University of Nigeria Research Publications

UDOH, Nse Sunday

Nam

e

PG/M.SC/99/26853

Title

Optimal Transportation Algorithm (OTA) for the Distribution of

Petroleum Products in Nigeria Using the Distribution of Petrol in Akwa

Ibom State as a Case Study

Facu

lty

Physical Science

Dep

artm

ent

Statistics

Dat

e December, 2001

Sign

atur

e

OI"I'1MAL 'I'RANSYOK'TA'I'ION ALGORITHM (O'I'A) IWR 'I'I-IE DISI'HIBIITION O F I'E'I'HOLEIJM PROL)lJC"l1S IN

NIGERIA USING ?'ME DIS'I'RIBUrI'ION OF' I'E'I'IIOL IN AKWA IBOM ST,Al'E AS A CASE STUDY

9

UIIOH, NSE S f JNDAV PCIMSc I99126853

' C '

DEPARTMEN'I' OF S'I'A1'ISTICS UNIVERSITY 01; NIGIEHIA, NSlJKKA

BEING A PROJECT SUBMITTED IN PARTIAL, FULFILLMENT OF ?'I-iE REQUIREMENTS FOR THE AWARD FOR MASTER O F

SCIENCE DEGREE IN SI'A'I'IS'I'ICS OF 'I'HE UNIVERSITY OF NIGERIA

DECEMBER, 2001

'I'he work embodied in this projcct report is original and has not been

submitted in part or lhll for any other depees of this or any other

IJlt 1'. L. CHIGBU M H E. P. OKIJUIKE

(SUPERVISOR) (HEAD OF DEPARTMENT)

NAME: ........ : ............... SIGNATIJKE: . . . . . . . . . . . . . . . . . . . (EXTERNAL EXAMINER)

torxh bcilrcr ol' my acsdcn~ic success. Your ingenuity and ~hori)ughncss

toiling period. My heart goes lo collcagucs and liicnds; I3usscy, U . J.,

Ihssey, K. S., Ossai Ii., Ugbe 'I'homas, Ajaroagu Jude, Mlbn Ildoh,

towards the success 01' this programrl re

LIST OF TABLES

Table 2.1 Sources of Petroleutn Products in Nigeria ....................................... 12

Table 2.2 Cost and Requirement table for the distribution of Petroleum Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

'Table 2.3 Application of North West-Corner Method to the Metro Water district Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Table 2.4 Application of the Least-Cost method to the Metro water district problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Table 2.5 Solution of the Metro water district problem using .. the Least-Obst Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Table 2.6 Application of Vogel's Approximation Method (VAM) '

to the Metro water district problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

'Table 2.7 Application of Russell's Approximation Method (RAM) to the Metro water district problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Table 2.8 Summary of the Selected basic Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

'I'able 3.1 Population and sanrpled filling stations in the five Cities .................................................................. of Akwa lbom State 36

l'able 3.2 Constraint Coefficient for the Distribution of Petrol . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Table 3.3 Incon~plete Cost and Requirement tables ......................................... 43

. . . . . . . . . . . . . . 'Table 3.4 Initial Cost and Requirement table for the distribution of Petrol 45

......... Table 3.5 Complete Cost and Requirement table for the Distribution of Petrol 47

Table 4.1 Cost and Requirement table for the Distribution of Petrol in five Cities of Akwa lbom State (AK5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Table 4.2 Application of VAM to the distribution of Petrol in AKS . . . . . . . . . . . . . . . . . . . . . . 60

Table 4.3 Application of R A M to the distribution of Petrol in AKS . . . . . . . . . . . . . . . . . . . . . . 63

' I 4 . 4 Summary o S SCICCICC~ illi t i i l l basic . . . . . . . . . . . . . vnriablcs wing R A M .

I I J I I . 3 I . Nclworh ~qxescnht ion ol ' t lx djswi1)utiorl ol'pclrol in A!iw;r I born State w~th storecl ;111o~;1tion

? > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iincl supply lo deslinrrtion.. 3 b

9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I l l 4 I It~grcdicnl ol'OlZ/MS actrvity. 52

ava~lab~lity and sull'icicncy in llic p e ~ r ~ l w ~ i i industry was soughl in this

working cli1lk1-enlly with each separate ~ n c h d 1i)r constr-uc~ing i~~ilial basic

as sources and five ciocs i l l A k ~ v a 1lm11 S~a1c ;IS ~ C S I ~ I I ~ ~ ~ I ~ I ~ S . 1'11~ rcsid~

uccordi~ig lo their nceds ul mini mu111 cost. It dso csposcs the i~isul'l'icicncy

was i n v c s ~ ~ g a ~ c d in comparison by pcrlbrming it " 1 1 - i d " lest I'or optilnality

L I S I I I ~ ILARI'S r ~ ~ r l l (ir~ilial basic I'casiblc solu~ioi l) . 'l'hc: choice ol'VAhl is

advar~~agcous on economic utilizalion ol' avail:tblc ~C:;C~CII.CCS d w r than

d i . a \v i~g I ~ I O I I I du111my source, I [ d s o C I I S U I . C S supply it1 I ~ I C bcari.~blc

lnr~~i~nurn cos1 ol'ttansportalion.

1 5 S c q ~ ol'lhc Study.. . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . 5

2 .2 . 1 Sourc.cs ol' I 'c lrc) lwni 1'1-oducls Supply. . . . . . . . . . . . . . . . . . . . . I I

s i i i

-1.7.3 I I I I L I I I ~ I I I - I - V V L I S . . . . . . . . . ..74

CHAPTER ONE

GENERAL OVKHVIEW

1.0 INTH0I)UC'I'lON

Worldwide, petroleum and energy in general are indispensable for human sustenance and

industrial production. I n the specific context oS Nigeria, the petroleun~ industry assumes

a special importance. Ninety percent (00%) of the country's foreign exchange earnings

come fiom this sector while about sixty five percent (65%) of Federal Government's

income are derived from this sector: see Federal Government's Special Committee

Report on the Review of petroleu~q products supj~ly and distribution (October, 2000).

Petroleum products supply and distribution has a direct impact on all aspects of the

Nigerian economy from peasant hrming in the rural area, to the workplace in the urban

centres, up to the most sophisticated industrial production process. The distribution of

petioleurn product is a typical transportation problem in Operations Research or

management sciences, which in this research work we seek to solve, optimally

'I'tle transportation (or distribution) pr-oblen~ was an early example of linear network

optimization and is now a standard application for industrial lirnis having several

~l~anulitctur ing plal~ls, wateliouscs, salos LCI ri101 ics, ~ C ~ O L S i l t d dis~l.il)utio~~ OUL~CLS, SCC'

Wagner (1969), chapter 6. 'l'ht: transponation problem is &tined asfhat in which it is

requiRd to find a method of apportioning a homogeneous product, which are available in

given

'l'hc supply ol; l'clrolcurn products namely I ' lwt icr Moror Spirir (I'MS) -

I k l i nc r~cs i ~ r thc cour~wy at I'orl I larcourl 1 & I I, Warri and Kadi~na wlrilc

0 1 ' ~ p p l y srood a1 74% ol' I'MS 4:3911 01' A(;() and 4S':h 01 ' 1)I'K wl~rlc X'h

tlccd, 1)l'MC has now rcsorlcd lo trucking ol' products by road liom 11ic

sou~hcnl l lcpo~s 10 I-illing S ~ a ~ i o n s in othcr parts of ~ h c coun~t.y. Ir)dus~ry's

ini l l~on lilrcs, 10 rnrllion Ii~rcs fur 1WK and 8 million li~rcs t'vr AGO.

( 'I l A l N l i l l :A( "I'KON 'I'his i s a scqucnce ol' co~npcllsal~ng chnnges in

o~hcr basic variahlcs (allocarions), in ordcr lo coiliinue salislying rhc

The data used for the analysis of this work is limited to the distributiodsales of petrol by

independent marketers in Nigeria. This is because every city in the country has at least a

t Filling Station owned by an Independent Marketer. Also, the Department of Petroleum

Kesoirrces store data on supply/distribution of pekol from Depots to Filling Stations

owned by different Lndependent Marketers in all states of the federation.

1.6 REVIEW OF RELATED Ll'rElL41'UHE

In an assign~nent problem, a special type of transportalion problem where the sources

now are assignees and the .destinations now are tasks, Bolat (1999) notes that the

assignment of' arriving flights to the available gates in an airport can have a major. impact

on the etliciency with which flight scliedules are maintained. A robust (dummy) feasible

assignment was found so that a small variations on the announced flights schedules do

not disrupt the assignment. I t was also found that the performance of the optimum

alborithm is very much affected by the utilization level of the gate. Klincewicz (1990)

considered the solution of freight transport problem (FTP) using Facility Location

Technique. The basis of the heuristic for the problem just mentioned is found in the

classic uncapacitated 'facility location problem (IJFLP). In UFLP, there is a set of

customers indexed by m = 1, . . . . . . ., M and a set of possible Picilities indexed by

n = I ,. . . ., N. There are no capacity constraints. If facility n is

o~)cr~cil, a I'ixed SCILIP C O S ~ I;,, IS incut-red. I I ' C U S I O I I I ~ ~ 111 IS assIgncd lo opcn

Iicrlity 11, iin assig~r~llerrt cost C,,,,, 1s incurl-ed. 'l'hc problcin i s 10 choose

open I:clcilil~cs md . C U S I O I I ~ L ' ~ assignmenls 1 1 1 ~ ~ n i n i ~ n i z ~ lixcd selup costs

plus nssignincn~ cosls, sul~jecl to cach cusloincr being assigned lo csaclly

consolidalcd tcnninal to transport Pi-oducts li.om various sources to various

costs Instcad ol'rnakirlg direct shipments, each sourcc can shop ill bulk LO

one O r rnorc coilsolidutcd terminals. 13ar ~ ~ u l ( 198 1) developed a branch-

progranimillg problem having a linitc: number of fkasiblc solutions. 'l'he

1~1sic ct)i~ccpt ui~~lcrlyi~lg the bra~lch-;lrld-.t)ound ~ccllnicluc is Lo "divide"

a d "Conquc~-". Sincc the original "large" problerr~ is loo dill'icul~ lo be

solved d~rcctly, i t is dividzd inlo smaller and s~nallcr sub-problz~ns i~nlil

pur~itio~liilg 1 1 ~ cntire set of lkasiblc solutions into srnilller and s~nalla-

subscls. 'l'hc conquering (I'athorning) is done partially by "bounding" how

good the bcst solution in ~l lc subscl can be and then discarding [he subscls

if-its bound ind~cates that il cannot possibly contain an oplinlal solution l'or

8

On the practical applications of linear programming to distribution, Zierer et a1 (1976)

researched on a linear progamming formulation of Shell's distribution network with three

, products handled by three transportation systems between four sources of product and a

large number of transshipment point and terminals with emphasis no cos! logistics,

ranking of various alternatives, and some applications o mixed integer fbrmulations.

Klingman et a1 (1975) presented a specialized method for solving transportation problems

wi!h several extra linear constraints. The method id basically the primal simplex method,

specialized to exploit+hlly the tol;ological st,-ucture embedded it1 the problem. Also, on

comparison of shortest path. algorithm, Golden (1973) presented some computational

experience to support the claim that a version of 13ellman's algorithm for the shortest-

path problem outperforms Dijkstra's algorithm for the shortest-path problem, for a

certain class on Networks. Still on solving transponation prob!em and its variants,

Appa(1973) showed that either; -

. I . the objective function is unbounded, or

. . 11. the problem is infeasible, or

... 111. the problem can be solved by solving a related transportation

Problem for the fifty four (54) out of eighty four (84) unimodular linear programming

problems considered in detail. Scrinivasan e f d (1973) on his work "assigning uses to

sources considered a special class of transpol-tation

pioblcr~~s 11.1 whii.11 111c rlcccls ol'cach uscr arc lo be supplied c111i1-cly by one

01 ' lllc ;ivailablc sources". A bra1lch-und-110~11d algori~hn~ li)r linding the

C I C S I I ~ C C I w lu~ io r~ W;~S prcsc~~\cd i111d ~ I ~ U S I I . ; I I U ~ 1 ~ 1 ~ 1 1 an L ' X ; I I I I I ~ ~ . MOIX 011

h c 111c111od 01. solvii~g trarqwrt;\~io~i l)rol)lc~ii IS tlic worh 01. W;I~CIICI-

( I L)72) wllo dcscr~ilxd tl ncw tra~lsl)or\i~tiorr ~)r'ol)lc~n n~oclcl ~vtlicti ~ 1 s t ~ ciicll

L O ~ L I I I I I ~ 01' 111c cost matrix i l l 1~11.11 ; I I I ~ was ideally S U I I ~ ~ LO C~rnpuLcrs wlth

t i I Lhlas (1965) co~~sidcrccl :I solu~ion 10 ;I I ~ I -g~- - s i . ak

lralq,orlatlol1 problc~n, whcrc troll1 il largc- sciilc ~ra~tqxmation problcrn, a

~ r r i u l l u ~ ~ o w was derived. 'I'lir: l irs~ ~ - ( ~ I c I N was SOIVL'LI by ~nc~llod ol'

aggrcgallorl uird thc 01I1cr by r-l [.>I-owss 01' p i ~ ; l l disagrcgit~ior~, dclx~lcling

01) 111c 0p1im11 S O I U ~ ~ O I ~ LO thc I ' O ~ I I I C C . '1'1-1~ IWO s11h-l1iol)Ic1ns wcrc tlicn

L I S C C I lo~,cLhcl.

. . ( ; I ) I I a I r l y o o " n i l o l i i o ~ i I I o r ~ y problcrn,

011 the application of transportation problem to solve other related problems (production,

shceduling, inventory, etc.) Bowman (1956) suggested that the same problems may be

placed into a transportation method framework and further, that many transportation 9

problems may be extended to include multiple times period where this is meaningful for

the reap of several advantages. A generalized scheduling problem was transformed into

the standard lorrn of the transportation table.

Finally, in a premier. work on the flow of motor spirit from refinery to consumers by

Shragah (1954), there was a reorientation of the distribution pattern of motor spirit based

on existing depots. This was necessitated by the great expansions in the United Kingdom

(UK) retinery capacity. In his work, several factors were considered as components of the

system's storage capacity such as rate of daily issues, size of the economic transportation

units used for moving products, the frequency with which replenishments must arrive to

keep pace with issues, insurance cover against break down or delays, trading value placed

on possessing further reserves to ensure maintenance of business against erratic

disrup~ion, for exaniple, labour disputes. The assessment of daily issues was based on

peak arid the storage capacity allowed for absence of replenishment on Sundays and ibr

24-hours minimum service.

pclrolcurn products in Nigeria by NNI'C. 'I'Ilc clussical ~ransporlation

h e i n i t l d lxisrc variables in a classical 1ranspo1:;won problem ~nodcl arc

~woducts, into the Nigerian ~narket hiis been through NNI'C - PI'MC. 'l'he

1mi111y iiS ;L resi11.I 01' thc poor pcrlim111r1g st;itc 01' the rc l i~~cr i~ '~ . IIL'IICC',

pcr day (kpd) and optimum capacity uchievcd.

TABLE 2.1: SOURCES OF PETKOLEIJM I'IiOL)IJCrT'S IN NIGERIA

SOURCE: Federal Goverm~~ent's Special Commitlce lieporb on tlrc Review of Petrolcum Products Supply and Distribution (Octobcr 2000)

The primary products (white product) produced by thc four rcfincl-ics are

. . 1 I Dual Purpose Kerosene (DPK) - Kerosene

... I I I Autoinolive Gas Oil (AGO) - Diesel, etc.

'I'he value chain of a local Rcfincsy begins with i~ ip i~t supply, which mi les niilinly li-om

cride oil delivered by pipelines from ihe explo~ation lields. The govemmcnt allocates thc

crude oil to NNPC at a Goverlllncnt psc-determined price ol'Bf9.50 per barrel as at ycar

2000, even when the international price is rnuch higher. Covcrnn~ent allocates 3,000,000

bpcl to NNPC for refining to satisfy dorllestic nccd. Ilo\vcvcr, ~ h c rcfining capacity of thc

bpd wds sold by NNPC at the international market price. 'l'l~c proceeds were applied toward

the importation of producls by the N N P C lo salisly thc donicslic consumplion. Similarly,

during 1l1c first q ~ ~ a r t e r oi' the ycar 2001, 572,625 barrels pcr clay (bpd) or 76.35% ul' the t

cruclc oil purchased was allocr~tccl lor domcsl~c rclin~ng, wliilc t l ~ c balance of 177,375 I~ptl or

23.65% was cxporled. This has revel-sed Llw t r c ~ ~ d ohm-vccl during year 2000 w h e ~ c 76'5: oi'

thc c~ucic oil purchased by the corporation WiiS ~mporlccl and only 24% was processed

locally o w n g to the problems with the clo~nestic Kclincrics. 'The NNPC consigns all he

petrolcum products to PPMC Sol- sales and tlislrib~~lion.

2.2.2 SrI'OKACJS FACll~l'J'IES

, . I hrce main groups own the major storagc and clispenstng Iicilities nation widc. 'l'hey arc

NNPC', Major Marketers and the Indcpcnclcnt Markctcrs. I ' k coinbinccl capacity of thc

sloragc: Depots for products reprcsenls 71 clays, 99 days and 108 days nation-wide

'sufficiency at thc consumption levcl ol' 18, 8 ;111tl 10 ~iiillion lilres pcr day ibr I'MS, DPK

ancl AGO, rcspeclivcly.

'The hciltlies are:

1. 23 Depots/hcilities ownecl by NNPC and localctl in various parts ofthc country.

. . I Apapa Major Markctcrs I'acilitics

I'lic total storage capacity i s as contained in Appendix: 2A ." It is obvious

'I't~c~~cl'Orc, lllc solution LO 1111: prot>lc~ns Lllill invariably rcsulls in scarcity

I I

( 1 ) Sx,, 5 S, I - ; I ,?,. . . .,I11 (Supply) . . . . . . . .2.2

( I V ) X l 1 4 l i w all i and J . . . . . . . . . . . . . . . . . . . . . . . .3.4

I I I

x dl J 1,2 . . . . . . . . , I ) . . . . . . . . . . . . . . . . . . . . 2.6 I

1 1 1 LLISC 01 ' L I I I I M I ~ I I C ~ ~ hysic111 ( ~ I I C I ~ C S, w dl I ~ C ~ I ~ C S C I I ~ S i~ ~ ) O L I I I ~ I . ; I I I I C ~ i l u n

csact rcquirc~nc~~l), a i fu~n~i~y sourcc or durnlny dcs~inic~iou car1 bc

I I I L I O ~ L I C L ' ~ 10 I L I ~ C I I I C S I ~ I C ~ i n O I ~ C ~ 1 0 COIIVCII I I I C i n c q ~ ~ ; ~ I ~ ~ i c s 10

C C ~ U ~ I I I ~ I C S iind saiisl-jl ihc l'cusibiliiy condi~ion; see, libr csuinplc, I lillicr &

( 2 ) I : ~ i ~ s i b l ~ solution p r o l m - t y : A 1lcccssiir.y aild sul'l'icrwt condilion

The current row 0 which contains the coetticients of the variables in the

objective function can be obtained without using any other row simply by

calculating the current values of Ui and Vj directly, where U; and Vj- are as

defined in section 2.4,

The leaving basic variable can be identified in a simple way without

(explicitly) using the coefficients of the entering basic variable. As a

result, the new RF solution also can be identified immediately without any

algebraic manipulations on the rows of the simplex tableau.

Etticiericy and convenience: for a trarisportation problem having m

sources and n destination, the simplex tableau would have m + n + 1 rows

and (m -+ 1) (n + 1) columns (excludipg those to the left of basic variables

(XiJ) column) where m is the number of rows and n is the number of

columns. But the transportation simplex tableau would have m rows and n

columns (excluding the two extra informational rows and columns)

showing description, destination and demand for rows and source supply

for columns.

2.3.3 LOCATION OF DEPOTS

Appendix 2 H shows the map of Nigeria showing sources (Depots) and Refineries in

Nigeria. By ignoring the geographical layout of Depots and

prublw ilS S I W W I ~ in Appendix 2C but with m ; 2 sources and n = 5

ticstlrlations. lkstina~ion here refers to citics withirl a parliculur 'I'ranspor-t

i .':r,crltial %one (,'I'DZ) wherc li lling stalions arc si~uatctf. Sources and

dcstrrlat~ons 111 Appendix 2C arc each rcyrcserltcd by a nodc. 'l'hc arrows

I illking the sources and destination represent the ruutes betwce~l thr: source

arid dcst I nations.

V, -- 1,argcst uni t cost slill rcnlrrining in that column.

\ I , M~rIt~ple ol'or~g~nrrl row I [hilt has been subtracted (drrcctly or-

I I I J I ~ L ' C ~ I ~ ) li'o111 orig~rlal row O (contilming Lhc coeJ'Iic~cnts of the

Ixis~c variables) by thc s~rnplex method dur~ng all rtzrutio~ls leading

lo tllc currc111 simplex 1, '1 L ) 1 cilu

V , Mulriplc ol'originill row 111 1.1 that hm bccr~ sub~rirc~ctd (drrcctly

oi ~ l ~ d i r w ~ l y ) l'rorn osrgi~ml row 0 (cor~tair~ing the coc~'Iic1c1~1s 01'

1 ~ 1 s k V ; I I . I ; I ~ ~ C S ) lw 1 1 1 ~ s11111)h.s tn~-tI~o(l ~ I I I V I I I ~ ~ , ;\!I I I C I : I I I ~ I I S I L Y I ~ I I I I , :

lo I I K cut r.cllt si~l~l.)lcx lilblcai~.

111 rlu~~lbcr- o 1 ' 1 . o ~ il l a trilnspor~;llIw si~nplcx tihlcau cqual to the

~~urnt)cr oi'sourccs

11 r~i~rnbcr ~ S C O ~ U I ~ ~ I ~ S i n ;L ~ T ; L I I S I ) ~ ) I . L ~ ~ I ~ ~ I I simplex I L L I ) I ~ ; I U W ~ I C I I is

q i ~ d 10 ~ l i c I ~ L I I I ~ L X ~ o l ~ d c s ~ ~ r u ~ ~ o ~ i s .

IL'I u huge pos~livc nulnbcr (gol~cn I'onn thc big Eul I I I C I ~ W ~ ) Io

icprcscI11 ;I 1111gc urlil COSI li)r C O I I V C ~ ~ C I I C L ' .

2 .5 'I'll 14; f l ' I~ANSI'OI~I '~ \ ' I ' ION A1 ,GOl~1 ' l ' l l k1

A C L ~ ) ~ ~ I I I ~ , 10 I llIl!er 22 l , i c l x r ~ ~ w ~ ~ ( l9(j5), chaplctr 8, lhc lrimsp)sla11o1i

1)ioOIc1~ IS g c ~ ~ c ~ ; ~ l I y ~ o ~ ~ s i d c r c d thc mosi rnlpor-tant special typc ol' lillcar

pogra1l1111111g 1)roblcnl uld has ;I \~Oldilllg S O ~ U I ~ O I I SCI 01 ' ; ~ l ~ ; , O i l l ~ l l l l ~ I V C I I

bclow

S ' l ~ l ~ ~ l ) I INI'I'IAI .l%A'I'ION

Isor a I I ~ I I ~ S ~ I C ) I I ~ ~ I I ~ , I I problcnl with ni sources ;lnd 11 dcs~inu[iorls, ~ h c nulnhcs

01. ~ ~ L I I ~ C I I O ~ I L ~ ~ ~ 0 1 ~ ~ l ~ i k 1 1 1 1 ~ IS 111 1 11. 1 I O W C V C ~ , 1 1 1 ~ 1 1 ~ 1 1 1 1 ~ ) ~ ~ L)I. ~ M S ~ C \ ~ i ~ i t i l ~ ~ ~ ~ ! i

IS c q ~ l IL) 111 n - I . 'l'hc proccdurc for cons~ruc~ i~ lg a11 irlr~ial 131; soltilion

S ~ I C C I S IIIC 111 1 II I SIC v;~riilbl~s onc at a lime. Al'icr cad1 sclcction, a

valuc tha~ wll siltisly onc additional collslralrll (thcrcby eli~ninating tI1:1t

coIIstr;Ll~It sow or C O I L I I I I I I 1'1.0111 l'urthc~. c o ~ ~ s ~ d c ~ x t i & li)r poviding

( a ) 'l'hc Nar~hWcsl Car-ncr Mc~hod

( d ) li.ussclI's App~uxitnulion M e ~ h o d (HAM)

I ~~ . .... 1 5 Supply -X.,;,-.. . - @ . - I ~ o u r c e I . 5 o . . -

X : I p X2j . 60 - . - - .. 50 1 ' P' x,i -

, I:-- . - & , . - - . .. -. 50 . . --

'I'hc sicps 01' [he solution arc its lidlow: X.12, and X.,, arc ~ h e variables

elcmcnc in rhc row (colulnn) limn rllc 111:sl sr11;lIlcsl unit (C,,) it, ~ h c

1.1 14 I I I5 I & ) 10.

M O M . .- -. -- - - --- -- - -- . . -

Supply 1 - h ~ Ili l'll.~.cr~cc: - . . - -. . . . . - . .- - - - -- 50 0

circled and lhc: S I ~ ~ I I I C S ~ unil cost i l l its row or coli~~nn is ericl~scd in u box.

'I'llc rcsul~ing sclcctiorl (and value) ol' thc variable having h i s unit c o s ~ as

the IIL'SI O;ISIC variable is i~lcficatcd i n t!ic lower right - hand comer ol' the

currcnl table, along wilh (he row or column ~hercby being clitiiir;::!t:d Iio111

I'ur~llcr consdcraiwn. 'I'hc tablc for the nest 1tcriiIto11 IS cx i~ l ly thc sarnc

except lor dclcting or ignoring this row or colirmn and subt~act~ng ihc cost

allocutro~~ tiom its supply or demand (wll~ch ever scmar~ls).

I t should bc notcd I I U I I<usscll's Approximation MdhoJ, which

Applying HAM to the Mutro Wntcr Distr~ct I'roblo~n in 1 lillicr ck

I,icbcr.man ( 1 995), chapter 8 , we havc thc following:

'B'AIiIJ; 2.7: AJ'I'I,ICA'I'ION O F l<USSl~:l J,'S AJ'l'l<OXl Niil'l' iON 1 1 ' 1 1 1 0 (HAM) ' 1 ' 0 ' I 1 M E'I'ItO WA'I'EIt DIS'l'I<lC'I' l b k t 0 U l , l ~ ~ M

----..-....-----..A -.--.---* ".-

Tcos t (~cf l s of Millions of Dollars pcr I Supply 1

- d

A,. - CI 12 -- (1 1 - V2 10 -- 23 -- 19 = - 23 -C1.{ --Dl- T. { -23 -23- -23 : - -23

'I'tic largest ncgalive value ( in absolutc knn) is -24. ScIcc1 XL? =- 0 and

~ ~ I L ' I L ' row 2. ' 1 ' 1 1 ~ remaining basic variublcs are X3? = 20, XS.\ = 30.

* denotes the selected basic v:rriables !'or which allocation is made i l l every

'I'able 2.8 glves the sulnmlrry f or thc selccted basic variables only

( c ) VO gel's Ap1)1'0xintitti0n I ~ C ~ J I O ~ (VAM) hits bccir ;I P O ~ L I I ; ~ ~

criterion lbr n~uny years partiidly bccausl: i t is rclativcly easy to in-rplclncnt

by I I ~ I K J . 'l'his is bccausc thc penally (di l'l'crcncc) represents tlrc mirlin~u~n

cxlra u r r i t cost i~rcurrcd by l'ailing to ~nakc: an allocutiorr to Lllc cell having

thc ~ ~ i l l l c s t uni t cost in that row or column. This criterion docs take cost

inlo account i n an cl'kclivc way using llrc ~.)cnally. Also, with this critcriotr,

i t is morc convenient to work with cost and rcquircrnc~r!~ L;.I~!L' ralhcr than

with conipl~tc tr;1nsporliltiorl sirnpl~x tableau. 11 is heuristic and usually

provides a better starting solution. It also yields an optirlrnl or ney optirnnl

slartirrg solution in most cases. For thcsc reasons, VAM shall bt: used Cdr

the a i l i ~ l y ~ i ~ in this work. In the Metro wlilcr district problem discussed in

sccliorr 2.5. I , i t yidlds iul oplirrrum trarrsportalion cost value ot'2,460.

(J) Itussell's Ap~)roxinw~iocl Method provides ntrorhcr cxcellcnt

cr;~criorr that is still quick to implcmcnt o n :I coinputcr (bul 1101 m;~rrunlly).

tl~hl~nct advantage ol'liusscll's approximi~tion rncthotf ovcr olhcrs i s Iha1 i t

is p~~ttcrncd directly alicr slcpl of the transportat~on sirnplcx method,

which somcwhar sirnplilics the overall coding ol'tht: cotnputer progrutl. In

parl~cular, ~ h c c, and v, values havc bcin dcl'incd ~n such n way that the

- -- r.clatrvc values ol'thc C , - U,--- V, cstirntrlc thc relative value!; ol'C!, - I!, -. V,

I Icncc, duc to thc aclvantagcs cshibitcd by V A M and It.4M, his work sl~i,:l

utilizcs the two methods i r l tlw Optimal 1'ra11~1)0rtiltio11 Algoritlrnr

(O'I'A) clcvcloj~cd in this work to dc~erniinc which ollc yields a bccter

optimal solut~on. In addition, V A M and RAM prodwc thc same {bur basic

vuriablcs out ol'a total ol'scvcn in the starling solution ol'tllc Mctro water

dish ict problm considered i n scction 2.5.1. Whcrcas NorthWest- Corner

Also, nmdc ol'data collection fbr this wol-I; is csha~~slivzly discussed, and

Lhe cho~ce ol'cilics with iill~ng slat ions as destination in Akwa lbom Stale,

Nigcria. All the L.ocal Govzrnlnent Arcas in thc Statc wcrc considcrd.

Attcnfion was paid Lo only thosc with [Jrban Status and prclkrencc was

givcn lo tliosc with high rated urban standard using a conibination 01'

hclors as ~ndcx. 1:actor.s that were considtxd arc thosc that rela~c with thc

dcm;~~~d/co~~sum~>t ion of I'etrol in the cities. 'l'hcy are:

( i ) l~conomic / industrial standal-d, .

.. ( i I ) 13i1sincss/corn1nerci;1l standard,

( , i i i ) Scal ol' Atluli~iislratio~l,

( i v ) I'opulation.

;.tnd rhc subur-bs. A W ' - I I lisi ol'tfic 1Tllirlg slations was obtained l'or each of

Slalior~s in a c i ~ y ) wer i assigned on N cards (ol'the same size and sl~apc)

to 111c ur~its (.tilling stations) in the population. 'l'hesc cards were p~lt in a

No. 5 is dr n\vn li,r irlstancc, this rncarls ' l l u t a rlulnccl lillirlg ,stitlio~l with

No.5 i n the li-ilne is included in the sarnplc. 'l'his is cultcd simple random

Data on the actual rnontldy allociltion of l'clro! fl-orn Depots to

I-~llir~g S~rtrorls In each ol' the live selected crties were obtained l'rom the

cornplcmcllt the overall data rlccdcd in this work.

0 1 ' the three common white pr.oducls of Ikh-olculn (I'ILIS, AGO, I)18K),

I'MS is clmscn for study and analysis in this work because of its relative

csscrl~iality ill the industrial lilk ol'lhc economy 2nd ils rccu~.rencc scarcity

over the past decade. Again, dul-ing the first quarlcr 01' year' 2001 under

co~lsrdcration, ill1 the rclinerias were operational ani opcmrd at i~nprovcd

capacrty ut~lrzat~on cornpard to tlre year 2000. This cnul)lcd NNYC to

rned and even L ' X C C C ~ the nattor~al demand for AC;O arid L)PK to a large

cxtc111. 'l'llis has brought about significant reduction in the quantities of

I'MS durrrlg the period under cons id em ti or^ was still signil'icanlly bclow the

the distributiorl ot'this I i~nr td available quaritity

I'ctr'ol is distl-ibulcd ill twenty-two Depots including Itcl'incries whcrc

01' puducls is by lrucli lo a f~nitc nurnbcr 01' l ' i l l i lq . ; stations i l l dillrcllt

pans oP tlic co~rn1r-y. In th is work, the Dcpots at Calnbar and Port I-larcourt

I<el'~ncrics arc used as sources ol'supply whilc the dcslirlrltiorls are thc total

nurlibcr ol' tilling stations in each of thc selectcd livc cilcs in Akwa lbotn

SL11c.

'1'0 reduce the shipping cost, u c h I>epot is allocated a ccrtaiti quantity 01'

' 1 ' 1 ~ problem dcpic~cd in Figure 3.1 is actually i t lincar programming

( thc Xij) SO i t L . !o

Subjccl to thc constrainls

'I'ahlc 3.2 h ) w s the constraint coefticicnts f i~r the distr~hu~iorl ol'l'clrol i n

thc l'ivc citics dcrivcd from rhe ubovc linear progl-ammi rig problem.

I I 1 City

I C ~ I I S ~ O ; . &

I t is rh~s spccial sLruc1ur.c in the pattcrll 01' tllcsc coel'licrcnls Ilrat cl~s~ingui:.'. *s this problem as n transporralion prublcrn. Any linear programming probl'crn lhat lits d~is special li)rrnulnrion is ol' the transportation prohlcm l y y , regardless of its physical conlcxt; set: Hillier k l,izbc~-niar~ (lc195), chapwr 8. I-lencc: thc obvious advantages 01' tl.i1lls~)Ortatio11 problcms as carlicr stated in scction 2.3.2. 'I'his justilies thc ilt i l imti~~l oStr;111sportiltio11 problem model in this work.

I lmx, the stalldard transportation problem rnodcl discussed in secticm 2.3

3.3.2 ASSt!ill I''I'I0N

, . l his work assumes that 111c tola1 supply lrom 1)cpols is dz!ivered to

spccilic dcstinutions (lilling stations) and as such do no1 lakc into accourit

. the c I ' I ~ c I 01' 110a1dilig and ~111~1gglirlg 01' ~ ~ O C I L I C L S LO ~.~cigltI)~rilig ~ou~i l r ics

o r dive~won l i j r black markel txlrnlng as clui~iicd by NNI'C since cxistirig

law;., ~askli)rcc(s) arid unil'orriicd pcrsonricl guararlluc tlic :hstmcc 01' such

opcralions or at worst is rcduccd to a bcarable minirnuni.

solurion. When thc total supply equals the total Jcniand

111 caw ol' unbidarrccd moclcl like ours wltcrc: tlclnand exceeds s ~ ~ p p l y mil

supply i n ordcr to corlvcrl thc inequali~y to cqualily and satisfy the

I'cii~lbil1ly coiidilion. 'l'hc l'rcqucrlt scarcity ol' I'cl~ol wil!)in a Ti)% acluitIIy

justilizs lhc need for thc introduction of' a dummy source to con-tplcn~cnt

the availi\blc sourcc: i l l ordcr to satisly the dcmand. 'l'abic 3.2 is an

ilicomplc~c cost and licquir-cmcnl ~ablc. S~ocli :!! Lkpots I'or i t lc pcr'iod

under considcralion arc

i '~~lil t)i ts 13,448,653 lilscs

'I'liis ~ot;il supply o f 8,882,000 Iiircs docs riot cqu;il the toktl cstirnatcd

in u rnanncr that will opiimally distribule lhc shortagc quan~ily 01' I'cirol

- -- - -. . - - - - - - --- - ---- - --- "- --- -.

-. . . . . . . . - Cost in bl per lilrc - .

. - . .. . - - - -.- ' - - . - --- . -- -- - - . -- .- A- . " - 1 I - LJYO I liUr

I OItON SIJI'I'I .Y

I*+!?;PJ! ;- . .- --- .---- A'%'!%- . . .- - . . . . ., . . . . .. . . . - . .- . - . .

C'a1nl)ur 1 1.10 1.10 1.20 1.20 1 1.40 1,090,300

bound. Ths upper bound is the quantity requested unless the request

exceeds the total supply remaining after the minimum needs are met.

Again, according to Wagner (1972),chapter 6, the numerical values of the

Xij are inherently approximate, since in most rcal applicalions the values of

'D, are only forecasts of requirements during the planning horizon. In this

iipplication problem, let D represent the estimated demand per month by a

filling station while other pwunders, are as defined in the expression for D

below. 'hu rr~athematical rclation dcrivcd for the estimation of the demand

of Petrol in this work is based on the number of sales day for a given

quautity of petrol and is given by 1

where 26 is the total possible number of sales day per month (excludmg

Sundays).

is the average number of sales day (i.e. duration of sales in

days) for a quantity of Petrol supplied.

Q is the quantity or allocatiori of Petrol supplied in a given

period.

'The average sales day in each city Liom field survey by the reseacher are

as follow:

[EY 1 AVEKAGl: ---- SALES DAY PER TRUCK 1 IKOT EKPENE UYO lKOT ABASl E E T OKON

Therefore, the estimated demand D are given bclow:

30000 1)- - 5 X 26 - 156,000 Ilucrs pcr month l'or lkoi Ekpnr; c~iy

3209800 *- 60 X 78 = 4,172,740 litres for the first quarter of 200lfor

lkot Ekpne city

where 3,209,800 is the total supply of Petrol for the first quarter of 2001 to

lkot Ekpene.

60 is the total actual sales day in the iirst quarter of 2001 to Ucot Ekpene

78 is thr: total number of possible salcs day in the first quarter of 2001 to lkot Ekpenz.

i;or llyo

D = 5069800 4 8 X 78 - 8,238,425 litres

lior I kol A busi L) = X 78 - 74,1100 I IUGS

Por Eket

For Oron 244800

D= 60 X 78 = 3l8,240

li tres

litres

Hence, the Cost and Requirement table is shown in table 3.4 'l'able 3.4: Initial Cost and Hequirenwnt Table for Distribution of Petrol

C -- --.

Cost in $4 per litre ====I

~ZL~L Minimum 3209800

Where destination 1 = lkot Ekpene,

2 = Uyo,

3 . = Jkot Abasi

4 - - Eke(,

5 - - Oron.

and source 1 - - Calabar,

2 -- - Port I4arcourt,

3(D) = Dummy Dcpot.

The imaginary supply quantity for this dummy source (Depot) is the

amount by which the sum of the demands exceeds the sum of the real

supplies.

The cost entries in the dummy row are zero because there is no cost

incurried by the fictional allocations from this dummy source. Finally, in

kot Ekpene, Ikot Abasi, Eket, and Oron destinations, the dummy sourct:

has an adequak (ficlional) supply to "provide" at least some of their

~ninilllwn uccd in additio~l Lo its oi l la rcqucstcd amount. Thcrcfore, since

for instance, lkot Ekpene's mini~num need is 3,209,800 litres, adjustment

must be made to prevent the dummy source from contributing more than

962,740 lilres. This adjustment is accomplished by grouping Ikot Ekpene

into two destinations; one having a &mad of 3,209,800 litres with a huge

unit cost " M for any allocation from the dummy source to ensure that this

allocation will be zero in the optimal solution, and the other having a

demand of 962,940 Litres with a unit cost of zero for the dummy source

allocation. The same adjustnxnt applies to lkot Abasi, Eket and Oron

destinations. Uyo does not require any adjustmen1 because its &maxi

(8,238,425) litres exceeds the dummy source's supply (4,409,785 litres) by

?,X28,540 litrcs, so ~ h c atnoutlt supplied IO l l y o Ii.olll reid soimcs will Lk' fit

least 3,828,540 litres in any feasible solution. Consequently its minimum

need of 5,069,800 litres is guaranteed This formulation gives the final cost

and requirements table shown in 'Sable 3.5

Table 3.5: Complete Cost and Requirement Table for the distribution

of petrol

- I

---

--.- ELI l

6

1.2

1.2

0

-- 3uMO

3.3 '1'HE OYTLMAL 'I'l~ANSI'OR'l'A'L'ION ALGORITIIM (OTA)

S'I'EP 1: 1NITlALIZATlO N

Construct an initial Basic I'easible (BE) Solution using

(i) Vogel's Approximalion Mcthod (VAM)

(ii) Russell's Approximation Method (RAM) .

Go 10 step 11

STEP I I : COMYAR l SON

(i) Compare Iht: optinla1 HI: solulion obtained using VAM and RAM.

( i i j Select Ihe method with the smaller value of the objec~ive function

(Z), one with a srnallcr minirnum co'st value,

Go to step 1 11

(i) Derive U, and V, by selecting the row having the,lugest number of

allocat ions.

(ii) Set its Ui = 0, where U, is the row having the largest rwnber of

allocation.

(hi) Solve the set oi'zqurriions Clj .- U, t V, l'or ~;u;il (ij) swh lIul X , is

basic.

(Note that this derivation of the 11, and V, values depends on wjlich

X!, vari;lblcs arc: bas~c: va~~;lblcs in ilic currcrst r3F solution. So this

derivation will need to be repeated each time a new BF solution is obtained).

(iv) If Cij - IJ, -Vj 2 0 for every (ij) is non-basic, then the current solution is optimal.

+

So stop, otherwise go to step iv.

STEP IV: ITEKA'I'ION

1. Detennine the entering basic variable: select the non-basic variable Xij having the

largest (in absolute te~ms) negative value oSC;, -- U, - V,.

11. Determine the leaving basic variable:

Identity the chain reaction required to retail1 fbasibility when the entering basic

variable is increased. The donor cell in the chain reaction having the smallest

allocation automatically provides the leaving basic variable (break tie arbitrarily).

The chain reaction ban be identified by selecting from the cells liaving basic variable:

a. the donor cell in the column having the entering basic variable,

I>. the recipient cell in the row having this donor cell, in the column having this recipient

cell,

c. and so on, until the chain reaction finds a door cell in the row having the entering

basic variable.

When a row or column has more than one additional basic variable cell, it

may be ~~wcssary to trace thc~n d l further to see which one must be

selected to be the donor or recipient cell.

(ill) Dwr-mine the ncw 131; soluliori:

Add the value of the leaving basic variable to the 'allocation fbr

each recipient cell. Subuuct this valur: from the allocation for each

donor cel I .

( iv ) Re'peat tht: optirr~alily lcsr (Step I l l ) at the end of every iteration,

until there is no basic entering variable. That is Cij - U; - Vj30 for

all i j. Consequently, no chain reaction is started off. Hence,

oplimality is attained.

4.1 INTRODUCTION

'There is a significant gap bctween model development and implementation

so the potential of Operations Research / Management . . Sciences (OWMS)

is most of the times largely unrealized. It is on this prcrnise that this chapter

considers the theory of implementation of (OWMS) procedures/models,' its

associated, problems and tinally the implemen~ation and application of

0'1'A developed in this work for the distribution of petroleum product

(petrol) in Akwa Ibom State. On the whole, this chapter presents a solution

to the distribution problem of petrol in Nigeria in Akwa Ibom State as a

case study.

4.2 MEANLNG 01; IMPLEMENTA'I'ION

It is donsidered to be the utilization by a client of the result of OR/MS

work, particularly formal projects. I t rcfers to the actual use of OWMS

output by managers that inlluznccs thcir decision processes. In this view an

OIUMS ~nodcl or projccl that does not change decision-making in some

way would not be considered to be implemented; see for example Schultz

and Slcvin ( 1 975), chapter I .

lrnplen~znbtion is a very old process indeed. Certiiinly il is as old as Lhc

history of technology. The process of technology consists of developing a

clwicc, li)r csw~~plc, thc ~ I L I ~ O I I I O I ~ I I C or radio, whiclr did not cxisl b c f ; ) ~ ~

and which ~hr: tcchnologicai cxpert dccmed to be a bet~cr choice than those

equipment that were available. The implementation process consisted of

the devclopm~~nt of this product arid selling it to the customers.

Once a product (model) is cerlilicd correct, then the. problem of

iniplemcntation becomes a problem of how to sell (utilize) it and what is

surcly the correct prescription (optimal value) for thc buyer (operators).

Irnplzmcnhtion occurs whcn an organisation has serious problems, and a

group of "experts" examine thc problcms and attempt through their

exani~natlon to help the managers of the organisation to makc: the best

choices to gain the really important goal of the organisation.

4.4 'I 'M LMl'LEMENr1'A'l'lON PROCESS

Implernt.ntat~on and hence implementation research, has been viewed from

a variety of perspectives including selling, involvement, mutual

icodcrs~;uiding, and organisutional change. Selling implies that

implementation is a marketing problem and that the product of Operations

liesearch (e.g. models) must be skillfully made, packaged, and sold to

potential users. involvement suggests that implementation requires

potential users to play an active role in the research process, becoming

involved as participants in management Science. Mutual understanding

reikrs to a state where the researcher and manager each understand the

other's stake in the project. Organisational change describes a view of

implementation that Swuses on behavioral changes that- occur in the

process of model acceplance. Each of these perspectives is important in

~dentifying implcmentaliim problems and even suggesting practical

solutions, but research on the process of implementation requires that these

ideas be integrated in a scheme that recognises their interdependence.

Radnor etal(1970) have found that top management support and

clienl - researcher relations are lactors related to implementation success

across different organisations.

In~plemcntation still rneans diSSercnt th~ngs to different people. One way to

classify the issue surrounding the implementation problem is to consider

the OWMS process and the role of implemenlation in it. In Figure 4.1, a

number of ingredients of OWMS activity are identified and classified as

inputs, agcnts, processors and outputs. They relate to the meaning of

implcmen~tion in the following ways.

OIUMS activity begins with the confluence of an organisation and its

problcms with the problem-solving sk~lls of managers and researchers. The

or~gin oS the aclivity, thcn, is problcnl idcntilication and the constraint

require for a solution. The solution is sought with certain organisational

goals arid constraints in mind. In addition, decision criteria must be

established to evaluate the acceptability of a proposed solution. The players

or agents in the activity are managers and researchers. Their roles have

traditionally been cast as managerldecision nrakers and rcsearcher/problem

solver (the later being the Operations Researcher or Management

Scientist). Although, in principle and incre?singly in practice, the roles art:

less distinct and would be reversed or played jointly. In addition to the

ultirnate user may be soincone olhcr than the manager. Those indvidlials

have bcen termed the clients of the reseilrch~r and have been considered as

a separate group by some. But for simplicity, we are considering the

manager and the researcher as 111~' two chief players in implementation

process. If a process has been identified, organisational

I lNPUTS _k I PROCESSORS k( OUTPUTS k

PKOJ EC'I'

liliSliAliCl ILKS MANAtiliRS

FIG 4.1 INCHEDlENT OF OWMS ACTIVITY SO URCL;:SCH UL II'Z AN11 SLE VIN, CI-IA P X R I

goals, constraints, and decision criteria specilied, and a munagerlresearcher

interaction established, then the rnodel -- building or problem - solving

phase of OlUMS activity can begin. At this point, the researcher brings

OlUMS niethodology to bear on the problem. In other words, the

researcher seeks a scientilic solution to the problem. Because model

building is ccntral to OIUMS method, this step usually involves the

develop~nent of some model of the problern and the solution otlkred is one

to the model and hopefully the problern. ldzntilication of the right problem

and development of an appropriate model turn out to be crucial

dctcrn~inants of OWMS success. Equally Impomnt, is the interaction and

ur~dcrsb~~ding of lilanagcr and rescrcrchcr. When this relationship is good

and when the understanding is mutual, the chance h a t thc manager will use

thr: r.csl;irrch rttsults wi l l be grcatly whar~ccd:

The output of the OWMS process is generally activity and specifically

projects, models, and solution. The piojCCtS and models can bc regarded

either as ends in themselves or as the mcms for influencing the

organisation's decision process. The notion of influe~~ce provides a key to

understanding the concept of implementation. The initial state of the

organisation would be a set of decision process (or procedures) and a

problem. At the outpui stage of OiUrrlS activity, the orgiu~isa~ion has a

solution to the problem. IS the solution is implemented, then the final state

ot' the organisation is a revised set of decision processes incorporating the

solution. Implementation, then, refers to the actual use of ORfMS output

by managers that influences their decision process. In this view, an OR/MS

model or project in some way would not be considered to be implemented.

Changes in decision making will presumably lead to changes in h e

manager's and the organisation's behaviour. The degree to which act&

change in the managers' or orgdnisation's behaviour is required for a

successl'ul implcrnentzltion may vary depending upon the researcher and the

situation.

4.5 OWMS IMI'LEM EN'I'ATION PROCESS THEORY

Three principal groups of decision-makers, i.e. managers, management

scientists, and clients, participate in the 01UWlS innovation diffusion

process through the set of dynamic c;o~nlnunic;auon I'eedhck loops

illustrated in Figure 4.2. It is shown that each group has theopportunity to

impede or even prevent implementation from taking place, as well as to

provide support ibr OlUMS products. The primary consideration that

arises from viewing implen~entation this way is to try to determine how

management scientist innovators can inllucnce both the management and

client groups to accept their products.

In addition to the influence thc participating groups have upon each other,

thcrt: are numerous otllcr variables Lfial a l l c l clicnt bchuvioul- towad

implementation. These are: unique product technical advantages, Top - management support for product, product urgency to organisation,

manager's behaviour supporting the product and management scientist's

behaviour supporting thc product. 'l'hc przssurc exertcd by these hctors

upori clienls to imple~nenl u~rd OWMS producls are counterac~cd

t somewhat by still other fiictors that cause resisbr~ce to such change such as

clients, hostility to management science in general, client's natural

resistance to change, extreme h~gh level of product complexity, client's

pecr group behaviour resisting the product and unsatisfiictory past

perforrnancr: of similar ideas. This conllict normally results in some

equilibrium level of resistance for each client affected by the product. It

should be noted that this concept ol'all'ecting factors and some equilibriun

.. level of' r&tance also applies to marlagcrs and Management Sc~entisls but

the rclativc: efi'ects of thc irdividual hcturs are not norm;r!ly thc sane on

tlwsc pnr~icipnls. For inst:,nCc, ;I p r o d ~ ~ c ~ ~h:rl scltms highly rclevitnt to the

rnanage~nent scientist and manager may not be perceived as relevant by the

clients at all. Similarly, thc sense of urgency or technical adequacy of the

product is almost always prceivcd differently by each of the three groups

involved in the implemenution process.

1'KEI.IMINARY WOIZK BY MANAGEMENT SCIl~N'l'lS'f

NO \I/ MANAGElvUiN'I'

NO . ACCW"'ANCL 4

CONDITIONAL

I

UNQUAL FED

-' No r Figure 4:2: The OlUMS product imple~neritalion decision process.

SOOf<Cli:S( Iki U L U A N D ,S',li VIN, ClfA1'77iIt 8

Thc problem of implementing OIUMS is an old one. OWMS is an applied

field and, although the dzvelopment of O~UMS methods may be thought of

as basic research, the primary thrust of activity in this area has always been

dii-ccted at application and implementation. The implementation problem is

also a fundamental one. Management decision making can only be

improved through OWMS if the methods and model arc: utilised in

organisations. Finally, the implementation problem is a continuing one

because future advances in OWMS methods must be matched by new and

useful applications. These are strong reasons for studying the

implernenthn process in detail and with considerable rigor. '

Another reason b r studying the implementation process is the potential

contribution of a research lo lhc undersliinding of olher fundanlental

processes of orpnisations, that is, lbr dding to organisation theory,

theories of change and knowledge oi'technological innovation.

.. 4.7 lMlbLEMEN'CA'l'lON 01; O'I'A IN 'I'I I E I)IS'l'l~lUU1'lON 0 1 ; lbE'l'HOL 1N AKWA 1UOh'l S1'A'I'lI.

Construct an initial Basic Feasible (UF) solutions using Vogcl's

Approximation Method (VAM) and Russell's Approximation Method

( IUM) for the given problem in Table 4.1

Table 4.1 COS'I' AND HEQUIIWMENT TABLE FOR TlfE L)IS'I'HIBU'I'ION OF l'Erl'KOL I N 5 CI'I'IES OF AKWA lSOM STATE (AKS)

A. COM PUrfA'l'lON OP IN I'I'IAL BASK I~KASIWLE (Blq SOLUTlON USLN G VAM

Table 4.2: AYPLICA'I'ION OF VAM 'TO 'I'LIE DLSTWUTION OF PETROL IN

4 1 4 W 5 Illinunate row 31) I

I . ' i l l 1 " I;; I " I .0 1.2 1.0 1.0 1.5

The solution procedure tbr solving the tableau from Table 4.2 is as explai~red and

illustrated in section 2.5.1 ( c )..$rice only one row remains under consideration, then the

procedure is completed by selecting every re~nairring variables associated with that row tc

be basic with the only feasible solutiorl. Hence, XZ3 = 299280, X26 = 300000,

X2h: 244800 and Z = 6533286

U. COR1IDtJ'l'A'I'ION OI; I N I'I'IAI- IIASIC' IiIIASIllLE (Ill:) SOLU'I'ION USING VAM

'I'he solution procedure for solving the given problem of Table 4.1 is as explained and

illustrated in section 2.5.1 (d).

'I'ablc 4.3: AYYLICA'L'ION 01; VAM 'L'O 'I'IIE I)ISTRIIJU'l'lON OF PETROL IN A IiS

1'0 calculate A,, = .C, - U, - V,, select the largest unit cost in row I . Hence; U I - 1.4 and

the largest 111 colun~n 1 is V1 =-= M, and so I'orth .See details of co~nputation in Appendix

4A.

'I'he liirgcst negative value ol' A,, (in absolute term) is XW -74,400. Delete column 9 since

its total demand is salislicd. Go to ~lcration l

The largest negative value of (,, ( In absolute tcrm) is X37 = 187,500. Delctt: column 7 since

its total demand has been sutisficd.

A. COMJ'U'I'ATION 01; INl'l'lAL BASK FEASlBLE (BF) SOLU'I'ION USlNC VAM

'I'hl: solution procedure for solving the given problem of Table 4.1 is as

explained and illustrated in section 2.5.1 ( d )

Table 4.3: AYYLICA'I'ION OF VAM 'I'O TI-JE 1)IS'I'tUL)UTION OF PETROL IN AKS

.. To calculatz'Ai, = Ci, - U, - Vj, sclzct the largest unit cost in row 1. Hence;

U1 = 1.4 and the largest in column 1 is V I = M, and so forth .See details of

computation in Appendx 4A.

'l'htt 1ilrg~'st negative value 01' A , (in absoluk tznn) is X3!, -74,400, Delclc

column 9 slnce its total demand is satislied. Go to iteration1

l'htt largest negative value of (, ( in absolute term) is X37 = 187,500. Delete

column 7 since its total demand has been satislied.

The largest negative value ol'Ai (in absolute term) is X35 =17,280. Delete column5. Go to

iteration 3 .

The largest negative value of Aij (in absolute te~m) is X33 = 4,131,565. Delete row 3. Go

dm The largest negative value of Aij (in absolute term) is Xz4 = 57,600. Delete,4. Go to

iteration 5 .

The largest negative value of' Aij (in absolute term) is X2*. Select Xzz = 962,940. Delete

colunm 2. Go to iteration 6.

- - - - . -L- .- ., . ; . . . - . . j bd 1 3 . 2 W A U ) O 410OhCXJ 2 4 4 X 0 0

The largest negative value of hij (in absolute term) is X28 (break tie arbitrary).

Select &&I =244,800. Delete column 8. Go to iteralion 7.

The largest negative value of Aij ( in absolute term ) is Xzl. Select X21 = 3,209,8000.

Dcletc column 1. Go to iteration h:

The la~gest negative value of A,, ( in absolute tcrm ) is Xln. Select XI) =1,090,300. De

row 1. Go to ilcration 9.

Destination

33 16560

Since only one row rmuins uilder consldcrallon, 111cn the procedure 1s complzted by

selcctrng every renla1111ug var~ablc ussociatcd with t l la t row to bc basic: with the only

Ikas~ble solution. ilence, Xz3 - 3,0 16,560, Xzc, - 300,000.

'I'able 3.4: SUMMARY OF SIE:l,IX'I'EI) LNI'I'IAL ISASIC VARIABLES USING

GO 1'0 STEP 11 of the algori~hm.

S'I'EP 11: COMPARISON

Cornpart: Ihc value of the objcclivc I'urlction, Z, ol' VAM and RANI. Then select the

method with a smallcr valuc of the objective function, Z, (minimum cost). Select VAM.

Go To STEL' 111.

S'I'EY LLl: 0I''l'IMALI'~Y TES'I'

t Select tlrc row wilh the 11ighL'sl i~ui~lbcr ol'alloca~ion. Sclcct row 2. SCL row 2; U 2 = 0; L ~ C I I

solve C , - U, -I- VJ for U, and V, for each (ij) such that X, 1s basic. The basic variables are:

XLI, X22, X23. X24, X Z ~ , &r,, &us X13. X33. X37, X39,

CZI = U 2 4- v, ; 1.0-0 -t: v,

Cz2 = U2 + v2 1 .0=@ + v2

C33 = U? 4- v3

1.2= 0 + v3

C24 = U2 + V', 1.0=0 -+ V',

CZ5 u2 -1- v5 1.0-0 + Vj

CZB = 112 + V b 1.2 - 0 .-t V(j

Czn = U2 + Vlc 1.5-0 + Vu

(213.- U, + v., 1.1 =U, + 1.2;

C33 = u3 + v3 0 = u3 + 1.2

UZ == 0 V I := 1.0

v?, I .0

V, = 1.2

V', = 1.0 V,, ' 1.0

V j = 1.0

V(, .-< 1 .2

V x - 1.5

U I = -0.1

u3 = -1.2

X37 C,7 - U3 4- v7 0 - -1.2 + v-, v7 '- 1.2

X Cs9 - l J3 f Vy 0;-1.2 + v9 vy - 1.2

'fable 1.5: Optimalily Solulion Of 'l'hc 1)istribulion Of Petrol Using RAM

1:ill thc value of U, and V, h r rows and colu~rms rcspzctively. Also, calculat~~ arid lill Lhc

value of C , - U, - V, for each non-basic variable X!, (that is, lor each cell without a circled

allocation) written in thc down right hand corncr of each cell as shown in 'I'ablu 4.5.

Apply the optimality test by checking the value of C,, - U, - V, given in Table 4.5, An

4.5. Therefore, the optirnality test identilies the set of allocations in this tableau ('Table

4.5) as bang oplimul, which concludes the Algorithm (OTA).

The interpretation is that lkot Hkpene can have optiinal allocation of 3,209,800 and

962,940 litres of petrol solcly from Port 1 larcourt RefinaylDepot to meet its total demuid

at the transportation cost of PI 1.00 per litre of petrol. Uyo would have optimal allocation

of 1 .O90,300 litr~s, 299,280 liuzs and 4,148,845 litres of petrol from Cdabu &pot, Port.

1itrrcour.t l<efir~y/Dcpot and thc: dunmy Dcpot rcspc;ctivcly and at the cost of44 1.00, N

1.20 and N 0.00 per litre respectivciy lo meet its nlinimurn need of 5,069,800 litrcs and

e x m 468,625 litres for reserved. This si~uation calls for the establishment of additional

Depot to supply Uyo with a lion share (4,148,845 litr'cs ) of its demand, which is hitherto

being supplied by the Duinmy Depot. i . , . t l

* --* . I

lkot Abas~ cily would have an opt~lnill allocation ol' 74,.800 litres from Port Hirrcourt

Itehery/Depot to meet its totill demand of 74,800 litres at the transport cost'of 441.00 per

litre. Also, Eket city would have an optimal allocation of 300,000 litres at the cost of

dummy source. lo meet its toial demand of 487,500 Litres. Again, this calls for the

estrrblislunent of a depot to meet the denland of the city of Eket. Oron city would also have

an optimal allocation of 244,800 litres. and 74,400 litres from Port Ehrcourt

licfinery/I>cpol and the durnrny Depot, respect~vely, lo meet its total demand need at the

cost ol'N 1.50 and 44 0.00, respcct~vcly. .

4.7.1 A 'I'HIAL OYI'INIALI'I'Y '1'ES'I1 USING IUM '

'I'l~e choice of a set of initial 13F solution from VAM in the execution of 0'1'A is a causc

!'or concern and is to be investigated to justify its comparative advantage over RAM. This

is because [he two rncthods were carlicr adjudged in seclion 2.5.2 lo bc better, scc, and as

such they formed the cornerstone 01' the O'l'A. In order lo effectively compare the

pzrt'ormance of the two methods, this trial teat kcornes eminent.

Sclect Lhe row, IJi, with thc highest number of allocation in table 5.6. Select row 2. Sel row 2; U2 = 0

Then solve C,, = U; + V, for IJ, and Vj for each (ij) such that Xij is basic. The basic

variables are:

X ~ I , X22, XU, x2.1, 2 , 2 , X . L ~ , X 3 , X ~ B , X37, X39,

Fill t h e values of Ui and Vj for each row and column respectively as shown in 'Table 4.6

(iteration 0). Also, calculate and fill the value of Cij - U; - Vj for each non-basic variable

X, ( that is, for each cell without a circled allocution) as shown in l'sble 4.6.

Table 4.6: OYTIMALI'I'Y TEST FOR THE I)l?jTKLBUTlON 01' YE'THOL USING RAM

Since zhere exist negative values of Ci, - Ui - V;, (XI& X18, &) we conclude that the

current L31': solutiorr is no1 opiirnal. Go '1'0 S'l'h;Ib IV ol'thc 0'1'A.

STEP 1V; I1'ElU'I'ION

1 . Choose Xij with the largest negative valuc: ( in absolute tenn) of

C,, - U, - V, as the entering basic variable. Choose Xi2 with

Cj2 = -0.9

2. Increase the basic enlcring variablc (&) fiom Zcro allocation. This sets off a

chain reaction of compensating changes in other basic variables, in order to satisfy

the supply and demand constrams.

Idcnlily the chain reaction by sclccting fiom the cells having a basic variable the

donor cells (X2? and X33) and the recipient cell (X23) (see the set of chain reaction

in iteration 0). 'The donor and

plus sign (+), respectively, at

recipients cells are denoted by minus signs (-) and

[he top right of the circled allocation. A box ( )

indicates the enlering basic variable. 'The value of C , - U, - V, is written in the

down right hand corner of each d l . A circle (0) indicates an allocation for a

particular variable, X!,.

COM1'UrI'ATlON 01: Ui AND Vi 1'OK l'CEHA'I'lON 1

Lel U3 - 0 (break tie ubitrar~ly)

u 3 - 0

v3 -: O

vz '= 0

v. .; 0 3

v, == 0

v 9 = 0

U, := 1 . 1

Uz -.- 1 .2

v, = - 0.2

V(, = 0

v', = - 0.2

Vg L. 0.3

F;ill lhe value of Ui and V, for iteration I as in Iteration 0. Also, calculale and fill the value

of C , -- IJ, - V, for each non-basic variablc X , (that is, lor cach cell wil l~ou~ a circlcd

allocation) as shown in iteration 1

'Fable 4.6: L'IWLQTION 1

Since we still have ncgaiivc vducs of U,, - U, - Vj (Xz2 and &), we conclude that ilw

current BF solution is not optimal. In the chain reaction, the tirst donor cell, X3s gives all

its allocation 17,280 litres lo the entering basic variable, &. This amount or allocation

hacornes a Pictor for compensating changes. Licnce,. the value of the entering basic

variable is added to all rccip~cnt cells (Xs3 and the entering basic variable) and subtracted

from the donor cells (X3$ and X23 ). This auto~mtically balances the dennand/supply

requirements. The 1cavi11g basic variable, XZ2 is the donor cell having the sntallzst

allocation (brcahng tics arbitrarily). Wc: can now procecd to iteration 2.

COMPUTATION OF (Ii AND VJ FOR 1TElW~ION 2

Select the row with the highest number of allocation and set it equal to zero.

Let UZ = 0; then solve Cij = U; + V, for Ui and V, tbr all basic variables;

Fill the value of Ui and Vi for iteration 2 as in iterationl. Calculale and fill the value of C ,

- U, - V, for each non-basic variable, X , (ihat is, tbr each cell withoui a circled allocation.)

S I I ~ we still have negativc v a l ~ ~ e of C 1 - IJ, - V, ( X2? ), we conclude that the current 131:

solution is not opti~nal. 'I'hc first donol- ccll X12 gives i t total allocation of 962940 litres to

the basic variable X22. This set ol't'a chain reaction and this allocation becomes a factor for

compensating changes in othcr basit: variables. Subsequen~ly, the value 01' the entcring

basic variable is added to the rycipient cell, ( X J ~ ) and subtracted from ~ h c donor cclls (&

and XZ3), This balances the dcmand/supply recpircrncnts. The leaving basic variablr: is

X3?. We can now proceed to ilc~alion 3.

COMPU'I'A'I'ION 01; lJi AND V, FOR II'LRAION 1 I I .

Sclect the row with the highest number of allocations and set it equal to zero. Let U2 = 0,

then derive the valucs of U, and Vj by solving C , Ui + Vj for all basic vuiablcs;

X13 : 1.1 - U j + 1.2 11, - - 1.2

X i 3 . 1.1=U1+1.2;U,"-0.1

X37: 0 " -1.2 + V7; v, " 1.2

X3'): 0 - 1 . 2 - v ; ' vy -- 1 2

X 1 . 1 - U 1 + 1.2 11 1 - -0.1

1;111 thc valucs of IJ, and V, I'or iteration 3 as In itzriltion 2 . Also, calculaie and till thc

value of C,, - U, - V, for each non-bas~c vur~ablc X,, as shown in iteration 3,

Since all the C , -- Vi -- V, values are nomagatjve in iteratuion3, the optimality test

identifies the set of allocations in this iteration us being opiimul, which concludes the

algorithm. The value oTZ at each iteration is obtained using equation (2.1).

4.7.2 1 ~ ~ 1 ~ ~ ~ ~ 1 ~ ~ 1 ~ ~ ~ 1 ~ 1 0 ~ 01; Cij - i i i - V, Values

'I'ht: C , - U, V, cluantitics dcrivetl dur~ng thc oplirnali~y test rncasurcs the ratc o fc lw~gc of

(he total cost. Because of the shili of 962,940 sllloclrt~on units from the donor cells to the

recipient cells at iteration 0, the tohl cosl changes by

2 ; 962,940 (0 - 1.0 - 0) - 962,040 (0.3) -= 192,588

'I'hus thc effkct of increasing the cnter~ng bas~c variable (EUV), X J ~ , from zero has bcen a

cost change at the ratc of 0.2 pcr u n i ~ increase i n XJ2. This cxplilins the incrcasc in thc

value o f% by 192,588 units in thc next iteration (iturationl).

In iteratwn 1, a 17,280 allocation unit is shrfied from the donor cells to the recipient cells.

'l'he total cost changes by % = 17,280 ( I .0 - O -I- 0 -- 1.2 ) 17,280 ( -0.2) = -3456.

'I'hus, the ei'l'ect of incrcas~ng the entering basic variablt: (ELIV), Xts, iiom zero has been a

cost change a1 thc ratc of - 0.2 per unil incrcasc in X25 This resulted in Lhc dccreuc i l l thc

vdue o f 2 by -3456 in iteration 2.

'I'he total cost clmges by

Z = 962,940 (0- t .O + 0 - 1.2) 462,940 (-0.2) - - 192,588

lhc valuc of 2 by -192,588 unitb i t ) ilzsa~ion

4.7.3 COILlPAHING I l l INI'I'IAL IMSICI VARIABLES

ALLOCA'I'ION OHTAINEI) FROM VAN1 AND 1UILl

r vAM

ANI) 1'1's

'I'hc transportalion problem algorithm rcquires that m -1 n-l ( 9 - 1 3 - 1 = 11 basic variables

be produced in this problem irrespective of the method or critcnon used in constructing

the slarting basic variables. 'Therclbre, the PI-oduclion of 10 out of 1 1 initial basic variablus .

that is common by both V A M and RAM showed the compe~i t iv~n~ss of these two

methods.

I lowcvcr, thcrc 1s only u discrclvr~~cy i l l Llie selcct~on ol'orle varrablc each by VAN1 and

RAM. While VAN1 had produccd X z j - 17,280 , H A M iw place produced X.,, 17,280.

'I'h~s result by V A M implies 1 1 ~ 1 thc city 01' lkot Abasi could get additional optimal -

allocat~on ol'pctrol by receiving an allocntwn 01' 17,280 litrcs oT petrol at Lhc Lransport cosl

of 441.00 pcr htre from Port Il;i~-court IZcl'ii~cryllkpo~ to lncct its demand, liAh.1 rdther

irt~pl~cd that tllc city of lkot Abasi could get addi~ional optimal allocation of 17,280 litrcs

to meel rt dcrnand from ~ h c l>umniy WLIIIX (13cpot) at the tiansport cost ol'N0 00 per 11tl.c

this problem for optimizing ~ h c available resources. I t is also obvious 111at 9 basic

var~ablcs: X21, XL2, X2j, &(,, Xlh3 X 1 j, 3 , Xs7, a11d X selcctcd

by bolh V A M and 1UM in stcp I ol' O'I'A are common to both V A M and R A M i n

occurrence and in magnitude (i.e. allocalions). In other words, the o p h u r n allocation 01'

pc~rol from Port I larcourt licli~iery/Dcpol to 5 dcstina~ions by classilica~ion are he same

using both methods. 'l'hz cities arc lkot Ekpene (Group I ) , lkot Ekpene (Group 2), Ikot

The op t~n~nl allocation for U y o (Xz,) dif1i.r using L ~ L ' two methods ( V A M and HAM)

While VAM gave Uyo 299,280 litrcs liom Port I larcourt Reiinery /llcpot a1 thc cost of

N1.20, M M gave i t 3,016,500 lilres at the same cost fiom the same source. The

culqxmllvc aclvrr~ililt;~ oL'VRR'1 uvcv ItAM ill tlris sil~rirtion is that thc dift'erencz in the

allocation could be gol1c.n liom Cillabar Depot at a cheaper cost of 441.10 per litrc.

r t I herel'orc, srncc Lhe aim ol' h'allspulwho~l publc111 algu~~tli~ir 1s v p ~ r i ~ u ~ l i dllucutio~~ ol'

resources at minimum cost, VAN1 is considered a better choice. Finally, the value of the

object furlction Z = 0,533,286 i n VAM and Z - 9,773,286 in RAM underscores the

seleclion of VAM in the execulion 01' O'I'A l i ~ r thk problcm.

The optilnality test using a set of initial basic fcusiblc solution ob~ain horn VAhl

concludes the optimalily test withoul any iteralion, herefore concluding the algorilhm

wrthoul any lclrlhcr proccdurc 01' clia~n rcac~ic)~~. A Lrial expcrilnenl was ped'orrned by

pcri'orining thc opti~nality test using a set ol' ini~ial basic solution from KAM i'or

comparison. 'l'hr: proccss which slartcd with an inilial value o fZ = 9,776,742 iinally ended

in 2 = 9,773:286 at iteration 3 in the accompanied chain reaction. 'The iinal set of basic

variabes a1 ilerdlion 3 are: , X13, X z l , X?Z, X2,, XZJ, X25, X2f,, XZO X3). &.

and X39, with a replacemenl ol' )(:is by Xzs. The allocalion for each of these variablcs

remaincd the same except for - 2,049,280, XI , =: 845,500 imd X33 = 4,148,845 where

X28 and Xi3 were slashed lo obtain a reduccd value of the objeclive Sunclion, Z in he

sequence ol' itemtion. Hence, thc simplilica~ion 01' computer code which I-lillier SL

I~ebern~an (1995), chapter 8, saw as all obv~ous advantage of RAM over VAM is

u~dwru~lecl by his work as L I I C min~n~um cost advantage of VAM collaborates thc

objwtive of 'l'ransportation Problem Algorithm.

~ ' l l A 1 " l ' ~ R l< lVl<

SUkllFlAHY A N 0 C O N ( ' L U S I 0 N

5.1 lNTHOl)ll<IrI'ION

'I'his chapter presents sal~cnt purnts and lindlngs drawn especially from the results of thr:

analys~s In Chapter ibur. Cunclu~on based on the results shall be made.

5.J SlJMMAIIY ANIJ (:ON('I.lJSION

A road nelwork distcibution of petl-uleun~ products (Petrol) and its impact on the

a'i/a~labilit~ and pump-pricc was studied in tl~is work. It was an attempt to seek solution to

the evcr crises-ridden Petroleum sector and its attendant scarcity and consequence

ovc~pr-icing ol'product by markclers. A means, which reveals the contribution or impact ol'

distribution network to thc petrolecl~n scctor is the 0'1'A devcloped in lhis work.

'I'he advantages of 0'1'A are:

( i ) I t produces initial bas~c 1i.asible solutmn, which 1s optimal, or ncar opurnal;

( i i ) It simplifies the optirriality test procedure by reducing or elimination thc number of

chain reactions required to al~ain ophialily;

( i~ i ) I t is corr~putatiol~ally el'licic~~l ;

( iv ) I t savcs time.

IiAM's optimal basic Scx4Al: solution obtained ut iteration 3 of 'l'ublc 4.6 in the trial

oplimalrly test is the same in occilrrcnce as VAhl's slarlirrp, basic feasible solution at

iteration 0 ol'l'ablc 4.5, hence thc choicc of VAM in the OTA. This selection is f u f l l ~ r

buttressed by the hllowing reasoaing bascd on this work;

(a) that the objcctivc ol'tl~is study imd tr:rnsport;ltiun problem in general is to

distribute the ava~lable quantity of pcwol among the cities at a minimum

cost ~ I ' . ~ ~ ~ I I S ~ C ) I . L ~ I I I C ) I ~ . V A M co~rlributes to this o1,jeclive.

(b) VAN1 had produced Xzj =- 17,280 as its optimal basic variable at $4.1 .OO

per litrc. This irl~plies that the city of lkot Abasi would get additional

durnrny (hlitious) depot, which in the red sense docs not e'xisl. Therefore,

VAM provides optimum utilimion of the available product and is said to

(c) Finally, Ihc use of VAM ~n the 0'1-A rcsultcd in a minimum cost of

N 6,533,286 whik RAM, used in the " 'friiil" optimalily test produccd its

minimum cost to be I4 0,773,286.

Again, the OTA applied in this work produced optimum allocation of

petrol to the five cities in Akwa lborn State considered in this work, subject's to

the dernancVsupply constrains For dctails of his, see scction 4.7

, However, the wide gap belween the demand of p e ~ o l and supply shown in this survey is

narrowed in prac~ice by the colnplerne~itary supply/distribution of petrol by Major

Marketers This is because the data used in the analysis only covers the

supply/distribution of petrol by lndependent Marketers, see section 1.5. The demand and

supply requirements of Uyo, Eket and Oron in Akwa Ibom State call for an urgent need

to establish additional Vcpot within or around Akwa Iborn Slate to cater for the high

denland of petrol. This will go a long way to ameliorate the suft'ering of the people

caused by frequent pelrol scarcity due to inadequate supply.

Secondly, the implementation of the result of this work would help stabilise the unitbrm

price policy of the Federal Governnient. 'rtlirdly, a tield survey undertaken in the course

of this work reveals lint the transporlation charge of N 1.00 per litre of petrol provided in

the cost structure of petrol has not been adhered to. What is obtainable is, the

Secondly, the implementation of the result of this work would help stabilise the uniform

price policy of the Federal Government. Thirdly, a field survey under taken in the course

of this work reveals that the transporzalion charge of N I .OO per litre of pelrol provided in

the cost structure of petrol has not been adhered to What is obtainable is, thc hrther the

destination, the higher the cost of transportation, and hence, the cost of petrol pr litre.

This also raises dust on the bridging policy that was meant to cushion this ef'fect by the

Petroleum Equalizatiori Fund (PEF) established by lltcree'9 of 1975 as amended by

Decree 32 ot' 1989

idowever, Wagner (1972), chapter 6, forewarried that h e plan's (any algorithm derived

and used) relative n~erit must be judged against whatever psactical altesliatives the

iridustsy can device, including of course, the current routing. And that the Xij solution

values rnay not represent amounts actually transported, but melely estiniale the order of

rnagnitutle of future ship~nents.

POSSlBLE GENEKA1,lSA'TON AND EX'I'ENSION

'I'he entire dislributiori network of petrolei~ni products tiorn Refinery (or jetty tariks used

for imported pe~rol) to Depots and finally to Filling Station in Nigeria should be studied

if the problern of petroleum products scarcity in the country is to be solved Moreover,

the identification of VAM as better than R A M in this work rieeds to be checked further

in this reyard.

The OTA of this work would need to be automated with of building a standard computer

program that can be called easily.

1. Agbese, D. (2001 ). Nigerian National P e l r o l c ~ ~ ~ ~ l ( '01-porattor~ 1<q)or1 011

Opcr;~lions: Newswalcli Mapzinc vol. 14, I . . ~ I ~ o s .

2. Appa, C. M . (1973).Thc 'I'rnnsposration I'roblcl~~ i111d Its Val-i~~nk,. O1< Quas~erly,

Vol. 24, 7'1-96

3. Bass, I<. S., Glover, F, and Klinglnan, D. (1081). A New Op~irnica~io~i Mc~liod for

l,argc-Scalc Fixcd-C'llargc 'I'l-al~spostatloli l'~.ol)lcn~. OIZ Socic~y. VoI.29, 448-

461.

4. Bowman, E. C. (1956). Psocl~~ct~on Sched\~ling by Ihe 'l'ransporldlon Mclhocl of

Linear 1'1-ogramming. 0 1 Z Socicly, Vol. 4, 101 -102.

.. 5. Halas, E. ( 1905). Solution ol' l;lsgc- Scale l'r;inspcv tat ion I'soblcm~s Ihrough

Aggl~cgario~i. OR Socicly, Vol. 13. 82-03.

6. Bolat, A. (1009). Assigning arriving Flights at ail Airport to the: Available Gatcs.

Journ. OR Society, Vol. 50.

7. Goldcn, B. (1073). Sliorksl-I'alh Algosilllm: A Comparison. OK ol' Amcsica, Vol.

2 1, 284-285.

8. liillicr, I;. S. and Liebernxrn, i;. J . (1995). lntroduclion lo Operations Rcscurcln 6th

ell. Mc. Craw-I-lill Book Co. Singapore.

9. Klingmlun, D. and Russel, li. (1975). Solving Constraint l'ranspoi-lation l'roblcm.

10. Klinl;cn,icz, .I. (3. (1990). Solving a Freighl 'I'r~~nspos~alion Psoblcm Using ITacilily

Localion Technique. OR Socicty, Vol. 38, 99-1 09.

I I. liadnor, ai~d Dii~id A. (1 970). lmplcnlcntalion 111 Opcralions Kcsearch and 'T & D

in Go\lcrnmenl and Business organizations. OR Society, Vol. 18, 967-991.

12. Keintild, N . V . and Vogel, W.R. (1959). Mathematical Programming. Prentice Hall,

Englewood ClilTs, New York.

13. Russell, E.J. (1969). Extension of Dantxig's Algoritl~n~ to Finding an Initial Near-

l87-~lC)l.

14.Schultz, l i .L, , and Slevin, D.P. (1975). Impleme~iting Operations

ResearchIManagement Science. American Elsevier Pub. Con1 lnc., New York.

I S.SI-inivasan, V, and Thompson, G . L . (1973). An Algol-ilhrn f u ~ Assigning Uses to

Sources in a Special Class-of Transportation I'rohlems 01t Society of America,

Vol. 2 1, 284-294.

16. Shragah, S. (1954). The flow of Motor Spirit from Relinery lo Consumers PR

Quarterly. Vol. 5.

17. l'aha, H.A. (1072). Operations Research: An Introduction 4th ed. Macrnillan Pub. Co.

* New York.

18. Wagener, U. A. (1972). A New Method of Solving Transpol-tation Problem. Or

Quarterly, Vol. 24,453-468,

19. Wagener, H. M. ( 1969). l'rinciple of Operations Research with Applicalions of Linear

Programming to Shell's Distribution Problems. Interfaces Vol. 6, 1 3-26.

20. Zierer, 't'. K . , Mitchell W.A. and White, T.R. (1976). Practical Applications of Linear

I'rogramming to Shell's Distribution I'roblerns. Inrerl'aces, Vol. 6, 13-26,

2 1 . Federal Government's Special Committee's Report on the Review of Petroleum

Products Supply and Distribution.(October 2000).

tll'PENDLX .'A: HO1,DING CAI'ACI'I' IES 01;. I'E'I'WOLEUM 1'ItOL)IICI'S

WAKKI * KADUNA * I K U A * *

SOoliltO1li: Ileport ol'llwSpecial Committee on the i<cvizw of f?ctroleunl produas Supply and>istribution (October, 2000).

92

* Derrotes Ketinery Depots where tankage is not strictly dedicaled to finished products. * * At lkeja, lank - 1i11.11~ belongs to ( I I C major J ~ ~ ~ I I ~ O L C I ~ S .

APPENDIX 2C

A NETWORK OF A TRAN SI'OI<'I'ATlON 1'1W13LEM MODEL W1'1'1-i TWO SOURCES AND FIVE DESTINATIONS.

UNIVEHSI'I'Y 01' NIGEHIA, NSUKKA DEI'AHTNIEN'I' OF STATISTICS

IDH03EC'I' O N 'TI1 E DIS'l'HI UU'I'ION O F I'E'I'HOLEU M I'HODUC1' (I'E'I'KOL) IN NIGEKIA.

A I-cscarch project is bcing u~dcrtahcn on the dis~r.ibutio~l of pctmleun~ produc~ - petrol.

I t is hoped that the result of thc research will be a contribution towards efticient

pctr~olcu~~~ proc)ucl dislribuliorl i l l L IX counlry. 1 Iwcby atlac11 a qucsliounairc, wllicll 1

I.CCIUCS~ you 10 conlplcle on bcl-lrdf of your Filling Sta~iori. T11e questionnaire is to be used

purely k'or academic purposes. No lialncs of pcr:sons and address involved are requircd

and lhere is no way some other person can identify whose information is recorded. In

any case the questionnaire will be trealed conlidentially.

Thanks for your co-operation

94

SECTION A

Please fill the appropriate answer(s) in the boxes and spaces provided. D o not include ini'ormation on petrol gottcrl from u~lotlicial supply.

Name o f Filling Station . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

What is the status o f your Cosqxmy? lndeperdcnt Marketer ( 1 Major M4r&fs 1-1

What is the total storage capacity o f your Filling Station? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4(a) t low many did your C O ~ I P ~ ~ I ~ Y registered with ? 01le I I TWO ( 1 . . . . . . . . . . . . . . . . . . . . . . . . '1'hrc.e Any other, please specify

(b) :I.icl, the Dc oi s *which your COIN ,an is registered with: Aba I] Calabar fi Port l - ~ a m u ~ ~ (-1 Any other, please

5 . Does N N I T pay you fbr "l3KIDGING" if'arly '? Ycs LEI NO 0 6 . Il'?cs in (5) is thc amount ec ual to the actual transportation cost incurred'?

Yen r -1 NO 1-1 7. liNo in (5) , what is the cost dii't'erence (44) per litw . . . . . . . . . . . . . . .

8. OII'k'ICLAL, AL1,OCA'I'ION 0 1 ; I'E'I'IIOLA (IN LI'I'IWS ) IWOM KEliINElUES / DEPOT Wl'L'IIIN "TIIE THANSPORT I)WFEIIEN1'IAL ZONE" ('l'DZ)

ABA

JUN. 2001

MAY .ZOO I

INDEPENDEN'I' MAKKKl'EJtS LIIi'I'INCS OF Y M S (I'WHOL) IN FIVE CITlES 1N A K W A W O M S'l'A'l'E FOR 'I'liE 1'ERlOI) JAN - APlG 2001.

UYO

- . . -- .- I KO'I'

... . -. -- - - - - - - - - . . . - , -. - . -- -. -- ---. - M O N I ' I I 01: I I I . . TOTAL

I'RODUC'I' Quantity (11l1.t'~) QUAN'I'I'I'Y

.- - - .---. - -- - -- -. -. - - - . . -. -. - - -- .--- ---- JAN. PMS 406600 562600 lW3. I'M S 16 1 7300 1 56000 1773300 MAR. IbMS 10 18500 504'700 1523200

JAN.. PMS

1 00000 I'MS . 43800

4 3 800 ~~0000 >J 60000

SOI/l<(X: Ministry of Petrolcum & Mi~lc ra~ Resources, Akwn lbom State.

AYYENDLX 3l3

INDEI'ENDENT MARKETERS LJF'rlNCS OF PETROL FOR THE PEKIOL) JAN. - MAR 2001

-. -- -- ~ D U ~ O I ' ~ - JAN -- --i- -- FIX .-~a . -- .-

CALAUAR 24537 148.71 6029733.734

--A - -- -- - . . ... MAK-/: 1 ~ 1 4 1 ~ 9779072.456 40345954.9 1

APPENDIX 3C:

INL)EIDENI)EN?' MAHKE'I'EHS Ll Irl'lNGS OF PETROL AND AVEIUGE SALES DAY 1"OH 'I'UE PElUOI) JAN. - MA& 2001

DESTINATION

UYO

OKON

UYO

3 0000 16000 1 5000 - -- - 35000 18000 16000 3 WOO 1 9000 1 7000

";OI/lUl: Field survey 200 1

1)erivatiou of 1Ji and Vj using Cij -111 - Vi for the coinputatios of initial basic feasible solulioe i e I U N l .

- 6 = 1.4, v I = M, a d so forth.

In iteration 1 : The largest negative value ( in absolutc term) is --M- 1.5, selecl

X ~ I , = 74440 and delete colulnn 9.

In iteration 2: ?'he largest negative valuc ( in absolute L~H-in) is -M- 1.2. Select

In iteration 3: The largest negative value ( in absolute term) is -M- 1.2. Select

X3s - 17280 and delete column 5 . (Break tie arbitrarily).

In iteration 4: The largest negative value ( in absolute term) is .-M- 1.2. Select

X,J .~ - 4 13 1565 and delete row 3.

ln,iteration 5: Tht: largest negative value ( in absoluie term) is -1.7. Select

X24 = 57600 and delete column 4 .

In iteration 6: The largest negative value ( i n absolute twn) is - 1 '6 . Select

9 X22 -. 962940 and delete colun~n 2.

In iteration 7: 'She la~gcst negative value ( in absolutc tern) is -1.5. Sclect

XzX = 244800 and delete column 8.

In iteration 8: The largest ~legaiive value ( in absolute term) is -1.3 Break tie arbitrarily.

Select X21 = 3209800 and delete column I .

In iteration 9: The largest negative value ( in absolute termj is -1.3. Select X13 = 1090300

and delete row I .

The remaining basic variables are Xz.1 =; 30 16560, Xzc, = 300000 and Z = 9776742.