lntetnattonal Journal ofProdu¢:ton Economws, 23 ( 1991 ) 165-174 16 5 Elsevier

The use of the PROMETHEE method in the location choice of a production system

l~..i~ Pave6 and Zoran Babl6

Facuttv of Econom,cs, Radovanora 13, 58000 Spht, ~ugoslavta

Abstract

The choice of location is often one of the most important aeclstons usually taken m the proceoure of pudding any produchon s~s~e'-~ I his paper considers the problem c,f 13catlon cho:cc for ~uzh producuon syslem~ v~h~re the basic and additional locatior factors can be identified

The basic |ocatlon faO-,.~ ar~ transportation costs production costs and duratmn of iransport The ad~ nonal factors ,~re oonleneck time, budding costs, mfrastruclu.¢, cost, .;. oat cos:g, we~th-r coudihotls, expansion poss,~,hty and trans- portatmn posstbdmes

ne importance of the basic location faOots will be solved by the use of a parametric approach fo~ tim mult,cr,ter,al transportatmn problem and tn this way the set of all efficient solutions ~,,.11 be found 2. tier that the locatmn choice t amo,ig fliese efficient soluUons~ will be foisted by the use of the PRGMETHEE method wh,ch takes into account the above mer,- t~oned ad~t'.tmnal location factors

L Introduction

Loc:,tton choice is one o f the mos t impor tan t d e o s l o n s made in the process o f p repa rauon for bu 'ddmg any k m a o f p r o d u c t m n system The im- por tance o f such d e o s l o n s hes primari ly m the fact that business costs and econom~r depend on iocauon to a v e w great extent Besides the cost d e p e ~ d e . . . . . . . . cat:c~, w~:e~ can by them~e!-e~_ be varying and unpredictable , there are o ther cir- , ~ u ~ t a n e e s which can be cons idered and which cannot always be accurately de t e rmined or quan- tified. In brief, this dec tsmn is affected b:¢ a num- ber of c i rcumstances whose inf luence is el'ten of mdef imte s trength and direct ion, subject to changes m the course of t ime. And, above all, it is difficult to reduce this inf luence to a c o m m o n denomina to r in the sense o f an accurate state- ment 1ol me cons idered Iocat,on ~olutlon.

In the nrocess of selectmn tt ,s necessary, first and foremo~" ~o identify the set o fmf luen tmt fac- tors releva ,t to the locahon chot_ce After that one has to es t imate the intensity and d~cectton of m- fluentml factors from the po in t o f vmw of their

effect on_ the location choice Id.cndl~catmn and es t ima tmn o f mfluentml factors are the condi- t ions for the evaluation of possible so luhon var- iants and for the choice of the most acceptable o~tes Whei~ solving the ia,~.er w.oblem ,t is nec- escary to use the appropr ia te me_hods and models m dependence on the kind ofmflue~t la l factors and c i rcumstances c h a r a c t e r l s t w fo~ a pa,'*,_o_,:,ar product ion sysLem

The p roduc tmn system considered m this pa- per ts character ized by tl-.e mfluentml basw and the a d d m o n a l location factc, rs. f h e tmsm loca- t ion factors are t ranspor ta t .~n costs, product ion costs and du ra tmn o f t ra~sport The addt tmnal factors are bott leneck tm:~, b~dd',ng o ~*~

aS,tu~.tlll~., costs , lobt_,tJts costs~ weather condi- tions, expansion poss~bfl,ty and translaortat~on poss~bflmes

The purpose of this p roduc tmn system ~s the product ion of a homogenous product (milk and similar products ) d l s tnbu :ed to a number of des- tmat,.ons and it :s supphed by raw matcr'.a!s cor , 'mg f~:.m a n , r n b e r of sources As the raw m , ~,:at is completely conta ined ,n the final

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166

product, It is irrelevant whether the transported goods are raw materials or final pr~duc~s Tl,e model of Iocatmn choice starts from the possdnl- lty to build one, tw,, or morc t:n,t~ o f the loiodat tlon syste,':, or~ the considered !oc~.,~on, thu~- the model 0£ !ocatmn choice is based on the assump- hon of tt~.e existence o t p dffferen* varlant~ for the location of r units o f the product ion system.

Although ~he assumptions Ior the apphcat lon of the considered model are very srec~fic this model can with some adjus tmems still have broader aophcat!en po~;lblhtles We find tLat 1" can be described as a mult~step and mu!t lcnter la l transportation problem and m this way the set o f all efficient solutions is to be found Aftcl that, the !ocat,on choice ~s to oe found by ~he u~e of the P R O M E T H E L method that takes into ac- could the aoove mentmned addi t ional location factors

system

Numerous and varied mfluent,,al factors and varied s~_tuatlons have affected the deve lopment of a number of mcth, ,ds and models for the Io- cation choice of a product ion system

Mathemat:cai models in mu!t,.cr,_tenal dec~ slon making are used to solve this kind of prob- lem, especially the methods for multlCrlteria! analysis from the family Electre and Proine:hee. The advantages e f these methods a ,e they en- able the selection o f one alternative from the set o f all alternatives, or at least one subset o f all possible a l tenmuves that are " the best"; they do not limit the number of criteria which .-an be used to rank the , ' t emauves , and finally daey allow tlzc poss~o'.,hty ot quantna t ive ana ~ua: ,-t~ve expression o f those criteria which can ha -e a dlf-

q he featarcz of the considered problem arc such that it is not possible to use any of~he men- tinned mode~s or rescz:cb, methods without hni- l tatmns We find that it is pt~ssible to solve the problem adequately by a combined application o~" . . . . . ~ ~,~" ~i~e methods which will objectively evaluate the Influence of a particular location factor which can be quanti tat ively expressed (m th~s case such factors are the transportation costs, plCflUCtlOn C¢oS~S ~n,t t ransoortat lon t~me on the

route source - pmdue tmn ws tem - ,;¢s*matlon) and some ot the methods which will be used for multlcrlterlal ranking of a~ternatlves based on the ,;stH~latlon Of relatwe ;r ~cortanee g~ve,,, by the decision maker

2 1 Mtdtwt tte~ tal It ansporta!ton, pt obte~,.

First it is necessary. ;o state the value of partlc- uiar measure c f !ocat!o~ criteria for every, var- .ant of the producUon system locat,.on, assuming that there are p different variants for the location of r units o f the producUon system and m sources ,,, ~upp!) ce.:x~e~ A, ,'t--- Io ,m), and n d~=~.ma- T,,~n, or demand centres E U = i , ,n). 5~.dc_-. it is assumed that on the route source-dest ina- tion t~e t ransported goods are held for some t ime m one of the product ion system units for ,b,. pur- pose o f processing. Whe~ tb* de~er,t~., ¢'MeIIIn- , c , ~s .ar r led or.q, all o t h o location factors are lmroduced into me a-mi~,~ . . . . . . . . . . . u" .-- factors is o f a quahtat~ve nature The goal of the analys~s ~s to state the total "va lue" o f part icular variants and th,e~_r ran~ of preference.

In the concrete model the measure of location op t imum are ~he t ransporta tmn costa, the pro- duct ion costs and the t ransportat ion t ime, where from we can define the three objecUve funcUons, l.e the e . . . . i. . ~ . ~ ,h , I ~ e l t tr lCtlOfl.3 w h t ~ t t m t n h ~ . $ ~ u t e t o , ~ . ~ , . n o -

ponat ion costs, the total production costs and the total tr" asportation t ime (or the ~ongest trans- porta ' ~on t lme) . But, as there is a posslbdlty o f summing up the transportat lon costs and the prodnct lon costs, and e~entuaily other costs which are .hffc~e,L fvi a.p,iJht.ulm va~ian~., tb~ pioblem is solved by the use of the two objee t r r t functions, the firs~ one mm~mazing the total d~- pendent costs and the second one mimmizmg the determined t ime dimension.

The objective function which includes the de- pendent costs can be showe as.

+ ÷ + ,, ( ' = e l y ` 4- t~. v , 4- ~ e . v

/ = l k = l /~= I j = 1 & = l j = l

where x,h = the quant i ty of goods t ransported t rom

the source A, to the proddct~on sy~:em Pt;

,No raret~lenee of a over b

:=g 1 Sh .pe of the preference function

i I p j d}

~nd=ff- ,increasing I S t r o n o pre fe rence erence ~preference l of a over b

6 ~7

I p~id)

I o I d !

/

(d)

0 t , , ~-,,,,,-..,- - - - - - ~ - - -~- d q P

4b)

(c)

I p ~ Id)

I

o i : d

P

U=(d}

(el I I Pj(d)

..... ? _ / q p

d ~ 2 b 2

1 I p j ( d J : l - e

b

F,g 2 Six generahsed cnter , a (a ) t~sual, , , t enon (b) Qua~ criterion, (c) Criterion w~th l ,eear pieference (d) L~ve.l critel,ou, (e) C r n e n o n wllh indifference area ( f ) Causs lan c r d e n s m

T A B L k 1

i" P2 B~ B2 Ba B4 B5

. , , , o - M M M M 0 ,1) 1 0 E 6 M M M M 0 206900

3 342 3 1 M M M M 0 4 1 6 t 0 M M M M 0 108500

J 338 3 2 M M M M 0 33 900

~ ~ 6 2 0 M M M M 0

3 316 3 238 M '~ I M M 0 Aa ~ 1 2 8 M M M M 0 41 800

0 ,~¢~ 2 26 2 182 3 1~$ 3 176 M P~ 0 M 0 8 1 2 12 0 9 M 2006vu

M 0 2 342 - . , 3 14 3 25," M P, 250 000

M 0 I 2 0 8 1 5 2..-I M

200 000 250 000 216 OOu 82 200 18 200 45 800 28 o00

168

i% = the quant i ty o f goods t r anspor ted f lora the p roduc t ion ~ystem P;, to the des t ina- t ion B,,

c,~ = the uni t cost o f t r a n s p c r t f rom the source A, to the p r o d u c t m n system P/,,

v~; = the uni t cost o f h a n s p o r t f rom the pro- duc t ]on system P~ to the des t ina t ion B,: and

c; = the p r o d u c t m n cost pet un t m tbe p]o- d u c t m n system Pt

Accordingly, he first stun ov the ,~ht-~ '~pc 1 side of the objective" func tmn oer. tcs d~e t rans- por ta t ion costs of the gOGds f rom ~,),e 5ot',rcc; to the p m d u c u o n system, tile second sum denotes I_he I_!aw, p o r t d l l O l l COSTS ~|~-)lll t h e L H t , t l u t l , a n i}=

terns to the destmations~ while the th i rd sum de- - : ~e~ the p r o d u c t m n costs m lhe p roduc t ion sys- t em ~rhe me thod for solving mul t i s tep p rob lems was proposed by the Soviet ma thema t i c i an , V A Mash. and before tha t an analogue ~dea had been proposed b~' th,: a m e r i c a n m a t h e m a t i c m n A Or- den I he O r d e n - M a s h me(hod reduces the prob- lem of th~s k ind to a classical t r anspor t a t i on

TABI E 2

Locdtlon Costs THee

Total a,,,~.ge Total Average

P~ 2492572 6 37 1090559 2 79 P2 2303766 5 3 ° 1597160 4 08 P~ 22735"5 5 81 1 ~ 4 9 0 ~'2 P~ ~325793 5 05 1710857 Z ~7

TABLE 3

Locaho~ ( ',S ~ Time

Total A',cragc ~o,a; Average

P~-P2 23735~2 6 07 1072252 2 74 2398280 t~ 13 887931 2 27 2411863 6 17 81568~ 2 09 2412501 6 18 813078 2 08

P~-P~ 2362785 6 04 1349452 3 45 23873"0 6 10 1206412 3 08 2392466 6 12 I'772S 0 3 01 239~o32 6 ,3 1165132 .98 2,111972 6 17 1092~ ~ 2 19 2412766 6 18 1090278 2 89

P~-~ 2395539 6 13 1334451 3 41 n433294 6 22 1141287 2 92

fABLE 4

Location (OSIS TP,IO

Tolm 4veragc Total Average

IB-P2-P~ 23~a214 5 9 7 1127503 2 8 8 2352050 601 908853 2 5 5 2371go0 ( ,06 8~0049 2 2 5 23~7629 6 0 8 847084 2 I I ~ ? " ~ 0 ~ C9 646344 2 16 2380332 o 10 845104 2 15

Y,-;'~-P4 2342655 6 00 959762 2 45 22~8350 6 9 3 842962 2 16 2380411 6 q9 730996 1 87

[i-1"~-524 2a35515 5 08 12~e963 3 16 2358590 6 0 3 1120163 2 8 6 2380650 6 0 9 1007296 2 5 8

Problem owing to ~ par t icu lar compos i t i on of the t r anspo r t a t i on t~ble

The objec t ive funct ion re la t ing to the t~me, i f its a im is to mm~mi~e the sum of mul t ip les of the t r anspor t ed quan t i t i e s and the t r anspo r t a t i on t ime, kas th is form

T = -- .~ t,hX~h + 'AID,; t= i /~=l ' = /

where g,; = the t ransponaO.on t ime from the source A,

to the p roduc t ion system l \ , x,~ -- the quant i ty of goods t r anspor ted f rom the

source A, to the p roduc t ion ~ystem & , t~, = the t r anspor t a t iop t ime f rom the produc-

t ion s~ . t em P~ to t ~e destma:zcm B,. and v~; = t n e quanut3 J g o o d s transp.3rted from the

p roduc t ion sysmm P; ~.o the dest,xlat,.on Bj One s n o u t d n o t e thq¢ " - ~ Woe,-~*, ~-llqt-tr9 is no~

taken ,.:to cons ide ra t toa Namely the process P~ ,e In T f u n c h o n . . . . . . . . ~-~ 7 r c . . . . . . . . un i t p roddc t lon is ldentmal in each p roduc t ion system

By in t roduc ing t~me as a second f u n c t m n the p rob lem is reduced to a b m n t e n a l t r a n s p o r t a t m n p rob lem with two h n e a r objec t ive f imetmns, whereby the eff iment solut ions o f this p i o b i e m can be found by lhe u~e of the mul tmHter la l m o - g r a m m m g m e t h o d for t r anspo r t a t i on p rob lem We pay soecml atgentloP to the p a r a m e m c pro~ g r a m m m g metL'od for t r an spo r t a t i on problem, w m ~ , v~as ,~ . . . . . . by V S r : n : : a s a n and G L T h o m p s o n [ ! ]

W h e n the calcula t ion on the bas:s of q u a n n t a -

169

t tve measures o f locat ion o p t i m u m is carr ied out all o the r i,n.q,_.ent!a! ;ac tors are :n t r - :duced in to the model We have to express tke va"qe of par- t icular va r ian t s in the un ique way thus e n a b h n g the c o m p a r i s o n o f inf luence o f var ious factov~ For so lvm ~ the p rob lem of th is k ind we can use di f ferent r ,ethods f rom the P R O M E T H E E a , d FLECTRFt t'~m~ ',~ u :~,ot-., ~n.. be compared ;_o the

d i m e n s i o n an dy'~s model for the locatioI, ci~oicc deve loped ~y 7 W. B n d g e m a . [2 ].

The PROI 'AETHEF m e t h o d [ 3 ] is apprc ' ~:a~c to t reat the m u l h c r d e n a p rob lems o f the ff ,low- mg type"

Max{f t ( a l , , I t ( a ) l a ~ K },

where ,Z is a f inite set o f p~.~,ble ac t , on ; (al:cr~ n a m e s ) , 2.,'_d f are ~ er**ena to ~e maxzmtrcd For each~ , , ~ . . . , f ( a ) is an eva lua t ion of '.~is ac t ion

W h e n we compare two ac t ions a , b e K we mus t be able t(, express the results o f th~s compar i son in t e rms o f preference We therefo~.c cons ider a preference f'~mcL~on P

P K × K ~ [0,1 ]

represent ing the ln tens! ty of preferei~. ~ ol ac t ion a w i t h a regard tn act :on t, I,~ pray[it c, th - :)rei'- erence ! ) n c t l o n will be a f e , ' - ' i o n of" the ¢l~ffer- ence be tween the two e . a l u a t ons d = f ( e ) f ( b ) , an0 ~ : : m o ,otonicali?, ,acreas lng

It is clear tha t th~s preference funct~or wd! ~aave toe shape as shown in P~g. 1

Six possible types of this func t ion ~.re pro- pGsed to the decis ion maker , t h e cffectl,,e ohotce is m a d e in te racm,e ly by the decis ion maxe r and the analyst accord ing to the i r feeling of the in ten- sities of preference In each case zero, ; ne or t ~ o par; mete rs have to be fixed ( l ) q is a threshold def in ing an indi f ference area, ( n ) p is a th resh- did def in ing a st '~et preference area a.~,d ( m ) s is a p a r a m e t e r the ~alue o f ~h~_ch lies "~e "~e~f p and q

These a i / g e n e r a h s e d cr i ter ia : 're show ~, m Fig 2 ( a ) - ( f )

Now we can def ine a preference Inde .

h

H ( a , b ) = .c .,, t , , ~ ~, / = l

vq~erc w, are ,,~e~_ght~ assoc, 'ated with each cr i ter ion

Fmaliy, fo~ ever), a e K , let us cons ider the two Iofiowlog ou t r ank ing flows

= !eavingf iow q ~ ( a ) = V H t a , b ) hE&

. en te r ing flow q)- ( a ) = ~ H ( b , a )

q~+(a) are the measures o f the ou t r ank ing cnarac te r of a The ac t ion is be ter if the leaving flow is higher, and the enter ing flow lower The P R O M E - i H E E i gives a typical preordc~ of the set of ac t ions in which so , r e act ions are compa- rable, some others are no~ Whe~ the decis loa w a k e r ~s request , ' ,g a c,?.pptetc ranking, the net ou t r ank ing flow may be cons idered

~ ( a ) = ~ + ( e ) - ~ - ( a )

and the h igher the net flow ti~e be t te r is .he ac- t ion All the ac t ions of K are no',~ complete , v r anked

3. Numerical example

3 1 T,)e !o~ at ton o to tce as multlc t l ierlal ',ratTspo, ration prob lem

The ca icu la tmn & t h e locat ion choice, wl?h the previously m e n h o n e d assumpt~,ons, ~a based on the requzrement for the eonst , ticvor~ o~ 1he pro- d u c q o n system with a total c a p a c l t / o f 450 00O produc t uni t s The choice is to be c'o~e among fore dd fe rcn : !ocations, wnereb~ tl1~ f" iowlpg var'av, ts arc cons idered • cons t ruOlon o f one p roduc t ion umt o~t!. a ca-

oacl ty of 450 000 produc t uni ts on one of the possible locat ions,

• cons t ruc t ions o f two p r o d u c , , o : uni ts with a capacity of 200 000 and 250 000 o, od~ct traits on two of the four poss~.ble loka*!ous,

• c o n s t r u c n o n of three p rodqc t uni ts w h a : a- pacity of 100, 150 and 200 t h o u s a n q produc*. uni t s on three o f t b ~ Ibur possible l o : a t m p s The posst_bllity o fcoPs t ruc t :ng the p lants on all

the four locat ions is ltot t aken Into eons l Je ra t ton because Lonsmact ion of a pi.,.nt w,lb ~ 2_~p.~ody

170

TABLE 5

The PROMCALC spreadsheet

Criteria C 1 C 2 Name Tran~,eosl Trans.time Min/Max mm mm Type 3 5 Weight 9 00 8 00

Parameters q - 010 p 0 40 0 20

C3 C4 C5 C6 C7 BOP Ttme S/Bot T. Build.cost Infr cost Labor.cost min mm mm mm mm 4 3 5 5 4 6 00 5 O0 9 00 8 00 8 00

0 20 - 500 i 0 I 0 50 5000 2000 50 3

C8 C9 CIO Weather Expansion Trans.p~ max max max

3 3 7 00 8'# 8 O0

0 : 0 0 80 0 60

Actions A I p l 688 279 A 2 1)2 6 16 4 08 A 3 p3 6.28 4,78 A 4p4 642 427 A 5 pl'~a 655 296 ~t 6p12b 062 245 A 7pl2c C-66 225 A 8pl2d 066 224 A 9p13a 652 373 AlOpl3b 6 59 3.33 ~ l l pl3t 660 325 AI2 pl3d 661 322 Al3pi3e 660 302 AI4pl3f 666 301 A.15 pl4a 6 12 342 ~,~6 pl4b 6 12 ~42 AI7 pl4¢ 6 12 342 Ai8 oI4d 612 341 Algpl4c 622 292 ~20 p123a 5 97 2 88 A21 p~23b 60~ 255 A22 pl23c 606 225 A23p123d 608 217 A2~, p124a 599 245 A25p124~ 603 216 A26 p124e 603 I 87 A27 p134a 3.98 3,16 A2Opl~4b 003 286 A29pI34c 609 258

2 60 113500 00 81000 00 80 O0 20 00 3.80 12900 00 72000 00 90 00 15 00 3 70 12900 00 74250 00 90 00 18 00 3 80 79600 00 76500.00 100 00 16 G0 3 00 33900.00 77000 00 153 00 21.25 300 3390000 77000.00 153.00 21 25 3 00 5000.00 77000.00 i 53 00 21 25 300 370000 7700000 15300 21 25 3 70 12900 00 78000 00 153 00 22 90 370 1290000 7800000 13300 22 90 3 70 3800 00 78000 00 153 00 22 90 3 50 42000 00 78000 00 15,3 00 72 90 3 50 13 tOO 00 78000.00 r 53 00 22.90 3 50 11800 00 78000 00 153.00 22 90 3.70 33900 00 7¢.000 00 162 00 21 80 3 70 5000 00 79000 00 !62 00 21 80

50 1820000 7900000 162 00 2t 80 3 50 1820000 7900000 :6200 21 80 3 50 18200 O0 79000 O0 162 O0 21 80 3 70 ~2909 00 76500 00 234 00 24.20 3 70 12910 00 76500 00 234 00 24 20 3 70 12900 00 76500,00 234 00 24.20 3 70 1290000 ~050090 234130 24 20 3 00 5000 OC 77060 I]0 243 0,3 23 30 3 c v' 5000.00 7700000 243 O0 23.30 300 500000 7/~)00 O0 24300 23";0 3o0 7960000 77750130 243.00 24 30 3.00 79600.00 7775000 743 00 24.3G 300 79600.00 / 1/3000 243.00 24.39

4 00 2 00 5 00 3 00 3 00 4 00 3 Ou 4.t:0 4 0O 4 00 5 O0 3.00 3.55 2.44 4 55 3 53 2 44 ~.55 35S , ¢4 455 3.55 2 4q 4 55 3 55 2 89 4 55 3 55 ? 89 4 55 3 55 2 89 4 55 3 55 2 89 4.55 3 55 2.89 4 55 3 55 2 89 4 55 400 333 411 40,) 333 411 4 t)0 3 ~3 4 11 400 ~33 4.11 400 333 4.11 3 44 "~ 7g 4.44 3 44 2 7~1 4.44 3 44 2 78 4.44 3.44 7,78 4.44 3.67 3 ~'0 4 22 36/ 3.00 422 3 67 3 O0 4 22 3 67 3 3/ 4 22 3 67 3,33 4.22 3 67 3 33 4 22

smaller than optimal ~hould be avnlded. ~uppiy of raw mdt. t ;als c.:ntre~ (a,) . demand

ef consumers centre~ ~'bj), capacity of produ, ~ion units (S~) cos~ oftran~poi~ation an~ pro- O~c?k,n (upper left comer) and transportation tl~ ~: (lower right come:) oer one variant am given m the "['able 1. This t ab~ i~ prep,gred fo, the c, oplie~t an of two step ~ anspon by Orden - Mash v.,c;nt)d, who.d:,) ,,~ ;he ero.-.a ~m~b,,,,~:2,~' the v,dues St and $2 would Chahge I~laces, i.e. A ~ ~ ~.>,3 ~}i)O and .,: ~. 20 v v00.

lr: ~he I ables 2, 3 and 4, the fgnc ion v,lues for

efficient solutmns are given. They ~ re calculated by a mult:2ntermi transportation problem in the case of constructing the production system t)n one, two or three locations.

In the case of the comuuction of one plant, as shown on the Table 1, in re, ms o f ~ s t s the loca- Liens P3 is the most acceptable, while in term~ (~f T) ~ location ~', has the advamage. At the same ';.r,, ~,-.~: of these two locations is inferior in terms o; the second considered cmenon, Le. the Iocat,.or~ Pt m terms ofcests and the h.~cation P3 m )~rms of time

TABLE 6

Correlation analysis

P r e h m m a r v S ta t t s tws

Criterion Minimum

C I Trans.Cost 5.97 C 2' Tmns.Ttme 1 87 C 3 Boil.Time 2 50 C 4 S/Bot.Time 3700 00 C 5. Build Cost 72000.00 C 6 Infr Cost 80 00

Maximum Average

6 88 4 78 3.80

113500.00 81000.00

243.00

6.31 3 00 3 39

27268.97 77413.79

175.66 22.07

3 63 3.03 4.31

C 7 Labor Cost 15.00 C 8 Weather 3.00 C 9: Expansto,: 2.00

I 0: Trans.Poss 3.00

COI~ , Z "b' t" .~n$

Correlation analysis for criterion C ~. Trans Cosl (min). C I. Trans Cost (min ~ 1.00 C 2. Trans. l'lme (mm) 0.131 C 3 BotI.Tim~ (mm) -0 ,143 C 4" S/Bot Time (mm) 0 087 C 5' Build.Cost (ram) d.l~3

Correlatzon analysis for criterion C 2 Trans Time (mm) C I TransCost (ram) 0 131 C 2 Trans Time (ram j 1.000 C 3 Boll Time (ram) 0 558 C 4" S/Bot Time t ~,,~) 0.136 C 5. Build Cost (rain) : -0 .204

Correiatlon analysis for ca ~tcnon C 3: Botl l tree (ram) C I Trans.Cost ~mln) - 0 143 C 2 Trans Time train) 0 555 C 3. Bml Time (ram) 1 000 C 4 S/Bot Time ~mrr) - 0 4 1 8 C 5: Build Cu~t (nm,) - 0 78fi

Correlation analysis for criterion C 4" S/Bot,Time (m~,l): C I TransCost (mln) ~' "0087 C 2 Trans Time (ram) 0 136 C 3 BotI.Ttme (ram) - 0 4 1 8 C 4. S/Bot Time (m:n) 1.000 C 5: Build.Cost (mm) 0 217

Correlation analys,s for crttertuJl C 5 ~,,,Id.Cost (mm) ' C I" TransCost (ram) 0 163 C 2" "! tans.Time (ram) --6.204 C 3' Bo~'l T~me (mln ~ -0 .286 C 4 S/Bol Time z qm) 0 317 C 5 Build Cost (m,n) I 000

Correlatlonan,dystsforcrlterlonC 6 infr.Cos~ (ram) C I. Trans.Cost ',rain) - 0 724 C 2. Trans Time (m~a) 0659 C 3. Boil Time (rain) : -0 .209 C 4: S/R~,! Time ¢lnin', - 0 096 C ~i' Budd.Cost (rain) 0 035

243.00 24.30

4.00 5 00 5.00

C 6 . C /" C 8 C 9" e l 0

C 6: C 7 . C 8" C 9 e l 0

C 6 C 7 C 8 . C 9 CI0

C 6: C 7 C 8 C 9 . ClO.

C 6: C 7 . C 8 C 9 CI0

C 6 . C 7 C 8 . C 9" Cl0

Infr.Cest Labor.Cost Weather Expansion Trans Poss

lnfr Cost Labor Cost Weather Expansion "1 tans Poss

lnfr Cost Labor.Cost Weather Fxp~n~ion Tr;,ns.Poss

lnfr Cost Labor.Cost Weather Expansion Trans.Poss

Infr Cost ! abor Cosl Weather Expansion Trars Poss

• '.a fr.Co~t I .thor.Cost ~teg~b.or Expanq~olJ Tra~s.Poss

(rain) (mm) (max) (max) (max)

(mmn) (rain) (ma~) (max) (max)

(mm) (.lax) (max) cmax)

(mm) (rain) (max) (max) (max)

(mm) (mm l ..'max)

(max)

(mtnl ~mln) (max) ?max) (max)

171

St.Dev.

0.38 0.67 0 35

29162.57 1581.51

5h34 2.29 0 26 0.54 0 34

: -0 .724 : -0 .360

- 0 079 : -0 .368 : 0.424

: - 0 659 -0 .643 -0 .088

0.629 -0 .502

- 0 209 -0 .188 - 0 208

0,427 -0 .352

- 0 096 -0 .107

0.345 0 176

: -0 .103

0 0~,5 0.422 0.825

-O t74 0 307

1.000 0.817 0.024

: - 0 104 0 079

172

TABLE 6 (Contmued) -

I” f+mmar~ sramr/es Criterion

- _“-. __-_.l_- wlcII-III --I “_ ^_I _-. __, _--

Mtntmum Maximum Avcragc St Dev.

Correlatton annlys~a for critcrton C 7: LaborCost (mm )’ C I: Tmns.Cost (mm) : -0.360 c 2: Tmns.Time (mm) : -0.643 C 3. kmll. 111ne (mm) : -0.188 c 4‘ S/Bet Ttmr (mitt) ’ -0 107 c 5. Build Cost (mm) 0.422

Correkrtton analysts for crrterton C 8. Weather (m’lx) Cl Trans COG (mm) -0079 c2 Trans Ttme (mm) -0 O%% c3 Botf Tune (m:ni 0 208 c 4. S/Rot 1 ime (mm) 0 345 c 5% Butld Cost (mm) : 0.875

Ccrrelatton analysis for crtterton C 9. Expanston (max) c I tram Coct (mm) -0.368 C2 Trans.Ttme (mm) 0.629 C3 Botl Ttme (r!Ua; 0 427 CA S/Bot.Ttme (mm) 0 176 k’ 5 Butld Cost (mm) -0 174

Correlation analysis for crtterton Cl0 TransPoss (mart)* C I Tram Cost (mm) 0.424 C? Tram Tnn~ tmint -0.502 C? Botl Ttme (mint * -0,352 C-t S,Bot Tnne i 1‘11111 * ’ ~no.. - u.r’rr c5 Butld C obl (mm) . 0 307

C 6: Infr.Cost (niin) 0817 c 7: LaborCost (mm) I .oOa c 8. Weaiher (max) 0.125 c 9. Expanston (max) -0.363 Cl0 Trans.Poss (mx) 0.485

C6 c7 c 8. CQ Cl0

I+& Cost Labor Cost Weather Expanston Trans Poss

wn) bin) (max) (max) (maxI

0 024 0 I25 I.000 0.194

-0.2OP

C6 c 7. r 8: c9 cio.

Infr.Cost (mm) -0.104 LaborCost I nun) -0.365 Weather ( max ) 0.194 Expansion fmax) I .ooo ?‘r.ns Pass Imax) -0.927

InF Cost (mm) 0 079 LI\oJr.Cost (mm) 0.485 Weather lnvix) -0.205 Fxpanston @tax) -0 927 Tram Poss (max) I 000

-- - -.._-

Nevertheless. not interfering with the prefer- ‘;I:SXS af the dltrisisn mraker, we think that the lo- cation PI has some advantacles over other loca- ttons, while the Costs are only insignificantly higher in rc:ati;s to those cf?he other locations. That is the reasou wh:~ ia the second variant we mvesttgatc the change of costs and transporta- tion time in case one more plant besides P, is in- stalled m one of the other 3 locations. The data of that calculation are given in the Table 3.

Everv subveriant contained in * thr variant shown on the Table 3, has efficient solutions ranking from the best one in terms of costs to the most accep?2ble one i0 ;cnr.s cftimn,. The lowest ttme IS achteved by the locat:~ of [Ants MI 2~ locattons Pi-P2, whereas the !cwtzst costs zrp 5

lated to the comhcat;on PI-PI. it IS also worth menttonmg that this vartant in terms of costs and time is more acceptable t.lan the previous one arrd il necessitates the calcutatton m IX% of!he plants being lor&d in three ot thu four possible loca- tions. The results oftii,s variant are shown in Ta- ble 4.

The decision maker is now faced with an ex- tremely coa&x problem, as it is necessary for him to choose from 28 different solutions. No one of these solutions can be said to be better than any o&her, i,e, this analysis has up to now oniy deduced the “best” 28 solutions from the great number of possible solutions of the ma!ticr;terial problem, It is obvious that any further analysis requires some communication with the decision maker in terms of his I;references and further cri- teria, i.e. location facto5rs. But to solve such a problem it is necessary to use one of the method of multicriterial ranking or one of the methods t*o:n the PROMETHEE family.

Tk FRCMETKF? method givrs ZPe possibil- ity to the decision makr ~4considering the prob- lem frrrm various aspects, mainly in order to present to himselfthe model as clearly as possible.

IABLE 7

PROIC/ETHEE ! partial ranking

173

Action Preferred to

A I/p2 no action! A 2/p2 no action t A 3/p3 pl p2 p4 A 4/p4 pl A 5/p12a pl3a pl3b pl3c pl3d pl4a

pl4c pl4d p123a p134a pi34b A 6/p12b pl2a pl3a pl3b pl3c pl3d

p13¢ pl3f pl4a pl4c pJ4d pl4e plg~a p123b pl34a ~!34b pl34c

A 7/pl2c pl2a pl2b pl3a pl3b pl3c pl3d pl3e pl3f pl4a pl4b pl4c pl4d pl4e p123a pi2~b pl23c p123d p134a p134b pi34t

A 8/p12d 912a pl2b p l 2 c ~ pl3a ~13b pl3c pl3d p[3e pl3f pl4a pl4b pl4c pl4d pl4e pl23a p123b pi23c p123d p134a p134d p134c

A 9/pl3a noacuon[ AI0/pl3b pl3d AII/pl3c pl3a pl3b pl3d Al2/pl3d no action v AI3/~!3e pl3a p!30 pl3d Ai4/pl3f pl3a pl3b pl3d pl3e AI5/pt4a pl3a pl3d AI6/pl4b pl2a pl2b pl3a pl3b pl3c

pl3d pl3e pl3f pl4a pl4c pl4u pl4e p123a pl23b p134a p134b p134c

Action Preferred 1o

AI7/pl4e pl3a pl3b pl3d pl4a pl3a pl3b p*3d pl4a p134a

AI8/pl4d pl3a pl3b pl3d pl4a p134a Ai9/pl4c pi2a pl3a pl3b pl3c pl3d

pl3e pl3f pl4a pl4c pl4d p123a p134a p134b pl34c

A20/p123a pl3a pl3b pl3c pl3d p134a A21/~123b pl3a pl3b pl3c pl3d pl3o

pl3f pl4a p123a pl~a p134b A22/,)I)3c pl3a pl3b pl3c ~t3d pl3e

pl3f pl4a pl4c pl4d p123a p123b p123d p134a p134b p134c

A23/p123d ~13a pl3b pl3c pl3d pl3c pi3f pl4a p123a p123b pl34a pl34b p134c

A24/p124a pl2a pl2b pl3a pl3b pl3c pl3d pl3e pl3f pl4a pl4b pl4e pl4d pl4c p123a p123b pl23c p123d p134a p134b p:4c

A25/p124b pl2a pl2b pl3a pl3b pl3c pl3d pl3e pl3f pl4a pl4b pl4c pl4d pl4e pl23a p123b pl23c p123d p124a p134a p134b pl34c

A26/p124c pl2a pl2b pl3a pl3b pl3c pl3d pl3c pl3f pl4a pl4b pl4c pl~d pl4e p123a p123b p|23c p123d pl24a p134a p134b pI34c

A27/~,34a p13~ pl3d ~"~/p~4b pl3a pl3b pl3c pl3d p134a A29/p134~ pl3a pl3b pl3c pl3d pl3e

pi4d p123a p134a p134b

Still, at the very beginning we come across the questien of criteria choice a~ wd! as their pref- erence and the complexity of pamcular criteria too.

Table 5 contains the basic data of the given probl~n% noting that the values for the first two criteria have been obtained by the preceding analysis (see Section 3.1), whi~e the values of other criteria have been obtained by special anal- ysis or estimation.

The criterion C3 denr~tes the maximally used transport time, while (-'4 denotes the quantity of ~.argo transpox~ in maximal ~.ime. The other cri- teria are: C~ - building tests. C~ - infra-structur~ costs, C7 - labour costs, C~ - weather condiuons,

C9 - possibility of production system expansion, ~" - transportation possibilities. ~.., i o

Table 6 contains ~, correlation analysis, i.e. the survey of correlation between a particular crite. rion ,~nd any other cr;terion. The importance of this correlation analysis consists in the shewing of the conflict among criteria, that ~s if there are any criteria to some other criterion used for find- ing out the advantage~ of a particular solution. As the tab!~ shows the criteria are well chosen be- cause the majority o f correlation coeffioents are either negative or with a small positive correla- tion. "i he criteria C6 and C7 possess certain sire. ilarit~ ,:,h-~,~ coefficien: correlation is 0,817.

I ' :hle l shows a partial ranking of each of 29

174

I AbLE 8

Actton Pht Action Phi

1. A 3 . p 3 (0.231) 16. Al4.pl3f (-0.048) 2. A 7:p12¢ (0.163) 17 Al3.pl3e (-0049) 3 A 8.p12d (0.162) 18. A20.p123a t -0.054) 4. A 4 p4 ( 0 1 5 8 ) 19 AI6 nI4b t-0.007) 5 A 2 . p 2 (0.152) 20. A 9 pl3a (-0068) 6. A25"p124b ( 0 1 2 6 ) 21 AI0pl3b (-0071) 7 A24 pt24a ( 0 1 1 7 ) 22 A29 pl34c (-0076) 8. A26 pi24c (0.112) 23 A28.1:t34b (-0.079) 9. A 6'p12b (0.085) 24. A27"p134a (-0.097)

10. A 5.p12a (0.065) 25. Alg.pl4e (-0.107) II. A21:p123b (-0.027) 26. AI2'pl3d (-0.114) 12. A22:p123c (-0.030) 27 AIT.pl4c (-0.118) 13. A I pl (-0033) 28. Ai8 pl4d (-0.119) 14. Ail:pl3c (-0.033) 29 Ai5 pl4a (-0.149) 15 A23.p123d (-0.038)

possible solutions, namely it shows which action is preferred to the other possible alternatives. As one can see from the table the best actions are a~, a25 and a26 that arc better than the other 2! actions.

Table '~ presents a complete ,anking, i.e. the renking of parti¢-!ar action due to their net flow, s~owing ~ 'mat the best action a26, according tu ,~,Mch the third efficient solution is chosen out of

a variant in which the three production sy¢tems are built on the location of Pi, P2 and P4. This table gives to the decision maker the choice pos- sibility of the best solutions or the eventual pos- sibility of comparing the given solution with the ones obtained on the bases of ~ifferent entrance parameters.

R e f e r e n c e s

I Snmvasan, V. and Thompson, G.L., 1977. Determining cost vs. time Pareto - opttm~! fromt~rs in mult~ - modu~ transporta|ton problems. Transp. Sci., i l ( 1 ).

2 Bnd$eman, P.W, 1963. Dimensional Analysis, Yale Uni. vers~t:r" Preps New Haven.

3 ~rans, J.P., Mares:hal, B. and Vmcke, Ph., 1984. PRO- METHEE" A new family of outranking methods ta MCDM, IFORS 84, North-Hel~and.

4 Babt~, Z., 1983. Vitekntertjalno programiranje kod pro- blema transporta. Zborotk radova SYM-OP-IS 83, H. Noel

5 B~bl~, Z. and Part6, l., 1989. Optimal location choice by the use of multlcritertal transt~ortatlon problem EilRO X, The 10th European C~Morence on Operational Re- search, Be!~a~e. Kre~l~, 1, ! 981. Prostoma ekonomqa - Osnove teonje lo- kaclje, razmjegtaja I organlza¢lje u prostoru. Informator, Zagreb.

7 Zelcnovm, M.D., 1987. Projcktovanje proizvodnih sts- tema. Nau~na knvga, Beograd.