
meant to imply criticism of the policy design method used by Richardson
and Pugh. Indeed the oppasite is the case and the work here should be seen
as 6 way of evtending the analysis provided by these authors
392 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
COMPILATION
OBJECTIVE FUNCTION
MAXIMISE OR MINIMISE?
PARAMETER RANGES?
LENGTH OF SIMULATION?
NUMBER OF ITERATIONS?
SIZE OF STEP?
NUMBER OF OUTPUT LINES?)
TOR HANDLING?
di
OPTIMISATION
/SIMULATION
FIGURE 1 THE STRUCTURE OF DYSMOD
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 393
The purpose of the ‘project model’ was to explore the evolution over time
of @ project involving @ set of tasks. The basic version of the model
focused on factors affecting project overruns, and subsequent
developments dealt with the introduction of more realism into the model
(for example disaggregating the workforce) and in examining the redesign
of parameter and structural based polices for control of the project.
Performance measures of totel project time and labour costs were
introduced as a basis by which to compare alternative policies.
An influence diagram of the model tailored to this presentation is given in
Figure 2 and a full listing of the model, including additional equations for
the purpose of optimisetion, is provided in Appendix |. Figure 2 is
somewhat self explanatory and no further discussion of the existing model
will be given here. Instead, the optimisation modifications to the model
will be described together with some policy experiments and results for
comparison with those achieved on the basic model. Comments will also
be made on these results where they provide evidence of the general
insights which cen be gain from the optimisation procedures.
2. Amendments to the Model, It will be seen from Appendix i that a
supplementery set of equations are introduced into the model (lines 98 -
111). Since the optimisation process consists of many simulation runs, it
is firstly very necessary to hove precisely calculated performence
measures rather than to rely on extracting such information from output
graphs. AUX1, ..., AUX4 therefore store for later calculation the exact
time of project completion and variables AUXC! end AUC2 register the
total cost from the project. Secondly, it is necessary to define a suitable
objective function. in the case of the project model an appopriate measure
is the trade off between the cost and the completion time for the project
as given by the equation for OBJ!
It is stated by Richardson and Pugh that (for such projects of the type
described in the model); “whether or not @ policy is an improvement
depends on how one weighs the additional cost against the saving in
completion time. A system dynamics model does not set or evaluate the
criteria for improved system behaviour eople make the value
judgements.” However, in optimisation it is perfectly feasible to explore
such judgements in the model. It will be seen that the equation for OBJ1
incorporates 6 parameter WEIGHT to achieve this. Varying WEIGHT
between model experiments allows the strength of the trade-off between
cost and time to be investigated. AUXS in the equation for OBJI is
included in order to penalise early completion and avoid trival results.
(Equations 105, 106, 107 and 110 given in Appendix | fecilitate global
sensitivity analysis and are not used in the experiments described here.)
394 THE 1986 INTERNATIONAL CONFERENCE OF THE.SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
reeeenctp ————D EFFECT OF SCHEDULING PRESSURE
SORKFORCE: ) ioicaTeD COMPLETION DATE | soreoueo COMPLETION DATE
v
ley P Tieton apparent —"———P eraction g_|
WORKFORCE. WORKFORCE PROGRESS: SATISFACTORY
wh aaa
woRkrorce 4 GENERATED REAL | 3s
GROSS
RATE 4- WorkForce UNDISCOVERED PROGRESS:
REWORK PRODUCTIVITY
RATE 7 ASSIMILATION :
TIME
WORKFORCE NORMAL
ADJUSTMENT — @SZPD) FRACTION
Bt UNDISCOVERED CUMULATIVE ‘SATISFACTORY
REWORK REAL
WORKFORCE PROGRESS ESAT:
SOUGHT NORMAL
TIME
To
t perecr t Cote PRODUCTIVITY
WILLINGNESS REWORK CUMULATIVE *\,
10 (Froev) DISCOVERY PERCEIVED
CHANGE RATE PROGRESS FINAL
PROJECT CURRENT
FRACTION DEFINITION PROJECT
COMPLETED ASSUMED DEFINITION
UNDISCOVERED
INDICATED ASSUMED ——) REWORK FRACTION RELY
WORKFORCE OF
TIME ¢ + : PRODUCTIVITY
To ERCE |
DISCOVER ASSUMED COMPLETE
REWORK CUMULATIVE
PROGRESS
GZTZ TIME REMAINING WEIGHT
t tT INDICATED
TIME SCHEDULED
COMPLETION L, EFFORT — PRODUCTIVITY
PERCEIVED
SsTE L REMAINING |
PERCEIVED
PRODUCTIVITY
INDICATED ———scHEDULE TIME
COMPLETION
DATE ¢——— 4. PERCEIVED
;
SCHEDULED
ADJUSTMENT PERCEIVE
TIME PROODUCTWVITY
GD J TEROD
FIGURE 2.
INFLUENCE DIAGRAM FOR PROJECT MODEL
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 395
3. Direct Parameter Policy Design Richardson and Pugh defined some
parameters which were used in their parameter based policy design
experiments, These ere listed in Table 1 (and highlighted on Figure 2)
together with the values allocated to them by Richardson and Pugh in their
base model run. This table also sets ranges for these veriables which
were used in the optimisation experiments. The conventions! approach of
system dynamics is to vary these paramters either one 4 a time or in
combination to establish their effect. Examples of the conventional
results obtained by Richardson and Pugh for project cost and time are
shown in Table 2, Run 1 on table 2 represents the results from the bese
run of their model and the second line represents the effect of a new
parameter set. Lines 3 - 6 in Table 2 give results from the optimisation
experiments. In each line the objective function weight wes. changed and
the parameter values given are those chosen by the optimisation software
Table 1. Parameters Defined for Parameter Policy Design Experiments,
Value used by Range set for
Rand P Base Optimisation
Model run Experiments
(weeks) (weeks)
TPPROD Time to Percieve Progress 6 o> 6
WEAT Workforce Adjustment Time 3 1-10
SAT Schedule Admustment Time 6 1-10
ASMT Workforce Assimilation Time 6 1-10
Table 2 Results of Conventional and Optimisation Experiments for
Parameter Policy Design
RUN WEIGHT ASMT SAT WFAT TPPROD TIME COST
Richarson) 1 me 6 6 3 6 61.500 6.247E+6
and Pugh ) 2 - + 3 4 2 3 60.250 S.B61E+6
23 050 1.0 3,673 1.008 4893 52.750 5.4696+6
4 0660 1.0 1,792 1.032 6.000 53.250 5.398E+6
Optimiser) 5 0.65 1:0 5.249 1.005 6.000 52.750 5.576E+6
»6 0.70 10 4951 1.001 6.000 52,750 5.567E+6
7 0.80 1.0 4951 1.001 6.000 52.750 5.567E+6
6 1.10 1.0 4950 1.000 3.270 5250 5.659E+6
99G THE 19860 INTERNATIONAL CONFERENCE OF TRE SYSTEM DINAMICS SOCIETY, SEVILLA, OCTOBER, 1986.
Some interesting fectors emerge from these results. Firstly, the
optimiser results indicate that it is possible to reduce both project time
and cost significantly from the base run; and from the position reached by
the individual parameter changes suggested by Richardsen and Pugh, by
manipulating the four basic parameters. Secondly, time end cost do not
appear to be sensitive to the value of the weight attached to these factors
in the objective function. Thirdly, that the approximate same time and
cost can be achieved by many different permutations of the perameters.
This result might be considered to contribute to the proof of the view that
there is no generally right combination of parameters for a model, which
is often claimed in conventional simulation.
It should also be noted here that WFAT and ASMT are chosen to be small.
The former implies that workforce imbalances should be eliminated es
quickly as possible and the latter thet the workforce should be
assimilated as quickly as possible. This may of course incur additional
training costs and highlights the need for this factor to be included in the
model. In fact, it is interesting to reflect that such low values of these
parameters might never have been tested out in conventional system
dynamics policy analysis, since conventional wisdom would have expected
instabilities to have been created. Conversely, the results imply thet the
scheduled adjustment time (SAT) and the time to percieve progress
(TPPROD) need not be low.
Finally, it should be noted thet identical results are obtained with weights
of 67 and 6.6. This results implies that model behaviour is not
necessarily always a function of policies and that what is really
important in system dynamics is the point along each continuous
parameter trajectory at which behaviour does change.
4, Table Function Policy Design. As an alternative to changing specific
parameters in the model, policy design can also be carried out by verying
table functions. Three of the table functions suggested by Richardson and
Pugh for this purpose are given in Table 3. These are also highlighted in
Figure 2. Table 3 defines each of these functions and gives the ranges of
‘the tables used in the optimisation experiments. In each case the first
boundary given is thet used in the base run of the madel by Richardson and
Pugh.
Run 1 in Table 4 shows an example of the results obtained by Richardson
and Pugh from changing TWCWF alone relative to the base model run and
setting this at the alternative boundary defined in Table 3. Runs 2, 3 and 4
in Table 4 show the results from optimisation when the previous
parameters, plus each table function in turn, are chosen by the software.
Run 5 allows e pair of table functions to change together. All optimisation
experiments were carried out a weight value of 0.6.
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 397
Table
Range Set for Optimisation
Experiments
TTDRW Table for time to detect rework
boundary 1 12/12/12/10/5/°.5
boundary II 6/6/6/5/3/0.4
TWCWF Table for willingness to change
workforce
boundary I 0/0/0/.1/.3/.7/,9/1
boundary II 0/0/.1/.9/1/1/1/1
TNFSAT Table for normal fraction
satisfactory
boundary I .5/,55/.63/,75/,.9/1
boundary II ,.6/,63/.7/.8/.92/1
TABLE 3. Table function defined for Policy Design Experiments.
TABLE FUNCTIONS COST ($) TIME
RUN |ASSMT| SAT | WFAT | TPPRCD|
(WEEKS;
Richard]
son and | 1 6 | 6,000] 3,000) 6.000 | TWCHF 0/0/,1/,9/1/1/1/1 €,5076E+6) 55.75)
Pugh
(12 1 |5,500}1.000} 3,018 | TTDRW 6.9/8.3/6.2/5,0/5/,5] 5, 1951£+6) 49,51
Opti- (|3 1 4,240}1,054) 6.000 | THCWF C/0/.1/,8/,8/1,0/.9 |5,566E+6 | 48,50)
Miser (4 1 |4,946]1,000]6.000 | TNFSAT .6/,6/.7/.8/.5 5.052E+6 | 51,25
C15 1 |7.418}1.001)6.009 (TTDRW 12/12/5/12/10/5/.5) |5,104E+6 | 47,75)
(
THCHF .1/.7/,6/,7/,9 )
TABLE 4, Results of Conventional and Optimisation Experiments for
Table Function Design,
398 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
{n general the results in Teble 4 again show significant improvement in
both time and cost for the project relative both to the base run and the
Table 2 results. Run 2 suggests an interesting policy alternative which
changes loop polarities. That is that the time to discover rework should
be @ concave function of the project progress rather than a convex
function. This function is chosen to be to close to its lower boundery
which suggests that the faster rework is discovered the better for the
project. This, in turn, suggests better quality control is needed and
indicates, realistically, that the cost of this should be taken into account
in the model. In runs 3 and 4 the optimisation also chose the alternative
boundaries of TWCWF and TNFSAT for which the same conclusions can be
drawn as for run 2. In run S the model developed in run 4 for TNFS was
taken as a starting point and the other two table functions optimised. The
results are the best of this group.
S. Structural Policy Design. Richardson and Pugh also introduced
structural policy design experiments into their model. An overestimation
parameter was introduced for this purpose and considerable discussion
was presented by these suthors as to whether overestimation should be
epplied to effort perceived remaining on the project or to the indicated
workforce. The final choice for locating this parameter was recommended
to be in the equation for the effort perceived remaining. in order to
investigate this issue using optimisation two overstimation parameters
were defined. As will be seen in the appendix OE1 wes defined in the
equation for effort perceived remaining and OE2 was defined in the
equation for the indicated workforce. An experiement was then conducted
allowing OE1 and OE2 to vary between 1 and 3 and allowing ASMT, SAT,
WFAT and TPPROD to vary as defined in Table 1. Table S shows the results
of this experiment with weight values from 0.4 to 0,65 (runs 2-7) together
with the Richardson and Pugh result from the bese model with OED
‘inserted in the effort perceived remaining equation at 6 value of 1.5.
in all cases of the optimisation results in Table 5, the project schedule of
40 weeks could be kept, and at « reasonable cost. (This cost could
possibly be improved even further by incorporating the teble function
related policies defined earlier). The results also shed considerable light
on the choice between OE1 and O£2. In the project model OE1 can be
considered to represent an overestimation of demand and OE2 an
overestimation of supply. When cost considerations are more important
that time time considerations (WEIGHT is small) there is no need to push
supply. Therefore, OE2=1 and OE2 1. When time is more important than
cost (WEIGHT is large) there is no need to push demand. Therefore, OE1 = t
end O&221. Inother words the two overestimation parameters were
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 399
never chosen to be effective together but a clear point emerged at which it
wes preferable to switch from one to the other. This situation is
somewhat analogous to the situation of monetary policy in economics and
Table 5 gives and answer to the question as to when to switch from
‘restrictive monetory policy’ to ‘expansionist monetory policy’ and
vice-versa.
RUN WEIGHT ASSMT SAT © WFAT TIPPROD OE1 © 0E2 TIME COST
Richardson 1 - 6.0 6,000 3.000 6,000 1,500 - 45,75 6,563E+6
Pugh
(2 04 1.0 1.216 5,489 6,000 2.138 1.000 40,00 5,462E+6
(3 0,45 1,0 1,000 5.055 6,000 2.271 1,000 40,00 5.416E+6
(4 0.5 1.0 1,000 5,433 6,000 2.320 1,000 40,00 5,454E+6
Optimiser 5 9,55 1,0 1,000 5.430 6,000 2.320 1,000 40.00 5,454E6
(6 0.6 1,0 1,000 10,000 3.000 1.000 2,116 40,00 5,569E+6
(7 0.65 1.0 1,000 10.000 3.000 1,000 2,116 40,00 5.H69E+6
TABLE 5, Results of Conventional and Optimisation Experiments for Structural
Policy Design,
CONCLUSIONS
This paper has presented the rationale for, and a case study of the merits
of, optimisation for policy design in system dynamics models. When
implernented through e good computer software interface the procedure is
entirely straightforward to perform. The results produced are, as in
conventional system dynamics, totally explainable in terms of the
underlying feedback structure of the model. However, the saving in
conputational effort required by the analyst in praducing policy insights is
enormous. This is not, however, to sey that the level of thinking is
reduced. Rather this is increased, since considerable skill is necessary in
formulating the model so as to provide the computer with the sufficient
scope to generate the maximum amount of model exploration.
400 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
REFERENCES:
Cavana, R. ¥. and R.G. Coyle (1982). DYSMAP User Manual, University of
Bradford.
Gustefsson, L. and M. Wiechowski (1966). Compiling DYNAMO snd
Optimisation Software. System Dynamics Review, Yol. 2, no. 1.
Keloharju,R. (1983). Relativity Oynamics. Helsinki School of Economics.
Mohapatra, P.J.K. and S.K. Sharma (1965). Synthetic Design of Policy
Dynamics in System Dynamic Models: A Model control Theory of
Approach. System Dynamics Review, Vol. 1, no.
Richardson, G.P. and A.L. Pugh ill (1961). Introduction to System
Dynamics Modelling with DYNAMO.
Sharma, 5. K. (1985). Policy Design in System Dynamics Models: Some
Control Theory Applications. Doctoral Thesis. Indian Institute of
Technology.
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
Appendix 1
® REVISED PROJECT MODEL
NOTE
NOTE REAL PROGRESS
TE
CRPRG.K=CRPRG. J+DT#RPRG. JK
CRPRG:
RPRG.KL=APPRG.K*FSAT.K
APPRG.K=WF.K*GPROD.K*ESPGP.K
NGPROD#EEXPGP.K
-K=NFSAT.K¥EEXPFS.K#ESPFS.K
NFSAT.K=TABHL (TNFSAT,FCOMP.K,0, 11.2)
TNFSAT=.5/.55/.63/.75/.9/1
NOTE
NOTE “EFFECTS OF EXPERIENCE §ND SCHEDULE PRESSURE
appopazerg
2
5
2
8
8
EEXPRP. K=TABHL (TEEXPG,FEXP. Ky 0,11 .2)
XPG2.5/.55/.85/.75/.87/1.
FEXP.K=EXPWF.K/WF.K
EEXPFS .K=TABHL (TEEXPF »FEXP.K» Ori 7.2)
TEEXPF=.5/.6/.7/.8/.9/1
ESPFS.K=TABHL(TESPFS, ICD.K/SCD.K,.9/1-27.05)
TESPFS=1.1/1.06/1/.98/.9/.83/.75
ESPGP.K=TABHL (TESPGP, ICD.K/SCD.K,.911-2,.05)
“leerane78/, 92/1/1.1/1.18/1.28/1.25
ZAvapaDvavz
=
TE
NOTE UNDISCOVERED REWORK
NOTE
R GURW.KL=APPRG.K#(i-FSAT.K)
IRW. J+DT#(GURW. JK-DURW. JK)
L
N
R DUR. KL=URW.K/TDRW.K
A TDRW.K=TABHL (TTDRW,FPCOMP.K»0,1,.2)
T TTDRW=12/12/12/10/5/.5
A CPPRG.K=CRPRG.K+URW.K
A FPCOMP.K=CPPRG.K/CPD.K
® CPD.K=TABHL(TCPD,FCOMP.K ,O+17-2)
T TCPD=800/830/900/1000/1140/1200
A FCOMP.K=CRPRG.K/1200
NOTE EFFORT PERCEIVED REMAINING
NOTE
A EPREM.K=OE1¥(CPD.K-ACPRG.K)/PPROD.K
C OEL
A ACPRG.K=CPPRG.K-AURW.K
A AURH.K=ADURW.K#ATDRW.K
A ADURM. K=SMOOTH(DURW. JK» TADURH)
© TADURW=8
A ATDRW.K=TABHL (TATDRW+FPCOMP.K»Or1,.2)
T TATDRW=8/8/7/5/3/1.5.
bi
N
a
c
4
7
a
PPROD.K=PPROD. J+(DT/TPPROD) (IPROD. J-PPROD. J)
PPROD=GPROD
IPROD.K=WTRP.KARPROD.K+ (1-WTRP.K)#NGPROD
‘TPPRO}
WTRP.K=TABHL (THTRP,FPCOMP.K +017 .2)
TWTRP20/.1/,25/,5/.9/1
RPROD.K=NGPROD#FSAT.K
402 — THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
5? NOTE
5@ NOTE HIRING
59 NOTE
BO A WF.K=
61 L_EXPWF.K=EXPWF. J+DT#WFAR. JK
62 ON
63 oc
64 oR OWFAR. KL=NENNF.K/ASMT
85 Cf ASNT=6
6G _L_NEWWF.K=NEWWF. J+DT# (HR. JK-HFAR. JK)
87 ON
sac
69 oR WES.K-WF.K)/WEAT
70°C WFAT=3
71 A WES.K=WCHF .KEIWE K+ (1-WCHF «KD #WE.K
72° @ WCHF.K=TABHL (TWCHF, TREM.K 10,2173)
730
7a A
7 6
76 NOTE
77° NOTE SCHEDULING
78 (NOTE
79 @ TREM.K=SCD.K-TIME.K
80 LL SCD.K=SCD. J+DT#NAS. JK
81 N SCD=SCDN
82 C SCDN=40
83 RNAS. KL=(ICD.K-SCD.K)/SAT
84 C SAT=6
85 A ICD.K=TIME.K+TPREQ.K
86 A_TPREG.K=EPREN.K/WES.K
+87 NOTE OBJECTIVE FUNCTIONS
88 NOTE
9 NOTE INDICATORS
90 NOTE
81 SCUMEFF . J+DT# (MF. JRESPGP J)
92
93 PMMACUMEFF . K
94 C CPMM=3000
95 NOTE
96 NOTE ###¥8¥EGUATIONS FOR OPTIMISATION ##¥HHeHHHRHH ER
87 NOTE *
98 A AUXL.K=CLIP(TIME.K,0.25,CRPRG.K»CPD.K) *
99 A AUX2.K=SAMPLE(AUX1 «Ks AUXI.K,0) *
100 A AUXS.K=CLIP(100,0.25+AUX1 :K~AUX2-K,1) *
401 A AUX4.K=SAMPLE (AUXZ.K AUXS«K 10) *
402 A AUXC1.K=SAMPLE (COST.K rAUXi-K 10)
103 AUXC2.K=SAMPLE(AUXC1.K ,AUX3.K +0
104 A AUXS.K=MAX(40-AUX4.K, 0). *
405 INIT=0- *
106 C _INCR=0.01 *
107 A TARG.K=(1+INCR)#INIT *
108 A OBJ1.K=WEIGHT#(1E+5)#AUX4.K+(1-WEIGHT)#AUXCZ.K+ #
109 X (1E4+5) #AUXS..K *
110 @ OBJ2.K=MAX(AUX4.K-TARG:K » TARG.K-AUX4.K) *
411 WEIGHT=0.5 *
412 NOTE *
$13 NOTE we. core ereeers aan
1140 -NOTE
415 NOTE CONTROL STATEMENTS
116 NOTE
117 C DT=0.25
118 C PLTPER=2
118 C PRTPER=40
120 C_LENGTH=80
121 PRINT 1)WF,SCD,0BJ2/2)CUMEFF .FCOMP,FPCOMP/3)COST,OBJ1+
122 © X AUXC1/4)AUXCZ, AUX1 y AUXZ/5) AUXS + AUX4s AUXS
123 PLOT MF=W(0,80) /SCD=S(30, 70) /CPPRG=P,CRPRGAR, URW#U( 0712003 /
124 -X PPROD=#(.671)
RUN
