The Application of Transportation Algorithm with Volume Discount on Distribution Cost (A case study of Port Harcourt flour mills Company Ltd.

Transcription

1 Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4(): 58-7 Scholarl Research Isttute Jourals, 3 (ISSN: 4-76) eteas.scholarlresearch.org Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4():58-7 (ISSN: 4-76) The Applcato of Trasportato Algorth wth Volue Dscout o Dstrbuto Cost (A case study of Port Harcourt flour lls Copay Ltd.) Eeze Da Da ad Opara Jude Departet of Statstcs, Io State Uversty, PMB, Owerr, Ngera. Correspodg Author: Eeze Da Da Abstract Utl recetly, heavy trucs could load up to ay capacty wthout lt. These trucs orally eceed the average loadg capacty of the truc. Ths was partally due to hgh trasportato cost. Drvers ad trasport owers together wth trasport users had to fd a way of copesatg for the hgh cost of trasport by creasg the truc load so as to aze proft. Ths had rpple effect o the state as a whole: creased road accdets, destructo of roads ad loger te beg spet o the road before gettg to destato. There s also the effect of creased cost of goods thereby creasg flato. Ths drves the atteto of the goveret to fd a lastg soluto to the probles. The goveret therefore wet to agreeet wth trasport owers to detere au loadg capacty of trucs. The purpose of ths paper s to fd out whether gvg dscouts o trasportato charges could ze total trasportato cost thereby creasg total reveue of both producers ad retalers. Ths study s focused o the Applcato of Trasportato Algorth wth volue Dscout o dstrbuto cost usg Portharcourt flour lls copay plc. Ths paper s teded to detere the quatty of Golde Pey Flour ( 5g bags), Golde Pey Seovta ( g bags) ad Wheat Offals (also 5g bags) that porthacourt flour lls copay should dstrbute a oth order to ze trasportato cost ad aze proft. A proble of ths ature was detfed as a Nolear Trasportato Proble (NTP), forulated atheatcal ters ad solved by the KKT optalty codto for the NTP. Thus, aalyss revealed the followg allocatos: bags of Golde Pey Flour should be suppled to Aba, 7 of the sae product should be suppled to Bayelsa. Allocate bags of Wheat Offals to Aba, ad 9 bags of the sae product to Eugu. Fally, allocate 4 bags of Golde Pey Seovta to Eugu ad bags of the sae product to Otsha. Hece, the total u trasportato cost for the above dstrbuto s N394,. Keywords: trasportato algorth, olear trasportato proble (NTP), KKT optalty codto, total u trasportato cost INTRODUCTION Whe cosderg trasportato, varous cosderatos are apparet. Ths cosderato cludes port selecto, lad oveet, ad port to port carrer selecto ad delvery oveet. I addto to these trasportato cocers, dstrbuto-related cosderatos ust also be gve atteto to such as pacg/pacagg, trast surace, ters of sale, port dutes, hadlg/loadg ad ethod of facg. Nevertheless, eve freght copaes proectg large volue oveets ca ecouter serous trasportato proble orgazg for dstrbuto. Uderstadg these trasportato probles especally that affects shppg costs s crtcal. Volue dscout, ore specfcally, targets shppg cost ad zg the latter, volue dscout ust be acqured. Networ odels ad teger progras are well ow varety of decso probles. A very useful ad wdespread area of applcato s the aageet ad effcet use of scarce resources to crease productvty. These applcatos clude operatoal probles such as the dstrbutos of goods, producto schedulg producto ad ache sequecg ad plag probles such as captal budgetg faclty allocato, portfolo selecto, ad desg probles such as telecoucato ad trasportato etwor desg. The trasportato proble whch, s oe of etwor prograg probles s a proble that deals wth dstrbutg ay coodty fro ay group of sources to ay group of destatos or ss the ost cost effectve way wth a gve supply ad dead costrats. Depedg o the ature of the cost fucto, the trasportato proble ca be categorzed to lear ad olear trasportato proble. 58

2 Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4():58-7 (ISSN: 4-76) I the lear trasportato proble (ordary trasportato proble) the cost per ut coodty shpped fro a gve source to a gve destato s costat, regardless of the aout shpped. It s always supposed that the leage (dstace) fro every source to every destato s fed. To solve such trasportato proble we have the strealed sple algorth whch s very effcet. However, realty, we ca see at least two cases that the trasportato proble fals to be lear. Frst, the cost per ut coodty trasported ay ot be fed for volue dscouts soetes are avalable for large shpets. Ths would ae the cost fucto ether pecewse lear or ust separable cocave fucto. I ths case the proble ay be forulated as pecewse lear or cocave prograg proble wth lear costrats. Secod, specal codtos such as trasportg eergecy aterals whe atural calaty occurs or trasportg ltary durg war te, where carryg etwor ay be destroyed, leage fro soe sources to soe destato are o loger defte. So the choce of dfferet easures of dstace leads to olear (quadratc, cove ) obectve fucto. I both the above cases solvg the trasportato proble s ot as sple as that of the lear oe. I ths wor, soluto procedures to the geeralzed trasportato proble tag olear cost fucto are vestgated. I partcular, the olear trasportato proble cosdered ths thess s stated as follows; We are gve a set of sources of coodty wth ow supply capacty ad a set of destatos wth ow deads. The fucto of trasportato cost, olear, ad dfferetable for a ut of product fro each source to each destato. We are requred to fd the aout of product to be suppled fro each source (ay be aret) to eet the dead of each destato such a way as to ze the total trasportato cost. Our approach to solve ths proble s applyg the estg geeral olear prograg algorths to t ag a sutable odfcato order to use the specal structure of the proble. STATEMENT OF THE PROBLEM The prces of coodtes are detered by a uber of factors; the prces of raw aterals, labour, ad trasport. Whe prce of raw aterals crease, so does the prce of the coodty. Trasportato cost also affects the prcg syste. It s assued that the cost of goods per ut shpped fro a gve source to a gve destato s fed regardless of the aout shpped. But realty, the cost ay ot be fed. Volue dscouts are soetes avalable for large shpets so that the argal cost of shppg oe ut ght follow a partcular patter. Ths paper therefore sees to develop a atheatcal odel usg optzato techques to brdge the gap betwee dead ad supply by dscoutg so as to ze total trasportato cost. The proble that wll be addressed ths study ceters o the trasportato probles epereced by freght copaes. Volues of goods to be shpped cur costs hece acqurg volue dscouts could effectvely lead to reduced shppg costs. The ey questo to be aswered s: How freght copaes could prove ther total output through effectvely reducg shppg costs through volue dscouts. LIMITATIONS OF THE STUDY Ths study s lted to the Port Harcourt Flour Mlls Copay ltd, Rvers State, Ngera. The result of the study could be replcated other copay. OBJECTIVES OF THE STUDY The a a of ths study s to desg atheatcal prograe that would prove the total output of freght copaes especally sce they deal wth shppg of goods by volue. Whether au proft wll be realzed wth dscouts o large volues eas to detere the best trasportato route that would lead to low trasportato cost ad the effectve trasportato of these goods. We wll also provde algorths ad dfferet soluto procedures to the dfferet cases that ght arse. TRANSPORT COSTS Trasport Costs ad Rates Trasport systes face requreets to crease ther capacty ad to reduce the costs of oveets. All users (e.g. dvduals, eterprses, sttutos, goverets.) have to egotate or bd for the trasfer of goods, people, forato ad captal because supples, dstrbuto systes, tarffs, salares, locatos, aretg techques as well as fuel costs are chagg costatly. There are also costs volved gatherg forato, egotatg, ad eforcg cotracts ad trasactos, whch are ofte referred as the cost of dog busess. Trade volves trasactos costs that all agets attept to reduce sce trasacto costs accout for a growg share of the resources cosued by the ecooy (Shetty; 959). Frequetly, eterprses ad dvduals ust tae decsos about how to route passegers or freght through the trasport syste. Ths choce has bee cosderably epaded the cotet of the producto of lghter ad hgh value cosug 59

3 Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4():58-7 (ISSN: 4-76) goods, such as electrocs, ad less buly producto techques. It s ot ucoo for trasport costs to accout for % of the total cost of a product. Thus, the choce of a trasportato ode to route people ad freght wth orgs ad destatos becoes portat ad depeds o a uber of factors such as the ature of the goods, the avalable frastructures, orgs ad destatos, techology, ad partcularly ther respectve dstaces. Jotly, they defe trasportato costs. Trasport costs are a oetary easure of what the trasport provder ust pay to produce trasportato servces. They coe as fed (frastructure) ad varable (operatg) costs, depedg o a varety of codtos related to geography, frastructure, adstratve barrers, eergy, ad o how passegers ad freght are carred. TRANSPORTATION COST ANALYSIS A typcal applcato of the trasportato proble s to detere a optal pla for shppg goods fro varous sources to varous destatos gve supply ad dead costrats order to ze total shppg cost. It s assued that the cost of goods per ut shpped fro a gve source to a gve destato s fed regardless of the aout shpped. However, actualty the cost ay ot be fed. Volue dscouts are soetes avalable for large shpets so that the argal cost of shppg oe ut ght follow a partcular patter. A trasportato servce curs a uber of costs: labor, fuel, ateace etc. ths cost ca be dvded to two: those cost that vary wth servces or volues called varable cost ad those that do ot vary wth servces called fed cost. LINEAR AND NONLINEAR Trasportato Revew The trasportato proble was foralzed by the Frech atheatca Gaspard Moge (78). Maor advaces were ade the feld durg World War II by the Sovet/Russa atheatca ad ecoost Leod Katorovch. Cosequetly, the proble as t s ow stated s soetes ow as the Moge-Katorovch trasportato proble as reported by The proble wth the producto capacty of each source fed wth costat ut trasportato cost was orgally forulated by Htchcoc (94) ad was subsequetly dealt wth depedetly by Koopas durg Secod World War. Aalytcal soluto to ths proble has bee gve by several authors. Strger ad Haley have developed a ethod of soluto usg atheatcal aalogue. George Datzg (95) adapted the sple ethod to solve the trasportato proble forulated earler by Htchcoc ad Koopas. Abraha Chares ad Wlla Cooper (954) derved a tutve presetato of Datzg s procedure called the steppg-stoe ethod whch follows the basc logc of the sple ethod but avods the use of the tableau ad the pvot operatos requred to get the verse of the bass. Marts, Luca, Craverha (5) preseted a study of a b-desoal dyac routg odel for telecoucatos etwor. The odel uses heurstc ethods to solve stablty probles. The routg ethods through heurstcs are copared wth the dscrete-evet sulato the dyac routg syste. The brach ad boud algorth approach s based o usg a cove approato to the cocave cost fuctos. It s equvalet to the soluto of a fte sequece of trasportato probles. The algorth was developed as a partcular case of the splfed algorth for zg separable cocave fuctos over lear polyhedral as Fal ad Solad. Pece-wse lear over approato s also the other approach to solve the olear cocave trasportato proble. Caputo et al (99) preseted a ethodology for optally plag log-haul road trasport actvtes through proper aggregato of custoer orders separate full-trucload or less-tha-trucload shpet order to ze total trasportato cost. They have deostrated that evolutoary coputato techque ay be effectve tactcal plag of trasportato actvtes. The odel shows that substatal savgs o overall trasportato cost ay be acheved adoptg the ethodology a real lfe scearo. Zagabad ad Male (7) preseted a fuzzy goal prograg approach to detere a optal coprose soluto for the ult-obectve trasportato proble by assug that each obectve fucto has a fuzzy goal. A specal type of o-lear (hyperbolc) ebershp fucto s assged to each obectve fucto to descrbe each fuzzy goal. The approach focuses o zg the egatve devato varables for to obta a coprose soluto of the ult-obectve trasportato proble. Surapat ad Roy (8) preseted a prorty based fuzzy goal prograg approach for solvg a ult-obectve trasportato proble wth fuzzy coeffcets. Frstly, they defed the ebershp fuctos for the fuzzy goals. Subsequetly, they trasfored the ebershp fuctos to ebershp goals, by assgg the hghest degree (uty) of a ebershp fucto as the asprato level ad troducg devatoal varables to each of the. I the soluto process, egatve devatoal varables are zed to obta the ost satsfyg soluto. 6

4 Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4():58-7 (ISSN: 4-76) Lau et al. (9) preseted a algorth called the fuzzy logc guded o-doated sortg geetc algorth to solve the ult-obectve trasportato proble that deals wth the optzato of the vehcle routg whch ultple depots, ultple custoers, ad ultple products are cosdered. Sce the total travelg te s ot always restrctve as a te costrat, the obectve cosdered coprses ot oly the total travelg dstace, but also the total travelg te. Lohgaoar ad Baa () used fuzzy prograg techque wth lear ad o-lear ebershp fucto (hyperbolc, epoetal) to fd the optal coprose soluto of a ultobectve capactated trasportato proble. Havg revewed soe past researcher s wor, we shall ow study the applcato of trasportato algorth wth volue dscout o dstrbuto cost, usg Portharcourt Flour Mlls Copay LTD as a case study. RESEARCH METHODOLOGY A typcal applcato of the trasportato proble s to detere a optal pla for shppg goods fro varous sources to varous destatos gve supply ad dead costrats order to ze total dstrbutg cost. It s assued that the cost of goods per ut shpped fro a gve source to a gve destato s fed regardless of the aout shpped. But actualty the cost ay ot be fed. Volue dscouts are soetes avalable for large shpets so that the argal cost of dstrbutg oe ut ght follow a partcular patter. Whe volue dscouts are offered, the obectve fucto or the costrat fuctos assue a olear for. We therefore use the olear ethod of soluto to solve such a proble usg Port Harcourt Flour Mlls Copay ltd, Rvers State of Ngera. THE KARUSH-KUHN-TUCKER (KKT) OPTIMALITY CONDITION FOR NONLINEAR PROGRAMMING PROBLEM Gve the o lear prograg proble: (NNP) f() s.t. g () =,, () h () = =,, l KARUSH-KUHN-TUCKER OPTIMALITY CONDITIONS NECESSARY Theore 3..: Gve the obectve fucto f :R R ad the costrat fucto are g :R R ad h : R R ad I = {: g (*) = }. I addto, suppose they are cotuously dfferetable at a feasble pot * ad g (*) for I ad h (*) for =,, l be learly depedet. If * s zer of the proble (NPP), the there est scalars ; =,, ad ; =,, l, called Lagrage ultplers, such that f (*) l g (*) h * (*) () g ; R KARUSH-KUHN-TUCKER SUFFICIENT OPTIMALITY CONDITIONS FOR CONVEX NPP Further, f f ad each g are cove, each h s affe, the the above ecessary optalty codtos wll be also suffcet. Justfcato Let be ay feasble pot dfferet for *. Fro the frst KKT codtos we obta f ( *)( *) l t g ( *)( *) h ( *)( *) Sce each g () s cove, ad h (*) ( *) =, we also have t g (*)( *) [g () g (*)] f (*)( *) g () Fro covety of f(), therefore, we get f() f(*) f(*) f() for ay feasble. LINEAR TRANSPORTATION PROBLEM Trasportato Model Proble Trasportato s a eaple of etwor optzato proble. It deals wth the effcet dstrbuto (trasportato) of product (goods) ad servces fro several supply locatos (sources) wth lted supply, to several dead locatos (destatos) wth a specfed dead wth the obectve of zg total dstrbuto cost; a typcal eaple of whch ths thess represets ( aalogy). Ths obectve s acheved uder the followg costrats;. Each dead pot receves ts requreet.. Dstrbutos fro supply pots do ot eceed ts avalable capacty. Ths goal s acheved cotget o avalablty ad requreets costrats. Trasportato proble therefore assues that the trasportato cost o a gve route s drectly proportoal to the uber of uts of the coodty trasported (Iyaa; 7). 6

5 Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4():58-7 (ISSN: 4-76) MODEL FORMULATION The forulato of the trasportato odel eploys double subscrpted varables of the for. Thus, the geeral forulato of the trasportato proble wth sources ad destatos, s gve by Mze Subect to (3) d d s C s,,3,,,,3,, FINDING INITIAL FEASIBLE SOLUTION TO TRASPORTATION PROBLEM The geeral forulato of the trasportato proble reveals that supply costrats ad dead costrats traslate to + total costrats. I the trasportato proble however, oe of the costrats s redudat resultg the fact that f, a balace codto, s + costrats are et the + equatos wll also be et. Oly + depedet equato, thus, est ad so the tal soluto wll have oly + - basc varables. The flow chart below llustrates the varous phases leadg to the optoal soluto of a trasportato proble Fg. Trasport ato Algorth Detere o tal basc feasble soluto Test the curret soluto for optalty Fd a better feasble soluto TRANSPORTATION TABLEAU The trasportato tableau s a uque tabular represetato of the trasportato proble. The varable gves the uber of uts trasported fro source to destato (whch s to be solved for) whle the ut cost for the trasportato fro to, deoted by C, s recorded a sall bo the upper rght had corer of each cell. Below s the for of the geeral trasportato tableau. d Optal stop Steppg stoe ethod Phase Phase Table : Trasportato Tableau C C C S C C C S To () Fro ( DESTINATIONS Supply C C C C C C S C C C C S X X X Dead d d d d S = d SOURCES METHODS FOR FINDING INITIAL BASIC FEASIBLE SOLUTIONS The frst phase of the solvg a trasportato proble for optal soluto volves fdg the tal basc feasble soluto. A tal feasble soluto s a set of arc flows that satsfes each dead requreet wthout supplyg ore fro ay org ode tha the supply avalable. Heurstc, a coo sese procedure for qucly fdg a soluto to a proble s a producer ost eployed to fd a tal feasble soluto to a trasportato proble. Ths paper eaes three of the ore popular heurstcs for developg a tal soluto to trasportato proble.. The Northwest corer ethod. The Least Cost Method. The Vogel s Approato Method SOLUTION PROCEDURES TO THE NONLINEAR TRANSPORTATION PROBLEM (NTP) I ths secto, we cosder a trasportato proble wth olear cost fucto. We try to fd dfferet soluto procedures depedg o the ature of the obectve fucto. Before gog to the dfferet specal cases, let s forulate the KKT codto ad geeral algorth for the proble. Gve a dfferetable fucto C : R R. We cosder a olear trasportato proble (NTP) C() s.t. A = b (4) where s ; s b ; d d 6

6 Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4():58-7 (ISSN: 4-76) A The KKT Optalty Codto for the NTP The trasportato table s gve as: C() C() s u C() s u C() C() s u d d d v v v where s the curret basc soluto. The Lagrage fucto for the NTP s forulated as z(,, w) = C() + w(b A) (5) where ad w are Lagrage ultplers ad R U() wr + The optal pot should satsfy the KKT codtos: z = C( ) w T A = = Specfcally for each cell (, we have z C() (u, v)(e,e (6) = where = ad w = (u, v) = (u, u,, u, v,, v ), e R + s a vector of zeros ecept at posto whch s. Fro the codtos (3.6) ad, we get, z C() (u v (7) z C() (u v (8) Geeral soluto procedure for the NTP Italzato Fd a tal basc feasble soluto Iterato Step I: f s KKT pot, stop. Otherwse go to the et step. Step II: Fd the ew feasble soluto that proves the cost fucto ad go to step. TRANSPORTATION PROBLEM WITH CONCAVE COST FUNCTIONS For large dstrbutos, volue dscout ay be avalable soetes. I ths case the cost fucto of the trasportato proble geerally taes cocave structure for t s separable ad the argal cost (cost per ut coodty dstrbuted) decreases wth crease of the aout of dstrbuto; ad creasg, because of the total cost crease per addto of ut coodty dstrbuted. The dscout. May be ether drectly related to the ut coodty.. Or have the sae rate for soe aout. Case : If the dscout s drectly related to the ut coodty the resultg cost fucto wll be cotues ad have cotues frst partal dervatves. Nolear prograg forulato of such a proble s gve by s.t d s C ( ) =,,, =,,, ; (9) Where C : RR Now before gog to loo for a optal soluto let s state a portat theore: Theore 3.3.: Let f be cocave ad cotues fucto ad P be a o epty copact polyhedral set. The the optal soluto to the proble f(), P ests ad ca be foud at a etree pot of P. Proof: Let E = (,,,,, ) be the set of etree pots of P, ad E such that f( ) = {f( ): =,, }. Now sce P s copact ad f s cotuous, f attas ts u P, call t, If s etree pot, we are doe. Otherwse, we have that, ; > where,,, are etree pots of P. The by cocavty of f t follows that, 63

7 Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4():58-7 (ISSN: 4-76) f () f f () f ( ) f () f ( ) (Sce for each =,, ; f( ) f( ) ad ) Sce s ze, addto we have, f( ) f( ) The above two relatos ply f( ) = f( ) Ths copletes the proof. Soluto Procedure Because of the above theore, t suffces to cosder oly the etree pots to fd the u; the followg s the procedure. After we fd the tal basc feasble soluto (they are + uber), let be the basc soluto we have the curret terato. Net let s decopose our to ( B, N ) where B - ad N are the basc ad obasc varables respectvely. Sce B >, the copleetary slacess codto gve (3.8) gves as + equatos; z C() (u v () B B Fro the above relato we ca detere the values of u ad v by assgg oe of u s the value zero for we have + varables, u ad v. The we calculate z for the o basc varables. Sce all are zero at the etree, the copleetary slacess codto s satsfed. Therefore f equato (3.) s satsfed for all o basc varables, s a KKT pot. Otherwse, f z (u v () We wll ove to loo for better basc soluto such that all the costrats (feasblty codtos) are satsfed. We do ths by usg the sae procedure as the trasportato sple algorth. THE TRANSPORTATION CONCAVE SIMPLEX ALGORITHM (TCS) Italzato Fd the tal basc feasble soluto usg soe rule le west corer rule. Iterato Step : Detere the values of u ad v fro the equato, C() (u v B Where B are the basc varables. Step : If C() (u v ) () (3) for all o basc, stop, s KKT pot. Otherwse go to step 3. Step 3: Calculate z C() u v (4) rl rl wll eter the basc. Allocate rl = where s foud as the lear trasportato case. Adust the allocatos so that the costrats are satsfed. Detere the leavg varable say Br, where Br s the basc varable whch coes to zero frst whle ag the adustet. The fd the ew basc varables ad go to step. Fte Covergece of the Algorth The feasble set of our proble s a o epty polyhedral set. Ad by defto, a polyhedral set P s a set bouded wth a fte uber of hyper plaes fro whch t follows that t possesses fte uber of etree pots. I each step of the algorth, we up fro oe etree pot to aother loog for a better feasble soluto plyg that the algorth wll terate after a fte terato. I addto sce for all ad, a {s, d }, P s bouded that guaratees the estece of u value. Case : I the case whe the volue dscout s fed for soe aout of coodty, rather tha varyg wth ut aout dstrbuted, the trasportato cost fucto wll be pecewse lear cocave yet creasg. To avod coplcato, assug that to each cobato of source ad destato, the terval whch the argal cost (cost per ut coodty) chages s the sae, the cost of dstrbutg uts fro source to destato s gve by C ( ), the the olear prograg forulato of the proble s gve by s.t C s d ( ) (5) 64

8 Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4():58-7 (ISSN: 4-76) where =,,, ; =,,, C ( ), a C ( ),a a l C () C ( ),al al C (),a a C ( ),a b a( s,d Ad. (, a,, a l,, a, a, b} s the partto of the terval [, b] to + sub tervals. Each C s lear the sub terval [a l ; a l+ ] l To solve ths proble, as we ca see fro the structure of the cost fucto, t's possble to drectly apply the algorth of the prevous secto for o dfferetablty of the total cost fucto hders as to do so. But, sce the fucto, also, has a sple structure ad dfferetablty fals at dscrete pots, t ca be easly approated usg dfferetable fuctos le Chebshev, trgooetrc or Legedre polyoals. We choose to approate t by the so called shfted Legedre polyoals. These set of Legedre polyoals say {p, p,, p r } s orthogoal [,] wth respect to weght fucto w() =, where the er product o C[; ] s defed by f, g = f ()g()d ; for all f, gc [; ]; where C[; ] s the space of cotuous fuctos o [; ] The frst four of the are, p () = p () = p () = p 3 () = ad the others ca be obtaed fro r d pr() = [( ) r ] r! d The, the space spaed by {p, p,, p r } s a subspace of C[:]. Hece, gve ay f() C[; ], we ca fd a uque least square approato of f the subspace. Note that every eleet of the subspace spaed {p, p,, p r } s at least twce dfferetable. 65 The least square approato of ay fucto f() wth r of these polyoals [; ] s gve by, f () = a p () + a p () + + a p () + a r p r () (6) where a pf ()d ; = ;,, r [p ()] d To approate our fuctos C ( ), the sae aer, we defe a oe to oe correspodece betwee [,b] to [,] by g : [, b] [, ] (7) g( ) = b That s, we substtute by b so that t's doa wll be [,] the we have, a C b ; b a a C b ; b b (8) C () Ĉ () C b a C b ; b Now, after approatg Ĉ by the shfted Legedre polyoals o [, ], assue we have foud t's best approato C ~ ( ). The, substtutg bac the C by b gves us the approato to C ( ) over [, b]. Therefore the best approato of C ( ) over [, b] wll be C ( ) = C ~ (b) whch has cotuous dervatves. Cosequetly, we solve the proble C ( ) a p ( ) l s s.t b (9) d b =,,, =,,, usg eactly the sae procedure as the prevous case. CONVEX TRANSPORTATION PROBLEM Ths case ay arse whe the obectve fucto s coposed of ot oly the ut trasportato cost but also of producto cost related to each coodty [Shetty; 959]. Or the case whe the dstace fro each source to each destato s ot fed. l l

9 Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4():58-7 (ISSN: 4-76) The proble ca be forulated as C() s.t A = b () where C() s cove, cotuous ad has cotuous frst order partal dervatves. THE CONVEX SIMPLEX SOLUTION PROCEDURE FOR TRANSPORTATION PROBLEM I the case whe the cost fucto s cove, the u pot ay ot be attaed ecessarly at a etree; t ay be foud before reachg a boudary of the feasble set. What precsely happes s that there ay be o basc varable wth postve allocato whle oe of the bass s drve to zero. To solve ths proble, we use the dea of the cove sple algorth of Zagwll [967] whch was orgally desged to tae care of cove ad pseudo cove proble wth lear costrats. Actually the orgal procedure s used to loo for a local optal soluto for ay other learly costraed prograg proble. We use the specal structure of trasportato proble the procedure so as to ae t effcet for our partcular proble. The ethod reduces to the ordary trasportato sple algorth wheever the obectve s lear, to the ethod of Beal whe t s quadratc ad to the above cocave sple procedure whe the fucto s cocave. We partto the varable = (,, ) to ( B, N ) where B s + copoet vector of basc varables ad N s ( ( + )) copoet vector of obasc varables, correspodg to the ( + ) ( + ) basc sub atr ad ( + )(( ( + )) o basc sub atr of A. Suppose we have the tal basc feasble soluto o. I the procedure what we do s to fd a echas whch o optal basc soluto at a gve terato s proved utl t satsfes the KKT codtos whch are also suffcet codtos for cove trasportato proble,.e. utl for each cell we have; f () (u v () f () ad (u v () Sce we have each basc varable B >, the above copleetary slacess codto ples that for each basc cell, we ust have f () (u v B B - basc varable. Sce we have + of such equatos, by lettg u = we obta all the values of u ad v as we have doe eactly for the cocave ad lear cases. Now for a o basc cell, at a feasble terate pot, we ay have:. f () (u v ; f () (u v (3). f () (u v ; f () (u v (4). f () (u v ; f () (u v (5) Or for all o basc, we ay have; v. f () (u v ; f () (u v (6) Fro the KKT codtos gve earler, the last case occurs whe s optal. But f the soluto falls o ether of the other three, t ust be proved as follows. Let IJ = { : s o basc varable} ad suppose that we are the th terato. We frst beg by coputg; 66

10 Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4():58-7 (ISSN: 4-76) z f ( ) u v IJ (7) rl z f ( ) st a u v st IJ (8) Here we do't wat to prove (decrease) a postve - valued o basc varable uless ts partal dervatve s postve. Therefore we oly focus o z postve values of the product Now the varables to be adusted are selected as; z z Case : If ad st rl (9) st Decrease st by the value usg the trasportato table as the lear ad cocave cases. Let y = (y, y,, y) be the value of (,, ) after ag the ecessary adustet by addg ad subtractg the loop cotag st so that all the costrats are satsfed. By dog so, ether st tself or a basc varable say Bst wll be drve to zero. Now y ay ot be the et terate pot; sce the fucto s cove, a better pot could be foud before reachg y to chec ths, we solve proble; f ( ) = { f ( ( )y : (3) ad get ( )y optal soluto of (3). Before the et terato, where s the If = y ad f a basc varable becae zero durg the adustet ade, we chage the bass. If y or f = y ad st s drve to zero, we do't chage the bass by substtutg the leavg basc varable by st. Case : If z z ad st rl (3) st I ths case the value of rl should be creased by ad the we fd y, where ad y are defed as the case. Note that: as we crease the value of rl oe of the basc varables, say, Bt wll be drve to zero, ad ths s the et crtera of the lear ad cocave trasportato sple algorth ad y would have bee the et terate pot of the procedure. But ow after solvg for fro (3), before gog to the et terato, we wll have the followg possbltes. If = y, we chage the forer bass substtute Bt by rl If y, we do ot chage the bass. All the basc varables outsde of the loop wll rea uchaged. z z Case 3: If ad st rl (3) st I ths case ether we decrease st as the case or crease rl accordg to case. THE TRANSPORTATION CONVEX SIMPLEX ALGORITHM Now we wrte the foral algorth for solvg the cove trasportato proble. Italzato Fd the tal basc feasble soluto. Iterato Step : Detere all u ad v fro f ( ) u v B (33) for each basc cell Step : For each o basc cell, calculate; z f ( ) = u v (34) rl z = st f ( ) u v (35) st z z If ad st rl (36) st Stop. Otherwse go to step 3. Step 3: Detere the o basc varable to chage. Decrease st accordg to case f z z ad st rl (37) st Icrease rl accordg to case f z z ad st r (38) st Ether crease rl or decrease st f z z ad st r (39) st Step 4: Fd the values of y, by eas of, ad, fro 3.5. If y = ad a basc varable s 67

11 Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4():58-7 (ISSN: 4-76) drve to zero, chage the bass. Otherwse do ot chage the bass. go to step. DATA ANALYSIS I ths sesso, we shall eae a practcal applcato of the above soluto procedures. Ephass wll be o the cocave trasportato proble. We shall eae the data obtaed fro the Port-Harcourt Flour lls copay Lted; apulate the data to sut our trasportato proble. DATA AND ANALYSIS The flour lls lted s a aufacturg copay located Port-Harcourt. The produce Golde Pey Flour (GPF), Golde Pey Seovta (GPS), Wheat Offals (WO) etc. These products are suppled to the followg states (locatos) Bayelsa, Otsha, Portharcourt, Kao, Aba, Eugu etc. For the purpose of ths thess, oly four (4) of these dead pots wll be cosdered; Eugu, Otsha, Bayelsa ad Aba. The estated supply capactes of the three products, the dead requreets at the four stes (states) ad the trasportato cost per bag of each product are gve below: Aba Eugu Otsha Bayelsa Supply GPF WO GPS Dead The proble s to detere how ay bags of each product to be trasported fro the source to each destato o a othly bass order to ze the total trasportato cost. A dagra of the dfferet trasportato routes wth supply ad dead fgures s show below. Supply dead. GPS (5) A Aba (). GPF (7) B Eugu (3) () C Otsha 3. WO () D Bayelsa (7) Forg the trasportato tableau. To for the trasportato tableau, let: = product to be shpped = destato of each product S = the capacty of source ode, d = the dead of destato, = the total capacty fro source to destato c = the per ut cost of trasportg coodty fro source to destato. p = percetage dscout allowed for trasportg fro to destato. If we suppose that dscout s gve o each bag trasported fro to, the the o lear trasportato proble ca be forulated as: 3 4 c S.t = = = = = = = 7 where c = 8 p c, 3 3 = 6 3 p 3 3 c = p, c 4 4 = 4 p c 3 3 = 6 3 p 3 3, c 3 3 = 8 3 p c 4 4 = 4 4 4, c 3 3 = 3 c = 6 p, c = 4 p c = p 3 3 p 33 33, c = 34 p If we allow the followg dscouts o each trasported product fro the source to each of the destatos, (p, p, p 3, p 4, p, p, p 3, p 4, p 3, p 3, p 33, p 34 ) = (.,.5,.3,.5,.,.3,.35,.,.5,.3,.5,.) Thus, the cost fucto c ca be epressed as c = 8., c 3 3 = c =.5,c 4 34 = 4. 4 c 3 3 = , c 3 3 = c 4 4 = 4.5 4, c 3 3 = 3. 3 c = 6., c = c = 4.3, c = we the develop the tableau as below; Aba Eugu Otsha Bayelsa S u GPF u WO u GPS u 3 d 3 7 v v v v 3 v 4 Total supply = 43, Total dead = 43, Hece the tableau s balaced Usg the vogel s Approato Method (VAM), we get the tal basc soluto. 68

12 Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4():58-7 (ISSN: 4-76) The soluto tableau s as show below Aba Eugu Otsha Bayelsa S u GPF u 9 7 WO u GPS u 3 4 d 3 7 v v v v 3 v 4 = ( B, B, B3, B4, B,, 3, 4, 3, B3, B33, 34 ) = (, 9,, 7,,,,,, 4,, ), thousads. The total trasportato cost becoes (8) + 9() + 7() + (6) + 4() + (6) = N43, The partal dervatos at are gve as: f () f () 7.96 ; 9. ; f () f () 6 ; ; 3 4 f () f () f () 5.78 ; 4 ; 6 ; 3 f () f () f () ; 8 ; ; f () f () 5.67 ; Now, we fd z f () u v B B B f () u v Thus, u + v = 7.96; u + v = 9.; u + v 4 = 9.65 u + v = 5.78; u 3 + v = 9.84; u 3 + v 3 = 5.67 lettg u =, fro the equatos above; we have u =, u = -.8, u 3 = -9.6 v = 7.96, v = 9., v 3 = 4.93, v 4 = 9.65 The, the reduced costs for the o-basc varables becoe z f () u v z f ( ) u v z f () u v z z 3 34 z f () u 3 f () u 34 z v 9.3 The reduced costs for the o-basc oes at a basc feasble pot. = ( B,, 3, B4, B, B, 3, 4, 3, B3, z B33, f 34 ) = (, (),, 7,, u 9,, v,, 4,.9, ) z f () wll be; u v z f () u v v It s obvous that the presece of egatve value for the reduced cost sgfes o optalty; hece we readust. Therefore should eter the bass sce t s the ost egatve reduced cost, after adustg the values left the basc. Aba Eugu Otsha Bayelsa Supply u (.7) - GPF 8 + WO (-.9) 6 (3.5) (4.53) GPS 6 8 (9.3) 4 (.6) d 3 7 v The basc varable wth the least value aog the corers havg sg the loop s the leavg varable. Hece, wth the least value of 9 s the leavg varables. Thus, we crease the corers wth + sg by 9, reduce the oes wth sg by 9. The adusted tableau becoes: Aba Eugu Otsha Bayelsa Supply u GPF WO GPS d 3 7 v

13 Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4():58-7 (ISSN: 4-76) z z z z f () u 3 f () u 4 f () u 34 f () u v v v v All are o-egatve, plyg that s a KKT pot. Hece, a optal soluto to our proble. Thus, the followg allocato should be ade: bags of GPF should be suppled to Aba, 7 of the sae product be suppled to Bayelsa. Allocate bags of WO to Aba, ad 9 bags of the sae product to Eugu. Fally, allocate 4 bags of GPS to Eugu ad bags of the sae product to Otsha. Total cost = (8) + 7() + (6) + 9(4) + 4() + (6) = N394, SUMMARY I soe occaso, there ay be dfferet ways to odel a partcular proble, but choosg the best approach reduces the coplety of the proble ad te to solve. Sce ay prograg proble wth costrat atr, structure the sae as the trasportato type proble, t ca be regarded as a trasportato type proble regardless of ts physcal eag ad because of ts sple for, odelg such probles as trasportato proble requres uch less effort to solve the odelg t dfferetly. I ths wor, the olear trasportato proble s cosdered as a olear prograg proble ad algorths to solve ths partcular proble are gve. The frst algorth s slar to that of the trasportato sple algorth ecept for the olearty assupto. The secod algorth s depedet o the sple algorth of Zagwll that we odfed to use the specal property of the coeffcet atr of the trasportato proble so that we ay tae shortcuts to ae proble solvg sple. CONCLUSION I ths research study, we were able to detfy the proble cofrotg Port-Harcourt Flour Mlls Copay Plc as a trasportato proble, forulate a atheatcal odel that represets the essece of the proble, detfy the fuctoal equatos of the proble as well as solved the proble usg the Karush-Kuha-Tucer (KKT) optalty codto for olear prograg proble. Fro the soluto obtaed, we were able to detere the u total cost of trasportato as N394,. Hece, the aalyss revealed that allocato should be ade as follows: bags of Golde Pey Flour should be suppled to Aba, 7 of the sae product be suppled to Bayelsa. Allocate bags of Wheat Offal to Aba ad 9 bags of the sae product to Eugu ad bags of the sae product to Otsha. We the coclude that gve dscouts o cost of trasportato could lead to creased productvty of producers. Ths s as a result of the fact that wholesalers ad retalers, wll have to pay less o trasport for buyg large quattes, subsequetly, cosuers wll buy at lower cost coparatvely. RECOMMENDATION The algorths used ths research wor are ot copared to ay other prevous algorths; therefore, the future, wor should be doe to:. Measure the effcecy of the algorth. Chec how ear the soluto of the approated proble of the pecewse olear trasportato proble s to the optal soluto of the orgal proble. 3. To pleet the algorth to cople real lfe probles. REFFERENCES Abad, P. L. (988). Deterg optal sellg prce ad lot-sze whe the suppler offers all-ut Quatty Dscouts, Decso Scece, page 9, Adby P.R ad Depster M.A.H. (976). Itroducto to Optzato Methods. Arcelus, F.J.; J. Bhadury ad Srvasa G (996). Buyer-seller quatty dscout strateges uder proft azg obectves: A gae-theoretcal aalyss, page Aucop, D.C. ad P.J. Kuzdrall (986). Lot-szes for oe-te-oly sales, Joural Operato Research Socety, 37, Baer, R.C. (976). Ivetory polcy for tes o sale durg regular repleshet, PIM 7, Baeree, A. (986). O a quatty dscout-prcg odel to crease vedor profts, Maageet Scece. 3, Bara, S. ad R.J. Terse (995). Ecooc purchasg strateges for teporary prce dscouts, Europea Joural of Operatos Research, 8, Basta, M. (99). A perfect lot-tree procedure for the dscouted dyac lot-sze proble wth speculato, Naval Research Logstcs, 3,

14 Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4():58-7 (ISSN: 4-76) Bazaraa M.S. (993). Nolear prograg: theory ad algorths, Joh Wley ad Sos, Ic, Sec. Ed.77 Beto, W.C. ad D.C. Whybar (98). Materal requreets plag (MRP) ad purchase dscouts, oural of atheatcs, page Beto, W.C. ad S. PARK (996). A classfcato of lterature o deterg the lot-sze uder quatty dscouts, Europea oural of Operatos Research, 9, Beto. W.C. (99). Qualty dscout decsos uder codtos of ultple tes, ultple supplers ad resource ltatos, IJPR, 9(), Caputo A.C. (6). A geetc approach for freght trasportato plag, Idustral Maageet ad Data Systes, Vol. 6 No. 5, page Efroyso M.A. ad Ray, T.L. (966). Brach- Boud Algorth for Plat Locato, Operatos Research, Vol 4, Ellwe, L.B. (August, 97). Fed Charge Locato-Allocato Probles wth Capacty ad Cofgurato Costrats, Techcal Report No.7-, Departet of Idustral Egeerg, Staford Uversty, Fal J.E. ad Solad R.M. (97). A Algorth for Separable Nocove Prograg Probles, Maageet Scece Vol. 5, page Frederc S., Hller ad Gerald J. Lebera (99). Itroducto to atheatcal prograg, McGraw-Hll, Icoprato, Secod Edto. Gddgs A.P., Bauet T.G. ad Moore J.T. (). Optalty aalyss of faclty Optalty aalyss of faclty locato probles usg respose surface ethodology, Iteratoal Joural of Physcal Dstrbuto ad Logstcs Maageet, Vol. 3 No. page MCB Uversty. Gray (97). Eact Soluto of the Ste Selecto Proble by Mced Iteger Prograg, Appled Matheatcal Prograg Techques,E.M.L. Beale(ed.), Aerca Elsever Publshg Co., New Yor. Htchcoc F.L. (94). The Dstrbuto of a Product fro Several Sources to Nuerous Localtes, oural of Matheatcs ad Physcs Vol., page Ie Tras.,, (994). Optal prces ad order quattes whe teporary prce dscouts result crease dead, Europea Joural of Operato Research. 7, 5-7. Iyaa S.C. (7): Operato Research: Itroducto. Supree Publsher, Owerr. Kara P. M. (). Lear Prograg ad Theory of Gaes, New cetral boo agecy (P) Ltd. Khuwala B.M., (97). A Acet Brach ad Boud Algorth for the Warehouse Locato Proble, Maageet Scece. Vol.8, Koopas T.C. (947). Optu Utlzato of the Trasportato Systes," Iteratoal Statstcal Coferece, Washgto, D.C., Vol.5. Lau, H.C.W.; Cha, T.M.; Tsu, W.T.; Cha, F.T.S.; Ho, G.T.S ad Choy K.L. (9): A fuzzy guded ult-obectve evolutoary algorth odel for solvg traportato proble. Epert Syste wth Applcatos: A Iteratoal Joural. Vol. 36, pp Letcher, R.F. (969). Practcal Methods of Optzato, Uversty of Dudee, Scotlad, U.K., Lohgaoar, M.H. ad Baa, V.H. (): Fuzzy approach to solve ult-obectve capactated trasportato proble, Iteratoal Joural of Boforatcs Research. Vol., pp. 4. Mars D.H. (98). Faclty Locato ad Routg Models Sold Waste Collecto Systes, Ph.D. Thess, The Johs Hops Uversty. Mohaed A. (996). A teratve procedure for solvg the ucapactated producto dstrbuto proble uder cocave cost fucto, Iteratoal Joural of operatos ad aageet, Vol. 6. No. 3 page 8-7. Murtagh B.A., Ss J.W. (995). Iproved odelg of physcal dstrbuto, Iteratoal Joural of Physcal Dstrbuto ad Logstcs Maageet, Vol. 5 No.8 page. 47-5, MCB Uversty. Reep J. ad Leaegood S. (Jue ). Trasportato proble: A Specal Case of Lear Prograg Probles, Operatos Research Socety of Aerca. Sa G. (969). Brach- ad- Boud ad Approate Solutos to the Capactated Plat Locato Proble, Operatos Research, Vol. 7, page 6-. Shafaa A; Goyal S.K. (Aprl 988). Resoluto of Degeeracy Trasportato Probles, Operatos Research Vol.39, No Shetty C.M. (959). A Soluto to the Trasportato Proble wth Nolear Costs," Operato Research. Vol. 7. No.5. page

15 Joural of Eergg Treds Egeerg ad Appled Sceces (JETEAS) 4():58-7 (ISSN: 4-76) Sos (March, 96). Nolear Prograg for Operato research, Socety for Idustral ad Appled Matheatcs, Vol., No.. Solad R.M. (974). Optal Faclty Locato wth Cocave Costs" Ops. Res. Vol., No.. page Spelberg K, (967). A Algorth for the Sple Plat Locato Proble wth Soe Sde codtos Operatos Research, Vol.7, page Stewart T.J. ad H.W. Itha (968). Two stage Optzato Trasportato Proble. Strger J. ad Haley K.B, (957). The Applcato of Lear Prograg to Large-scale Trasportato Proble, Process Frst Iteratoal Coferece o Operato Research, Oford Taha, H.A. (99): Operato Research: A Itroducto, 5 th ed. (Maclla, New Yor. Vdale M.L. (956). A Graphcal Soluto to the Trasportato Proble," Operato Research, Vol. 4, page Wllas A.C. (March, 96). A Treatet of Trasportato Probles by Decoposto, oural Socety for Idustral ad Appled Matheatcs, Vol., No. page Zagwll W.I. (967). The Cove Sple Method, Theory Seres.Vol.4, No.3, -38. Zagabad, M ad Male, H.R. (7): Fuzzy goal prograg for ult-obectve trasportato probles. Appled Matheatcs ad Coputato, vol. 4, pp APPENDIX (TORA SOFTWARE PACKAGE OUTPUT) TRANSPORTATION MODEL (VOGEL S METHOD) Iterato ObVal 43. D D D3 D4 Supply Nae v=8. v=. v3=6. v4= S u=. 8.. S u= S3 u3= Dead 3 7 Iterato ObVal 394. D D D3 D4 Supply S S S3 Nae v=8. v=. v3=6. v4= u= u= u3= Dead 3 7 supply 7 5 7