1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
|
<?xml version="1.0" encoding="UTF-8"?>
<sect1 id="sect-advanced-analysis-simulation">
<title>Simulation Analysis</title>
<sect2 id="sect-advanced-analysis-simulation-intro">
<title>Introduction to simulation analysis</title>
<para>A simulation is the imitation of the operation of a real-world process or system. The behavior of a system is studied by generating an artificial history of the system through the use of random numbers. These numbers are used in the context of a simulation model, which is the mathematical, logical and symbolic representation of the relationships between the objects of interest of the system. After the model has been validated, the effects of changes in the environment on the system, or the effects of changes in the system on system performance can be predicted using the simulation model.
<footnote>
<para>Adapted from Banks, Carson, Nelson and Nicol (2001), Discrete-Event System Simulation, 3rd ed.</para>
</footnote>
</para>
<para>Gnumeric includes a facility for performing Monte Carlo Simulation. Monte Carlo simulation involves the sampling of random numbers to solve a problem where the passage of time plays no substantive role.
<footnote>
<para>Definition from Law and Kelton (1991), Simulation Modeling & Analysis, 2nd ed, pp. 113.</para>
</footnote> In other words, each sample is not effected by prior samples. This is in contrast to discrete event simulation or continuous simulation where the results from earlier in the simulation can effect successive samples within a simulation experiment. The Monte Carlo simulation will be enabled through the use of the Random Number functions as described in ??? <!--<xref linkend="CATEGORY_Random_Numbers"/> --> and the results presented along with statistics for use in analysis.
<footnote>
<para>Gnumeric random numbers are generated using the Mersenne twister MT19937 pseudo-random number generator as implemented in the GNU Scientific Library.</para>
</footnote>
</para>
</sect2>
<sect2 id="sect-advanced-analysis-simulation-setup">
<title>Setting up the simulation model</title>
<para>The remainder of this chapter will illustrate use of the simulation tool using an example from Banks et. al.
<footnote>
<para>Adapted from Banks, Carson, Nelson and Nicol (2001), Discrete-Event System Simulation, 3rd ed. pp. 42-45.</para>
</footnote>
A classic inventory problem is the newsvendor problem. A newsvendor buys papers for 33 cents each and sells for 50 cents. Newspapers not sold are sold as scrap (recycled) for 5 cents. Newspapers are purchased by the paper seller in bundles of 10. Demand for newspapers can be categorized as “good,” “fair,” or “poor” with probability 0.35, 0.45 and 0.20 respectively, with each day's demand being independent of prior days. The problem for the newsvendor is to determine the optimal number of papers to purchase when the day's demand is not yet known.</para>
<para>The daily profit equation for the newsvendor is:</para>
<informalequation>
<alt>Profit = [(Sale revenue) - (Cost) - (Scrap value)]</alt>
<mathphrase>Profit = [(Sale revenue) - (Cost) - (Scrap value)]</mathphrase>
</informalequation>
<para>To set up the model, this example will use two tabs in Gnumeric, a tab labeled 'Profit' to calculate profit, and a tab labeled 'Demand Tables' to store the various tables needed to calculate the demand for any given sampling.</para>
<para>For the Profit tab, set up the profit tab as in <xref linkend="fig-simulation-example-profit-table" />.</para>
<para>At the top of the Profit' tab, the Profit table will be entered . There are three variables: Sale revenue, Cost and Scrap value, and they take the per unit coefficients of 0.5, 0.33 and 0.05 respectively. Enter the coefficients in cells B13 through D13. In cells B12 through D12, enter the equations for sale revenue, cost and Scrap value that are in the list below. In cell E12, enter the equation for Profit</para>
<para>Next, we add the values for the decision variable, which is the amount purchased, and the amount sold. </para>
<itemizedlist>
<listitem><para>B12: =B13*min(B16,B20)</para></listitem>
<listitem><para>C12: =C13*B16</para></listitem>
<listitem><para>D12: =D13*max(0,B16-B20)</para></listitem>
<listitem><para>E12: =B12-C12+D12</para></listitem>
<listitem><para>B13: 0.5</para></listitem>
<listitem><para>C13: 0.33</para></listitem>
<listitem><para>D13: 0.05</para></listitem>
<listitem><para>B16: 50</para></listitem>
</itemizedlist>
<figure id="fig-simulation-example-profit-table">
<title>Profit table for newsvendor example</title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/analysis-simulation-profit-ex1.png" format="PNG" />
</imageobject>
<textobject>
<para>
This screenshot depicts the profit table from the newsvendor simulation example after the set of entries in cells B12 through E13 and Cell B16 have been filled in.
</para>
</textobject>
</mediaobject>
</screenshot>
</figure>
<tip>
<title> Using SIMTABLE for parameter values.</title>
<para>Sometimes, there is a need to try a number of different values for a single parameter. In <xref linkend="sect-advanced-analysis-simulation-simtable" /> the <function>SIMTABLE</function> function will be used to automate the use of a set of values for a parameter such as purchase quantity. For now, set the purchase quantity to 50 in cell C16.</para>
</tip>
<para>Next, create the demand tables from which the demand will be generated. In the tab 'Demand Tables' enter the values of the probability in cells B4 through B6 (B4: 0.35; B5: 0.45; B6: 0.2). In cells C4, C5 and C6 enter the cumulative probability values (C4: 0.35; C5: 0.8; C6: 1) as shown in <xref linkend="fig-simulation-example-demand-newsday-type" />. </para>
<itemizedlist>
<listitem><para>B4: 0.35</para></listitem>
<listitem><para>B5: 0.45</para></listitem>
<listitem><para>B6: 0.2</para></listitem>
<listitem><para>C4: 0.35</para></listitem>
<listitem><para>C5: 0.8</para></listitem>
<listitem><para>C6: 1.0</para></listitem>
</itemizedlist>
<figure id="fig-simulation-example-demand-newsday-type">
<title>Probability distribution for type of newsday</title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/analysis-simulation-demand-ex1.png" format="PNG" />
</imageobject>
<textobject>
<para>
Screenshot depicts Demand Tables tab of simulation worksheet with the cells B4 through C6 filled in with probability distribution for type of newsday.
</para>
</textobject>
</mediaobject>
</screenshot>
</figure>
<para>The next table is the daily demand for newspapers based on the type of news day. The table Distribution of Newspapers Demanded is in cells A11 through D18 of the Demand Tables worksheet as shown in <xref linkend="table-analysis-simulation-daily-demand"/> and contains the daily demand distribution values. The cumulative distribution tables in cells A21 through G29, shown in <xref linkend="table-analysis-simulation-cumulative-demand"/> are derived values from the Distribution of Newspapers Demanded using values in the top Distribution of Newspapers demanded table.</para>
<table frame="all" id="table-analysis-simulation-daily-demand">
<title>Daily newspaper demand distribution table in Demand Tables worksheet</title>
<tgroup cols="4"><tbody>
<row>
<entry>
<para/>
</entry>
<entry>
<para><emphasis>A</emphasis></para>
</entry>
<entry>
<para><emphasis>B</emphasis></para>
</entry>
<entry>
<para><emphasis>C</emphasis></para>
</entry>
<entry>
<para><emphasis>D</emphasis></para>
</entry>
</row>
<row>
<entry>
<para><emphasis>11</emphasis></para>
</entry>
<entry>
<para>Demand</para>
</entry>
<entry>
<para>Good</para>
</entry>
<entry>
<para>Fair</para>
</entry>
<entry>
<para>Poor</para>
</entry>
</row>
<row>
<entry>
<para><emphasis>12</emphasis></para>
</entry>
<entry>
<para>40</para>
</entry>
<entry>
<para>0.03</para>
</entry>
<entry>
<para>0.1</para>
</entry>
<entry>
<para>0.44</para>
</entry>
</row>
<row>
<entry>
<para><emphasis>13</emphasis></para>
</entry>
<entry>
<para>50</para>
</entry>
<entry>
<para>0.05</para>
</entry>
<entry>
<para>0.18</para>
</entry>
<entry>
<para>0.22</para>
</entry>
</row>
<row>
<entry>
<para><emphasis>14</emphasis></para>
</entry>
<entry>
<para>60</para>
</entry>
<entry>
<para>0.15</para>
</entry>
<entry>
<para>0.4</para>
</entry>
<entry>
<para>0.16</para>
</entry>
</row>
<row>
<entry>
<para><emphasis>15</emphasis></para>
</entry>
<entry>
<para>70</para>
</entry>
<entry>
<para>0.2</para>
</entry>
<entry>
<para>0.2</para>
</entry>
<entry>
<para>0.16</para>
</entry>
</row>
<row>
<entry>
<para><emphasis>16</emphasis></para>
</entry>
<entry>
<para>80</para>
</entry>
<entry>
<para>0.35</para>
</entry>
<entry>
<para>0.08</para>
</entry>
<entry>
<para>0.06</para>
</entry>
</row>
<row>
<entry>
<para><emphasis>17</emphasis></para>
</entry>
<entry>
<para>90</para>
</entry>
<entry>
<para>0.15</para>
</entry>
<entry>
<para>0.04</para>
</entry>
<entry>
<para>0</para>
</entry>
</row>
<row>
<entry>
<para><emphasis>18</emphasis></para>
</entry>
<entry>
<para>100</para>
</entry>
<entry>
<para>0.07</para>
</entry>
<entry>
<para>0</para>
</entry>
<entry>
<para>0</para>
</entry>
</row>
</tbody></tgroup>
</table>
<table frame="all" id="table-analysis-simulation-cumulative-demand"> <title>Cumulative demand distribution table in Demand Tables worksheet</title>
<tgroup cols="8"><tbody>
<row>
<entry>
<para/>
</entry>
<entry>
<para><emphasis>A</emphasis></para>
</entry>
<entry>
<para><emphasis>B</emphasis></para>
</entry>
<entry>
<para><emphasis>C</emphasis></para>
</entry>
<entry>
<para><emphasis>D</emphasis></para>
</entry>
<entry>
<para><emphasis>E</emphasis></para>
</entry>
<entry>
<para><emphasis>F</emphasis></para>
</entry>
<entry>
<para><emphasis>G</emphasis></para>
</entry>
</row>
<row>
<entry>
<para><emphasis>21</emphasis></para>
</entry>
<entry>
<para/>
</entry>
<entry>
Cumulative Distribution
</entry>
<entry>
<para/>
</entry>
<entry>
<para/>
</entry>
<entry>
<para>Values</para>
</entry>
<entry>
<para/>
</entry>
<entry>
<para/>
</entry>
</row>
<row>
<entry>
<para><emphasis>22</emphasis></para>
</entry>
<entry>
<para>Demand</para>
</entry>
<entry>
<para>Good</para>
</entry>
<entry>
<para>Fair</para>
</entry>
<entry>
<para>Poor</para>
</entry>
<entry>
<para>Good</para>
</entry>
<entry>
<para>Fair</para>
</entry>
<entry>
<para>Poor</para>
</entry>
</row>
<row>
<entry>
<para><emphasis>23</emphasis></para>
</entry>
<entry>
<para>40</para>
</entry>
<entry>
<para>0.03</para>
</entry>
<entry>
<para>0.1</para>
</entry>
<entry>
<para>0.44</para>
</entry>
<entry>
<para>0</para>
</entry>
<entry>
<para>0</para>
</entry>
<entry>
<para>0</para>
</entry>
</row>
<row>
<entry>
<para><emphasis>24</emphasis></para>
</entry>
<entry>
<para>50</para>
</entry>
<entry>
<para>0.08</para>
</entry>
<entry>
<para>0.28</para>
</entry>
<entry>
<para>0.66</para>
</entry>
<entry>
<para>0.03</para>
</entry>
<entry>
<para>0.1</para>
</entry>
<entry>
<para>0.44</para>
</entry>
</row>
<row>
<entry>
<para><emphasis>25</emphasis></para>
</entry>
<entry>
<para>60</para>
</entry>
<entry>
<para>0.23</para>
</entry>
<entry>
<para>0.68</para>
</entry>
<entry>
<para>0.82</para>
</entry>
<entry>
<para>0.08</para>
</entry>
<entry>
<para>0.28</para>
</entry>
<entry>
<para>0.66</para>
</entry>
</row>
<row>
<entry>
<para><emphasis>26</emphasis></para>
</entry>
<entry>
<para>70</para>
</entry>
<entry>
<para>0.43</para>
</entry>
<entry>
<para>0.88</para>
</entry>
<entry>
<para>0.94</para>
</entry>
<entry>
<para>0.23</para>
</entry>
<entry>
<para>0.68</para>
</entry>
<entry>
<para>0.82</para>
</entry>
</row>
<row>
<entry>
<para><emphasis>27</emphasis></para>
</entry>
<entry>
<para>80</para>
</entry>
<entry>
<para>0.78</para>
</entry>
<entry>
<para>0.96</para>
</entry>
<entry>
<para>1</para>
</entry>
<entry>
<para>0.43</para>
</entry>
<entry>
<para>0.88</para>
</entry>
<entry>
<para>0.94</para>
</entry>
</row>
<row>
<entry>
<para><emphasis>28</emphasis></para>
</entry>
<entry>
<para>90</para>
</entry>
<entry>
<para>0.93</para>
</entry>
<entry>
<para>1</para>
</entry>
<entry>
<para></para>
</entry>
<entry>
<para>0.78</para>
</entry>
<entry>
<para>0.96</para>
</entry>
<entry>
<para>1</para>
</entry>
</row>
<row>
<entry>
<para><emphasis>29</emphasis></para>
</entry>
<entry>
<para>100</para>
</entry>
<entry>
<para>1</para>
</entry>
<entry>
<para></para>
</entry>
<entry>
<para></para>
</entry>
<entry>
<para>0.93</para>
</entry>
<entry>
<para>1</para>
</entry>
<entry>
<para></para>
</entry>
</row>
</tbody></tgroup>
</table>
<para>When these values are entered, the final results will look like <xref linkend="fig-simulation-example-daily-demand" />.</para>
<figure id="fig-simulation-example-daily-demand">
<title>Completed probability distribution tables in Demand Tables worksheet</title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/analysis-simulation-demand-ex2.png" format="PNG" />
</imageobject>
<textobject>
<para>
Screenshot of Demand tab for the newsvendor example with probability distribution for type of newsday in cells A21 through G29.
</para>
</textobject>
</mediaobject>
</screenshot>
</figure>
<para>Finally, back in the Profit tab, the demand data will be filled in through the use of references to the Demand Tables tab as shown in <xref linkend="fig-simulation-example-profit-sheet" />. </para>
<para>In the following cells, enter the equations below in the 'Profit' tab:</para>
<itemizedlist>
<listitem><para>B17: =rand()</para></listitem>
<listitem><para>C17: =if(B17<'Demand Tables'!C4,"Good",if(C19<'Demand Tables'!C5,"Fair","Poor"))</para></listitem>
<listitem><para>B18: =rand()</para></listitem>
<listitem><para>B20: =lookup($C17,$B23:$D23,$B24:$D24)</para></listitem>
<listitem><para>B21: =E12</para></listitem>
<listitem><para>B24: =lookup(Profit!$B18,'Demand Tables'!E23:E29,'Demand Tables'!$A23:$A29)</para></listitem>
<listitem><para>C24: =lookup(Profit!$B18,'Demand Tables'!F23:F29,'Demand Tables'!$A23:$A29)</para></listitem>
<listitem><para>D24: =lookup(Profit!$B18,'Demand Tables'!G23:G29,'Demand Tables'!$A23:$A29)</para></listitem>
</itemizedlist>
<figure id="fig-simulation-example-profit-sheet">
<title>Profit table for newsvendor example</title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/analysis-simulation-profit-ex2.png" format="PNG" />
</imageobject>
<textobject>
<para>
This screenshot depicts the profit sheet for the newsvendor example after references to demands are included in cells B17 through B21 and cells B24 through D24.
</para>
</textobject>
</mediaobject>
</screenshot>
</figure>
<para>When done, the Profit spreadsheet will be setup with a profit equation, decision variables, and random events as shown in <xref linkend="fig-simulation-example-profit-sheet" />. The <function>rand()</function> functions in cells C17 and C18 return a random value between 0 and 1, which are used by the <function>lookup()</function> functions in cells B20, B24, C24 and D24 to calculate a randomly determined daily demand. Next, this sheet will be used for analysis through the use of simulation.</para>
</sect2>
<sect2 id ="sect-advanced-analysis-simulation-running">
<title>Running the simulation</title>
<para>To run the simulation, from the Gnumeric toolbar, select Tools → Simulation. In the Risk Simulation dialog box that appears, the first tab is the Variables tab. There are two entries in the Variables tab: Input variables and Output variables (<xref linkend="analysis-simulation-variables-dialog" />).</para>
<figure id="analysis-simulation-variables-dialog">
<title>Variables tab in simulation dialog box</title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/analysis-simulation-variables-dialog.png" format="PNG" />
</imageobject>
<textobject>
<para>
Screenshot of the simulation dialog with input and output variables filled in for the newsvendor example.
</para>
</textobject>
</mediaobject>
</screenshot>
</figure>
<para>Input variables are the cells which hold the functions based on random numbers of the type described in Section A.14. In this case, they are the cells B17 and B18 in the Profit worksheet, which hold the rand() function. Later, when the quantity purchased is a parameter set by the <function>SIMTABLE</function> function, cell B16 which holds the purchase quantity will be added to the range of input variables.</para>
<para>Output variables are the results of interest, or the dependent variable. In this case, the dependent variables are the demand and the profit, which are in cells B20 and B21.</para>
<para>The next tab is the Options tab . There are four settings in the options as shown in <xref linkend="fig-analysis-simulation-options-dialog" />. </para>
<figure id="fig-analysis-simulation-options-dialog">
<title>Options tab in Simulation dialog box for newsvendor simulation example</title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/analysis-simulation-options-dialog.png" format="PNG" />
</imageobject>
<textobject>
<para>
Options tab in simulations dialog showing Rounds and number of iterations.
</para>
</textobject>
</mediaobject>
</screenshot>
</figure>
<para>The second pair of options are the number of iterations and the Max time. In a simulation, each iteration is the equivalent of a sample. A sample from a random distribution is taken for each of the input values (as specified in the Variables tab) and the resulting output value(s). The more iterations, the better the estimate of the output value. However, this also takes more time to run. A Max time value is specified in seconds where the simulation will end without output if an individual simulation takes longer than the Max time allotted. If this occurs (see <xref linkend="fig-analysis-simulation-maximumtime-dialog" />), the options are to either increase the Max time value, or decrease the number of iterations. A more drastic option is to change the model so that fewer calculations or samples of random numbers need to be made.</para>
<figure id="fig-analysis-simulation-maximumtime-dialog">
<title>Maximum time for simulation exceeded message box</title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/analysis-simulation-maximumtime-dialog.png" format="PNG" />
</imageobject>
<textobject>
<para>
Dialog to warn that the maximum time for the simulation was exceeded.
</para>
</textobject>
</mediaobject>
</screenshot>
</figure>
<para>The next tab is the Summary. There are two boxes in this tab, the Simulation Summary and the Summary of results (see <xref linkend="fig-analysis-simulation-summary-dialog" />). In simulation summary, there is a description of the simulation parameters.</para>
<note>
<para>Due to the random nature of the simulation, the output may vary between simulation runs). </para>
</note>
<figure id="fig-analysis-simulation-summary-dialog">
<title>Summary tab for simulation tool</title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/analysis-simulation-summary-dialog.png" format="PNG" />
</imageobject>
<textobject>
<para>
Simulation summary tab inside of simulation dialog box including summary of simulation options and summary of results.
</para>
</textobject>
</mediaobject>
</screenshot>
</figure>
<itemizedlist>
<listitem><para>Simulations: Number of rounds as determined in the Simulation Options box.
</para></listitem>
<listitem><para>
Iterations: Number of iterations in a single simulation round.
</para></listitem>
<listitem><para>
# input variables: Number of random numbers sampled for each iteration.
</para></listitem>
<listitem><para># output variables: Number of outputs recorded for simulation</para></listitem>
<listitem><para>Runtime: Runtime of simulations in seconds.</para></listitem>
<listitem><para>Run on: Date and time simulation was run.</para></listitem>
</itemizedlist>
<para>In the summary of results window, there are summary statistics for each round of the simulation. If multiple rounds were done, the results of each round can be browsed by using the 'Prev. Sim.' and 'Next Sim.' buttons below the Summary of results box. For each output and input variable, the summary shows the Min, Average and the Max value across the iterations for that round of the simulation. Note that for the input variables, this shows the random number that is the average, max and min. If the statistics on intermediate values, such as a cost distribution, was desired, these intermediate values should be added to the list of output variables.</para>
<para>The last tab is labeled 'Output'. This tab identifies the location where the output table will be generated. There are two sets of options, first the Output Placement then Output Formatting as shown in <xref linkend="fig-analysis-simulation-output-dialog" />.</para>
<figure id="fig-analysis-simulation-output-dialog">
<title>Output options tab for simulation</title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/analysis-simulation-output-dialog.png" format="PNG" />
</imageobject>
<textobject>
<para>
Output tab inside of simulation dialog box showing options for output placement and output formatting.
</para>
</textobject>
</mediaobject>
</screenshot>
</figure>
<para>The default output placement is 'New sheet'. This will create a new sheet in the Gnumeric workbook labeled 'Simulation Report (1)', where '1' can be replaced with another number if a tab labeled 'Simulation Report (1)' already exists. The option 'New workbook' creates a Gnumeric workbook named 'Book2.gnumeric' with a tab labeled 'Simulation Report.'</para>
<para>The third option is to embed the output table into an existing worksheet. This is done by specifying the 'Output range'. Note that the output range must be large enough to include the entire table, including heading information. For a single round this requires 11 rows and 16 columns. For example, the range Profit!A24:P35 would contain the statistics for one round with the three input variables and two output variables. As input and output variables change, or the number of rounds of the simulation change, the number of rows required will change.</para>
<para>For output formatting, their are four options.</para>
<itemizedlist>
<listitem><para>'Autofit columns' automatically makes each column long enough to include the largest entry in that column. Note that column 'A' in the resulting spreadsheet used to save run information such as date and time and is kept narrow.</para></listitem>
<listitem><para>'Clear output range' is in effect if the Output Placement option chosen is Output range. It clears the selected cells in the spreadsheet before putting the output table in its place.</para></listitem>
<listitem><para>'Retain output range formatting' retains formatting for cells such as number formatting.</para></listitem>
<listitem><para>'Retain output range comments' retains comments that have been placed in output cells. This is most useful when the input and output variables remained the same.</para></listitem>
</itemizedlist>
</sect2>
<sect2 id="sect-advanced-analysis-simulation-output">
<title>Simulation output</title>
<para>The simulation output provides statistics on the output and input variables for each round. The statistics are calculated over the iterations in a single round of the simulation. These statistics for each variable are:</para>
<itemizedlist>
<listitem><para>Variable type and name - input variables are labeled as '(Input)'.</para></listitem>
<listitem><para>Min – Minimum value of variable among iterations of round.</para></listitem>
<listitem><para>Mean – Arithmetic mean of variable among iterations of round.</para></listitem>
<listitem><para>Max – Maximum value of variable among all iterations of round.</para></listitem>
<listitem><para>Median – Median of variable among iterations of round.</para></listitem>
<listitem><para>Mode – Mode value among iterations of round. For the input variable, this will be “#N/A”.</para></listitem>
<listitem><para>Std. Dev. - Standard deviation of the variable.</para></listitem>
<listitem><para>Variance – Second moment of variable.</para></listitem>
<listitem><para>Skewness - Third moment of variable.</para></listitem>
<listitem><para>Kurtosis – Fourth moment of variable.</para></listitem>
<listitem><para>Range – Difference between min and max of variable among iterations of the round.</para></listitem>
<listitem><para>Count – Number of iterations in round.</para></listitem>
<listitem><para>Confidence (95%) - 95% confidence interval of value, centered on mean.</para></listitem>
<listitem><para>Lower Limit (95%) - Lower limit of 95% confidence interval of the value, centered on the mean.</para></listitem>
<listitem><para>Upper Limit (95%) - Upper limit of 95% confidence interval of the value, centered on the mean.</para></listitem>
</itemizedlist>
<figure id="fig-analysis-simulation-report-screen">
<title>Simulation output example</title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/analysis-simulation-report-screen.png" format="PNG" />
</imageobject>
<textobject>
<para>
Simulation output spreadsheet including input and output variable statistics.
</para>
</textobject>
</mediaobject>
</screenshot>
</figure>
<para>The output will include a heading, then a table for each round of the simulation. Judicious choice of output variables will also include any intermediate values of interest in the simulation in this table. Each row of the output table has statistics of the values of a variable over the iterations of the simulation as shown in <xref linkend="fig-analysis-simulation-report-screen"/>. </para>
<para>The output will be of the input variables and the output variables that were variables tab of the Simulation window . For the input variables, the output will be the statistics of the random variable used in modeling the input variables. For the output variables, the statistics will be of the output variable. These statistics, in particular the standard deviation and confidence interval, should be examined to ensure the simulation was at a precision adequate for the purpose. Some notes on how to use these statistics for refining the simulation design can be found in <xref linkend="sect-advanced-analysis-simulation-iterations"/>. </para>
</sect2>
<sect2 id="sect-advanced-analysis-simulation-simtable">
<title>Using SIMTABLE</title>
<para>The <function>SIMTABLE</function> function is intended to change a variable in the simulation so that each round of the simulation can be used to evaluate a different scenario. This automates the use of simulation for what-if questions or to create a set of possible outcomes to a situation.</para>
<para>In this example, we will use the <function>SIMTABLE</function> function to find the optimal quantity of newspapers to buy. For the purchase quantity in our spreadsheet, we will replace '50' with the following formula in Profit!B16:</para>
<programlisting>
Profit!B16 = SIMTABLE(50,60,70,80,90)
</programlisting>
<para>Each entry in the list of the <function>SIMTABLE</function> arguments is a value that will be used for the purchased quantity. Each entry corresponds to one round of simulation, as used in <xref linkend="fig-analysis-simulation-options-dialog" />. In this example there are 5 entries to the SIMTABLE list, so '5' will be entered into the 'Last Round #' option in the Options tab of the Simulation dialog. </para>
<figure id="fig-analysis-simulation-simtable-screen">
<title>Simulation output example using SIMTABLE and several rounds </title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/analysis-simulation-simtable-screen.png" format="PNG" />
</imageobject>
<textobject>
<para>
Simulation output example using SIMTABLE and several rounds.
</para>
</textobject>
</mediaobject>
</screenshot>
</figure>
<para>When this simulation is run with 5 rounds, the summary of results will have one entry for each round, with each round using a different entry from the <function>SIMTABLE</function> function for the purchase quantity. The results for the various rounds can be previewed using the 'Prev. Sim.' and 'Next Sim.' buttons. The output also has one table for each round of the simulation.</para>
<para>As seen in <xref linkend="fig-analysis-simulation-simtable-screen" />, each value in the original SIMTABLE statement corresponds to a simulation round, with the Purchase Quantity taking on the value from the SIMTABLE list. The analyst can then record the Profit statistics (mean, variance, skewness, kurtosis, 95% confidence intervals) and determine if the simulation results are of sufficient resolution for the analysts purposes.</para>
<para>The use of SIMTABLE to change parameters within the simulation provides a convenient method to do what-if analysis, and analyze the results as a whole.</para>
</sect2>
<sect2 id="sect-advanced-analysis-simulation-iterations">
<title>Determining the number of iterations</title>
<para>In simulation, one major question is how many iterations are needed to reach a chosen level of precision in the results. Simulation as a tool provides an approximation of the actual relationship between the input and output variables. The precision of the approximation is based on the number of iterations of the simulation done. More iterations in the sample lead to greater precision. But the relationship between iterations and precision depends on the relationship between the variables in the precision. In addition, the analyst must decide which output variable is the variable of interest, and what degree of precision is required. The next step is to determine a sufficiently large number of iterations
<inlineequation>
<alt>$R$</alt>
<mathphrase>R</mathphrase>
</inlineequation>
be used to satisfy:
<informalequation>
<alt>\[ P\left(\left|\left(\widehat{\Theta}-\Theta\right)\right| \leq \epsilon \right) \geq 1-\alpha \]</alt>
<mediaobject>
<imageobject>
<imagedata fileref="figures/analysis-simulation-confidence-interval-equation.png"/>
</imageobject>
<textobject>
<para>The probability that the difference between the actual mean and the sample mean is less than epsilon is 1 - alpha.</para>
</textobject>
</mediaobject>
</informalequation>
</para>
<para>Where
<inlineequation>
<alt>$\widehat{\Theta}$ </alt>
<mathphrase>&THgr;-hat</mathphrase>
</inlineequation> is the estimate of the mean,
<inlineequation>
<alt>$\Theta$</alt>
<mathphrase>&THgr;</mathphrase>
</inlineequation> is the actual mean, &egr; is the specified error, and (1-&agr;) is the probability that the estimate is within &egr; of the actual value (i.e. the (1-&agr;) confidence interval).
Common values of (1-&agr;) are 95% and 99%.
The Simulation Report from Gnumeric includes values for the 95% confidence interval as shown in <xref linkend="fig-analysis-simulation-report-screen" />.
</para>
<para>The general procedure is as follows:<footnote>
<para>Adapted from Banks et. al. Discrete-Event System Simulation, 3rd Edition, pp. 414-416.</para>
</footnote>
</para>
<orderedlist>
<listitem><para>Run simulation for a sample of
<inlineequation>
<alt>$R_0$</alt>
<mathphrase> R<subscript>0</subscript></mathphrase>
</inlineequation> iterations.
The default value in Gnumeric is 1000, set in the options tab of the Simulation menu, <xref linkend="fig-analysis-simulation-options-dialog"/>. </para></listitem>
<listitem><para>Take the sample variance
<inlineequation>
<alt>$S_0^2$</alt>
<mathphrase> S<subscript>0</subscript><superscript>2</superscript></mathphrase>
</inlineequation> from the simulation output spreadsheet and determine the sample standard deviation
<inlineequation>
<alt>$S_0$</alt>
<mathphrase> S<subscript>0</subscript></mathphrase>
</inlineequation>
(see <xref linkend="fig-analysis-simulation-report-screen" />). </para></listitem>
<listitem><para>Using
<inlineequation>
<alt>$z_{\alpha/2}$</alt>
<mathphrase>z<subscript>&agr;/2</subscript></mathphrase>
</inlineequation>
as the z-value of the
<inlineequation>
<alt>$(1-(\alpha/2))$</alt>
<mathphrase>(1-(&agr;/2))</mathphrase>
</inlineequation> percentile of the standard normal distribution, set the initial estimate of the number of iterations required as the smallest integer
<inlineequation>
<alt> $R$</alt>
<mathphrase>R</mathphrase>
</inlineequation>
such that
<equation><title>Iterations required for simulation</title>
<alt> $R \geq \left(\frac{z_{\alpha/2}S_0}{\epsilon}\right)^2$</alt>
<mediaobject>
<imageobject>
<imagedata fileref="figures/analysis-simulation-interations-equation1.png" format="PNG"/>
</imageobject>
<textobject>
<para> R is greater than equal to z of alpha divided by 2, times standard deviation, divided by epsilon. Quantity squared.
</para>
</textobject>
</mediaobject>
</equation>. Note that if
<inlineequation>
<alt> $R_0$ </alt>
<mathphrase>R<subscript>0</subscript></mathphrase>
</inlineequation> is small, it would be more appropriate to use the student's t-distribution of
<inlineequation>
<alt> $t_{\alpha/2, R_0}$ </alt>
<mathphrase>t<subscript>&agr;/2, R<subscript>0</subscript></subscript></mathphrase>
</inlineequation> instead of
<inlineequation>
<alt> $z_{\alpha/2}$</alt>
<mathphrase>z<subscript>&agr;/2</subscript></mathphrase>
</inlineequation>.
</para></listitem>
</orderedlist>
<para>In this example, to estimate the profit to within
<inlineequation>
<alt>$\epsilon = 0.05$</alt>
<mathphrase>&egr;=0.05</mathphrase>
</inlineequation>
, first run the simulation with 1000 iterations and a purchase quantity of 50 results in the following </para>
<para>
<informaltable frame="all">
<tgroup cols="4"><tbody>
<row>
<entry>
<para/>
</entry>
<entry>
<para>Mean</para>
</entry>
<entry>
<para>Variance</para>
</entry>
<entry>
<para>Confidence (95%)</para>
</entry>
</row>
<row>
<entry>
<para>Demand QUANTITY</para>
</entry>
<entry>
<para>59.19</para>
</entry>
<entry>
<para>152.4</para>
</entry>
<entry>
<para>0.64</para>
</entry>
</row>
<row>
<entry>
<para>Profit QUANTITY</para>
</entry>
<entry>
<para>7.85</para>
</entry>
<entry>
<para>2.51</para>
</entry>
<entry>
<para>0.08</para>
</entry>
</row></tbody></tgroup>
</informaltable>
</para>
<para/>
<para>Taking the variance of the table, and setting
<inlineequation>
<alt>$\epsilon = 0.05$</alt>
<mathphrase>&egr;=0.05</mathphrase>
</inlineequation>
and
<inlineequation>
<alt>$\alpha = 0.05$</alt>
<mathphrase>&agr;=0.05</mathphrase>
</inlineequation>, lookup
<inlineequation>
<alt>$z_{\alpha/2}$</alt>
<mathphrase>z<subscript>&agr;/2</subscript></mathphrase>
</inlineequation> from a standard normal table.
<inlineequation>
<alt>$z_{\alpha/2}=1.96$</alt>
<mathphrase>z<subscript>&agr;/2=1.96</subscript></mathphrase>
</inlineequation> so we have</para>
<para/>
<para>
<informalequation>
<alt> $R \geq \left(\frac{1.96 \times \sqrt{2.51}}{0.05}\right)^2 \geq 3856.8$</alt>
<graphic fileref="figures/analysis-simulation-interations-equation2.png"/>
</informalequation>.
</para>
<para/>
<para>Therefore, the minimum number of iterations is 3857. The simulation can then be re-run with 3857 iterations to create a 95% c.i for profit where
<inlineequation>
<alt>$\epsilon \leq 0.05$ </alt>
<mathphrase>&egr; <=0.05</mathphrase>
</inlineequation>
In this example with 3857 iterations, we get the following Simulation Report table: </para>
<para/>
<informaltable frame="all">
<tgroup cols="4"><tbody>
<row>
<entry>
<para/>
</entry>
<entry>
<para>Mean</para>
</entry>
<entry>
<para>Variance</para>
</entry>
<entry>
<para>Confidence (95%)</para>
</entry>
</row>
<row>
<entry>
<para>Demand QUANTITY</para>
</entry>
<entry>
<para>59.11</para>
</entry>
<entry>
<para>163.9</para>
</entry>
<entry>
<para>0.34</para>
</entry>
</row>
<row>
<entry>
<para>Profit QUANTITY</para>
</entry>
<entry>
<para>7.72</para>
</entry>
<entry>
<para>2.88</para>
</entry>
<entry>
<para>0.04</para>
</entry>
</row></tbody></tgroup>
</informaltable>
<para/>
<para>As expected, the 95% Confidence interval for Profit is less than 0.05. For the newsvendor example, the next step would be to look at the confidence intervals of the profit for all values of purchase quantity, and verify that this confidence interval is adequate for the decision to be made. </para>
<para/>
<para/></sect2>
</sect1>
|
