CBBNGING THE BASIS OF THE SUBPROBLEM
3. Computational Experience
MULPS. An experimental code, named MULPS (Multi-period Linear Programming System), f o r t h e veakly coupled l i n e a r programs was v r i t t e n i n FORTRAN using
The Results. The number of c y c l e s r e q u i r e d f o r o p t i m a l i t y and t h e CPU comput- ing t i m e a r e summarized i n Table 2. Table 3 i l l u s t r a t e s t h e degree of o p t i m a l i t y a t every cycle. I n Table 4 t h e CPU computing time up t o every c y c l e throughout the o p t i m i z a t i o n i s d e s c r i b e d in d e t a i l f o r Problem MAl. I n Figure 1 t h e t o t a l CPU computing time and t h a t p e r c y c l e a r e p l o t t e d f o r t h e corresponding number of p e r i o d s f o r t h e s i x problems G3A
-
G6B. Notice t h a t those problems have sub- problems of t h e same dimension, b u t a d i f f e r e n t number of p e r i o d s .I n Table 5 we compare both t h e CPU computing time and t h e amount of s t o r a g e r e q u i r e d in t h e system w i t h t h o s e by t h e d i r e c t simplex method(SEX0P) f o r MAl.
l'he ConcZusia. Prom Table 2 we n o t e t h a t t h e ' number of c y c l e s r e q u i r e d f o r o p t i m a l i t y is almost e q u a l t o , o r l e s s than t h e number of periods. For t h e purpose of comparing i t w i t h t h a t by t h e algorithms of column-generation scheme, we s h a l l r e f e r t o t h e e a r l i e r r e s u l t s of Glassey's a l g o r i t h m [ l o ] and of Ro and Manne's one [14].
I n Glassey [ l o ] t h e computational r e s u l t f o r almost t h e same model a s MAL derived from [7] was presented. The number of c y c l e s was r e p o r t e d t o be 31, which s h w s t o b e f i v e times l a r g e r than t h a t f o r W. I n Ho and Manne [14] t h e
two t e s t problems coded SCSOA and SC50B have 6 p e r i o d s and t h e dimensions a r e r a t h e r s m a l l e r than R1 and R 2 among our problems. The number of c y c l e s was r e p o r t e d t o be between 25 and 35, vhcih s h w s t o be s i x o r e i g h t times l a r g e r than o u r s . However, i t i s r e p o r t e d in t h e r e c e n t comparative s t u d y o f t h e i r method, Ho and Loute [13], t h a t t h e number of c y c l e s is g r e a t l y reduced. We could n o t t r a c e t h e same problems in t h e p r e s e n t experiment.
From Table 3 we n o t e t h a t t h e process of convergence i s f a i r l y f i n e and the "long t a i l " of convergence s c a r c e l y occurs. The degree of o p t i m a l i t y
a t t a i n s a very high p o s i t i o n a t a r e l a t i v e l y e a r l y c o o r d i n a t i o n c y c l e . The degree a t t h e f i r s t c o o r d i n a t i o n i s beyond 70% i n almost all c a s e s such t h a t t h e initial v a l u e s f o r t h e y-variables make t h e problem f e a s i b l e a t t h e i n i t i a l s t a g e . T h i s f e a t u r e s e e m t o b e s i g n i f i c a n t in a p r a c t i c a l u s e , and a near- optimal s t r a t e g y may work e f f e c t i v e l y .
From Table 4 we n o t e t h a t t h e CPU computing time p e r c y c l e tends t o decrease s l i g h t l y . A l l subproblems a r e optimized b e f o r e s o l v i n g t h e f i r s t coor- d i n a t i o n problem. Therefore, much more time i s consumed a t t h e f i r s t cycle.
Kxcept some s p e c i a l o c c a s i o n s , s o l v i n g t h e s u b p r o b l e m a r e skipped and t h e d i r e c t i o n - f i n d i n g p r o b l e m a r e solved only f o r t h e non-optimal blocks. We have obsermd s o f a r t h a t t h e number of non-optimal blocka g r a d u a l l y d e c r e a s e s according a s t h e c o o r d i n a t i o n proceeds.
Table 5 shovs t h a t t h e MULPS i s f o u r times f a s t e r than t h e d i r e c t method concerning t h e computing t i m e , and r e q u i r e s only a h a l f of memory f o r t h e d i r e c t simplex method in t h e case of M(U.
Acknowledgment
The a u t h o r wishes t o thank Professor Roy E . Marsten, The U n i v e r s i t y of Arizona f o r r e l e a s i n g the SEXOP program t o him and f o r ccmuuents and s u g g e s t i o n s vhich have r e s u l t s i n a n improved v e r s i o n of t h e paper, and a l s o P r o f e s s o r Leon S. Lasdon, The U n i v e r s i t y of Texas a t Austin, f o r t h e s u g g e s t i o n on using t h e SEXDP program w h i l e t h e a u t h o r was a t Case Western Reseme U n i v e r s i t y during t h e l a t t e r h a l f of 1974. The a u t h o r i s much indebted t o Messrs. M.Morita and H.Nishimoto, h i s s t u d e n t s a t Kobe U n i v e r s i t y of Commerce, f o r h e l p i n g t o code t h e MULPS and t o g e n e r a t e t h e t e s t problems.
REFERENCES
T. Aonuma, "A Nested Multi-level Planning Approach to a Multi-period Linear Planning Model", Working Paper No.35, Kobe University of Commerce
(Kobe, January 1977).
11
,
"A Two-level Algorithm for Two-stage Linear Programs", J o u . of Operations Reaaarch Society of Japan 21, 171-
187 (1978).I1
,
"An Extension of the Tvo-level Algorithm to Optimizing Weakly Coupled Dynamic Linear Syeteme", Working Paper No.41, Kobe University of Commerce (Kobe, March 1978).11
,
"Computational Experience in Using a Decompoeition Algorithm (MULPS) for Multi-period Dynamic Linear Programs", Working Paper No.44, Kobe University of Commerce (Kobe, July 1978).T.Aonuma and N.Nakahara, SEXOP/HITAC 8250
-
User's Manual-
(in Japanese), Research Report No.9, The Computer Center, Kobe University of Connnerce(Kobe, December 1977).
E.M.L.Beale,"The Simplex Method for Structured Linear Programming Problems", Recent Advmrcerr i n &athematical A.ogxwmringf e d . R . Gxuves and P . Wolfel, McGraw-Bill, New York, 1963, 133-148.
C.R.Blitzer, B.Cetin, and A.S.Manne, "Dynamic Five-Sector Model for Turkey, 1967-82". Memorandum No.70, Research Center in Economic Growth, Stanford University (April 1969).
A.M.Geoffrion, "Primal Resource-directive Approaches for Optimizing Non- linear Decomposable Systems", Gpnrr. Rea. 18, 375-403 (1970).
D.C.Gilrnore and R.E.Gomory, "A Linear Programing Approach to the Cutting Stock Problem Part 11", Opns. Rea. 11, 863-888(1963).
[lo] C.R.Glassey, "Nested Decomposition and Multi-stage Linear Programs", M g n t t . Sci. 20. 282-292(1973)
.
[ll] R.C.Grinold, "Steepest Ascent for Large Scale Linear Programs". SLIM Revieu 14, 447-467(1972).
[I21 G.H.Heal, Theoq o f Economic P h i n g , North-Holland, Amsterdam, 1973.
[13] J.A.Ho and E.Loute, "A Comparative Study of Tvo Methods for Staircase Linear Programs", mimeograph BKL-23156, August 1977.
[14] J.H.Ho and A.S.Hanne, "Neated Decomposition for Dynamic Models", k t h . P m g . 6, 121-140(1974)
.
[ U ] R.E.Marsten, h e r 'a
k m u t
for a x o P , Sloan School of Management, Massachusetts Institute of Technology (Cambridge, Februray 1974).[16] R.E.Marsten and F.Shepardson, " A Double Basis Simplex Method for Linear Program with Complicating Variables", Working Paper. College of Business and Public Administration, University of Arizona (Tucson, August 1978).
[17] T.Aonuma, H.Morita and E.Nishimoto, User's M a n u a l for MULPS/EITAC 8250
(in Japanese), Research Report No.13, The Computer Center, Kobe University of Connuerce(Kobe, December 1978).
[18] S.I.Gass, "The Dualplex Method for Large-Scale Linear Programs", ORC66-15, Operations Research Center, University of California (Berkeley,June 1966).
1191 C.Winkler, "Basis Factorization for Block-Angular Linear Programs: Unified Theory of Partitioning and Decomposition Using the Simplex Method1',RR-74-22, IIASA (Schloss Laxenburg, November 1974)
.
[20] T.AonumaS1'A Decomposition-Coordination Approach to Large-Scale Linear Pro- gramming Models
-
An Aspect of Applications of Aonume's Decomposition Method- ",
forthcoming.Problem Period
- -
Gilmore-Gomory
G3A 3
G3B 3
G4A 4
G5A 5
G6A 6
G6B 6
TABLE 1
Dimensions of Test Problems E n t i r e Problem
Rova Col.'a* %Density
%**
-
Subproblem
Rovs Col.'s %Density
- -
m e ' s Model
MAl 6 116 266 1.8 26 19-20 37-43 9.7-11.0
Refinery Prod.
RlA 6 60 186 2.7 30 10 26 20.0
RZB 6 Only t h e l i n k i n g matrix is d i f f e r e n t from R l A above.
R2A 6 90 198 2.5 30 15 28 15.0
*
Includea s l a c k v a r i a b l e a .**
L.V. denotes t h e number of l i n k i n g v a r i a b l e s .TABLE 2
Number of Cycles and CPU Computing Time Problem Periods Number of Cycles CPU Computing Time
- -
min. sec.
G3A 3 4 (o)* 5 40
*
A parenthesized figure denotes the number of times returned to Step 1 in Step 6. In Step 1 the subproblem is reopthized.TABLE 3
Degree of Optimality and Number of Cycles N d e r of Coordination Cycle
Problem 0 1 2 3 4 5 6 7 8
G3A
2
94.7 96.2 98.0 m.2G3B
2
0 18.5 71.5 71.5 E Z C4A2
89.6 93.1 97.3 97.5E Z
G5A
2
87.7 88.6 100- 100--
100%G6A
2
89.7 95.5 98.6 99.0 99.5 99.5 99.8E X
G6B2
76.6 87.6 96.6 97.3 99.3E Z
M A l
* -
0 32.6 71.9 87.2 96.8 X.2glA
* * * -
0E Z
ElB
* -
0 97.1 -%R2A
* * -
0 69.2 84.0=I
Note: An asterisk denotes that a feasible solutiol is not found yet.
2
denotes feasibility attained for the first time. 100 denotes a near 100.TABLE 4
CPU Computing Time up t o Every Cycle f o r MA1
Up t o Optimization Number 'of Cycle
of S u b p r o b . ' ~ 1 2 3 4 5 6
min. s e c . Computing
Time 2 37 4 49 8 30 11 27 1 4 1 7 1 7 07 20 00
P e r Cycle
-
4 49 3 41 2 57 2 50 2 50 2 53TABLE 5
Conzparison o f MUUS w i t h Direct Simplex f i t h o d
CODE MULPS SEXOP SEXOP /MULPS
PROBLXM CYCLES TIME ITER- TPIE RATIO I N
No. P e r i o d s S i z e ATIONS TPIE
M4IN
STORAGE USED
1 ) The t i m e s r e p o r t e d are in CPU seconds on a FACOM M-160 (comparable t o IBM 370/148).
i i ) The FACOM M-160 h a s 768 KB real memory and 1 6 MB v i r t u a l memory, which i s under OS IV/X8 (comparable t o IBM OS/VSZ ) . The FORTRAN IV HE compiler with OPTIMIZE(2) is used throughout (comparable t o IBM FORTX compiler v i t h OPT = 2 ) .
r: Per Cycle
/
Per Cycle min.
3 1 5 6 7
Number of Periods
P i g . 1 CPU Computing Time and Number of Periods f o r Problems G3A-G6B.