The solution of transport problems by the method of structural optimization

Transcription

1 Scentfc Journls Mrtme Unversty of Szczecn Zeszyty Nukowe Akdem Morsk w Szczecne 2013, 34(106) pp , 34(106) s ISSN The soluton of trnsport problems by the method of structurl optmzton Sergey A. Krgnov e-ml: sergey.krgnov@gml.com; krgnov@yndex.ru Key words: trnsportton problem, optmzton, the blnce of nterests, the compenston of dmges of the prtes Abstrct Currently do not tke nto ccount the possblty of constructng seprte supply of the product optml desgns for supplers nd consumers. Ths mpedes the development of optmzed trffc flow n the country. The proposed method of structurl optmzton llows to fnd compromse pln for optml delvery of products. Introducton The key feture of trnsport problems s tht the obects re homogeneous nd nterchngeble pln on how to use. Dstrbuton s only one type of resource, of whch ech unt dependng on the locton of ts orgnl locton nd plce of ts ntended use receves dfferent estmte of the totl cost of delvery. Trnsport problems re solved usng two methods: the smplex method nd the method of potentls. Both methods yeld the sme optml solutons tht re ndependent of the wll of reserchers to the nterests of ether supplers or consumers. When solvng rel-world trnsportton problems such optml soluton would be vrtully mpossble to mplement becuse provders nd recpents of products re well nformed bout mrket prces, trnsportton nd producton, nd no optml or expert methods of delverng ther products cn not force them to dopt economclly dsdvntgeous decson. The method of structurl optmzton, used for solvng trnsport problems nd developed by the uthor of ths work, helps to vod these shortcomngs nd fnd compromse soluton. When usng ths method, the phrse... the best structure... should be understood s n optml scheme of processes (optons for delvery of products), whch ensures complnce wth gven constrnts on supply nd demnd of products t the lowest cost of trnsportton. An lgorthm of the structurl optmzton method nd the results of ts use re llustrted n ths pper on the exmples of problem solvng number 85 nd number 71 from the textbook [1]. Exercse number 85. Mchnng Dvson hs t ts dsposl three mchnes: M 1, M 2, M 3 whch cn produce four knds of detls: D 1, D 2, D 3, D 4. Tme spent (n mnutes) s gven n the tble 1. Tble 1. The tme these mchnes need to produce one pece of ech type detls M M M The cceptble workng hours of mchne tools (ndctor A ) equls for the tool: А mn, А mn, А mn. The plnned producton of tools (ndctor B ) equls: В peces, В peces, В peces nd В peces. The tsk s to dstrbute the producton of prts for mchne tools so s not to exceed the tme lmt the use of mchnery to ensure the plnned volumes of producton prts re reched, tkng the optmzton crteron s the mnmum totl tme of usng the mchne tools. Zeszyty Nukowe 34(106) 59

2 Sergey A. Krgnov Soluton. Accordng to the optmzton crteron, the mthemtcl model tkes the followng form: x ) 12x 15x wth constrnts: 10x x 32 x 21 8x x 2x x mn 1) x 2x 10x 12x ) x 8x 2x ) x x (1) 4) x x x 200 (2) ) x x x ) x x x ) x x x where: mount of tme (mn) of tool (suppler) for producton of one detl of type (for consumer); х mount of detl type (for consumer), produced on tool (delvered by suppler). After solvng the problem wth the smplex method we hve: x ) = 5200, when the vlue of vrbles re: х 11 = 200, х 23 = 200, х 24 = 600 nd х 32 = 800. Consder soluton of ths problem provded tht the optmzton s crred out n the nterests of consumers of prts. In ths cse, the process of fndng the optml soluton wll consst of two steps: The frst step. Set prortes for ech consumer (the problem s consumers re detls) for the servces supplers (.e. the processng tme of detls): B P (3) Obvously, the greter the P () (Tble 2), the hgher the prorty of the provder detls. Therefore, the crteron of cost optmzton for the producton of detls cn be gven by the followng functon: Tble 2. The vlues of ndvdul prortes (P () ) for components ( ) M 1 40* M * 120* M * * ( ) P ( ) mx (4) where v = 1 for dmssble, under the ssumed constrnts, technologcl methods of delvery, wth hghest prorty. Step 2. Determne the optmum dstrbuton of volumes (X ) of detls n ech of the mchnes ccordng to the mxmum vlue of crteron (3) for every product. In tble 2, the vlues of these qunttes re mrked wth *. As for the detls of D 4 prortes of ts producton on mchne M 2 nd M 3 re equl, then the output of ths prt should be plnned on mchne whch s the lest utlzed n the producton, whch n ths exmple, hppens to be mchne M 2. The resultng optml (from the pont of consumers of these detls) dstrbuton of producton nd supply of detls for mchne tools s gven n tble 3. Tble 3. Optml dstrbuton detls to customers M M M The totl mnmum processng tme of prts x ) s: F ( ) mn. x However, the obtned soluton represents only the nterests of one sde detls of consumers who re nterested n reducng producton cost. Another spects of ths exmple re the prts supplers, whch, n turn, re more nterested n the rtonl (economcl nd unform) usng equpment owned by them. Therefore, for producers of the detls functon optmzton (4) becomes (5): A ( ) P( ) mx (5) The dstrbuton of the detls to the crter (5) under constrnts (2) n ths cse cn lso be relzed n two steps: Step 1. Settng prortes (P () ) 1 of mchnes 2 durng the processng of the detls of type tht re provded n tble 4. In tble 4 the sgn * mrks the best optons of dstrbutng of detls for producton on mchnes. 1 In ths exmple, the mxmum relese of detls of vrous types on prtculr mchne, but n generl lmtng shpments of smlr products ech suppler. 2 Assume tht the mchnes vlble for lesng. 60 Scentfc Journls 34(106)

3 The soluton of trnsport problems by the method of structurl optmzton Tble 4. Prortes for the producton of detls n ech of the mchnes M * ,33 M * 2000 M * Step 2. Allocte producton of detls for the mchnes of gven the vlues found prortes. For mchnes M 2 follow to be fxed s producton detls D 1 (P 2(1) = 1000) nd detls of D 4 (P 2(4) = 2000). Gven tht the producton of detls D 2 s benefcl for the speclzton of the two mchnes (M 1 nd M 3 ), the dstrbuton of producton detls D 2 between them should be mde proportonl to the vlues of the prortes of these mchnes n the producton of ech of the detls. So, f necessry, the producton of detls on two mchnes (k nd l) the dstrbuton of producton volumes of detls should be mde ccordng to formuls (6). B k( ) Pk ( ) B( ) ; Bl ( ) B Bk ( ) (6) P P k( ) l( ) In ths exmple, the dstrbuton of producton detls D 2 between M 1 nd M 3 wll be respectvely 274 unts [ /( )] nd 526 detls ( ). The dstrbuton of the plnned producton volume detls D 2 nd other detls re gven n tble 5. Tble 5. The optml producton pln for supplers of detls M M M Totl mnmum tme of mchne utlzton wll be n ths cse: x ) mn. If you compre the optons plns submtted n tbles 3 nd 6, berng n mnd the tme of mchne utlzton, t cn see tht the optmzton ccordng to crteron (6) relly reflects the nterests of the owner of mchnes (the mnufcturer): the totl tme of mchne utlzton ncresed by 1.25 tmes (6474:5200), s result of ncresed producton on M 2,.e. the mchne tht hs the hghest tme of utlzton. At the sme tme ncresed the volume of delveres of products from the mchne M 1 from 200 to 274 prts. It s obvous tht such lrge dscrepncy between the best possble solutons to ths problem for supplers nd consumers wll be the mn detls of the reson why none of the optml desgn of delvery detls wll not be relzed. However, both prtes nvolved n solvng ths trnsportton problem nd ts soluton should eqully consder the nterests (prortes) of both supplers nd customers of products. Therefore, the clculton of the bsolute prortes for the generl optons under consderton, the supply of products to solve trnsportton problems by structurl optmzton should be performed ccordng to formul (7): A B P P P (7) In ths cse, functon optmzton (v ) tkes the form: A B ( ) P mx (8) Note, tht ths method does not prortze the obvous dvntges to ny of the plyers n ths process. In ts fnl form of n lgorthm for solvng trnsport problems by the method of structurl optmzton cn be gven by: The frst step. Settng prortes (P ) of technologcl solutons to problems usng the formul (8), gven n tble 6. Tble 6. Uncondtonl prortes of technologcl optons for producton nd delvery of detls M ,3 M * * 2120* M 3 193,3 3200* The second step. Dstrbuton of the producton of prts for the mchne tools, ccordng to the mxmum vlue of P (n decresng order), whch re prortes n the tble mrked wth *. The optml soluton, obtned tkng nto ccount the bsolute prortes, s gven n tble 7. Tble 7. Combned optml producton nd delvery detls M 1 M M Zeszyty Nukowe 34(106) 61

4 Sergey A. Krgnov The totl mnmum processng tme of detls x ) mount n ths cse to: F ( ) mn. x The resultng optml pln not only reflects the needs of the detls to mnmze the processng tme, but wll lso mke one mchne (M 1 ) redundnt. Exercse number 71. A mnufcturer of buldng mterls hve three wrehouses (W 1, W 2, W 3 ) n dfferent prts of the cty provdes brcks to compny, whch constructs prtments n four dfferent vllges (V 1, V 2, V 3, V 4 ). In wrehouses locted respectvely 100, 50 nd 80 thousnd brck, nd constructon stes needs ccount for respectvely 40, 70, 30 nd 50 thousnd brcks. The costs of the constructon frm to delver one thousnd brcks from the wrehouses to constructon stes re shown n tble 8. The cost of storng thousnd peces of brck n wrehouses s 10, 12.5 nd 10 PLN. Tble 8. Costs (n PLN for 1 thousnd peces) for the vrous technologcl optons for delvery of brcks Wrehouse Vllges W W W Accordng to the pror rrngements vllge V 1 should receve no less thn 30 thousnd peces of brcks from the wrehouse W 1, vllge V 2 35 thousnd peces from the wrehousew 1, nd V 3 10 thousnd peces from the wrehouse W 2 nd no less thn 10 thousnd peces from wrehousew 1. Tkng nto ccount the submtted dt the pln of the brck trffc provdng wth the mnmum trnsportton costs nd storge costs should be drwn. One more queston to nswer s wht trnsportton nd storge costs would be f prelmnry rrngements fled. Soluton. Gven the prelmnry rrngements for mndtory supply of brck from wrehouses to consumers, brck wrehouses whch re free to dstrbute stock (А *), consttute (n thousnd peces): n wrehouse W 1 25( ); n wrehouse W 2 40(50 10) nd W 2 n wrehouse 80 thousnd of peces of brcks, nd the not yet met needs (B *) n buldng stes wll be respectvely: 10(40 30), 35(70 35), 10( ) nd 50. The overll cost of prelmnry greements reched 4950 thousnd PLN ( ). Tkng nto consderton the prelmnry greements, the trnsport problem tkes the form presented n the tble 9. Tble 9. Trnsportton costs of one thousnd brcks nd the remnng brcks n wrehouses (А *) nd yet not met needs for brck n vllges (n B *) Wrehouse Vllges А * W W W В * The frst step. Settng prortes (P ) of technologcl solutons to ths problem ccordng to formul (8). The prortes re presented n the tble 10. Tble 10. The vlues of the bsolute prortes of dfferent technologcl delvery vrnts for prevously not conducted delveres Wrehouse Vllges W W * W * 7.667* 1.800* The tble 10 the symbol * denotes gven vrnts of mxmum dstrbuton. The result of ths llocton s gven n tble 11. Tble 11. The prelmnry pln of optml supply of yet undelvered brcks Wrehouse Vllges W 1 W 2 40 W The second step. Snce the dstrbuton n the supply of brcks for the mxmum vlues of the bsolute prortes of the technologcl methods of delvery of brcks produced n the frst step of clcultons (Tble 10), dd not provde full sze requrements n the supply of brcks to vllge V 4 (P 3;4 = 2.167). The mssng brcks n need for ths vllge n order of prorty cn be suppled from stock W 3. Tble 12. The optml supply pln blnces the need for brcks Wrehouses А * W 1 25 W 2 40* 40 W 3 10* 35* 10* 10* 80 В * Scentfc Journls 34(106)

5 The soluton of trnsport problems by the method of structurl optmzton For technology optons dentfed n the tble 12 the symbol * needs nd supples of brck re equl wth the possbltes of meetng them. If we dd to the mounts n tble 12 gven the prelmnry rrngements whch hve lredy tken plce, the full mount of the optml supply of brcks tkes the form shown n the tble 13. Tble 13. The optml verson of the pln delvery of brcks (n thousnds) tkng nto ccount pre-rrngements Wrehouses А * W W * W В * 40* 70* 30* 50* As follows from the dt presented n tble 13 the possbltes of wrehouses W 1 nd W 2 hve not been fully utlzed. The remns of brcks n those stocks ccount for 25 nd 15 thousnd unts, respectvely. And the nnul storge costs mount to 400 thousnd PLN ( ). The totl mnmum cost of trnsportng the brcks n the optml verson of the pln wll be: x ) thousnds PLN It should be noted tht ths opton s the optml pln whch ws presented n tble 13 concded wth the verson of the pln clculted wth the smplex method. The ddtonl queston of Exercse 71 concerned the trnsportton nd storge costs f the prelmnry rrngements fled. The nswer to ths queston usng the structurl optmzton method cn be obtned on the bss of complete (wthout the prelmnry rrngements) vlues of the bsolute prortes lsted n tble 14. Tble 14. The vlues of the bsolute prortes of dfferent technologcl delvery vrnts for prevously wthout the prelmnry rrngements Wrehouses W W W Then the optml delvery pln formed n the order of decresng bsolute prortes,.e. by (8), tkes the form presented n tble 15. Tble 15. The optml progrm of brcks delvery wthout pror greement, drwn up by structurl optmzton method Wrehouses А * W W * W * В * 40* 70* 30* 50* Gven ths concrete verson of the pln, the trnsportton chrges wll be: x ) thousnds PLN At the sme tme storge chrges of the remnng brcks n the stock W 1,.e. 40 thousnd peces wll be 400 thousnd PLN (40 10), nd the totl costs of storge nd trnsportton 6050 thousnd PLN. It should be observed tht by solvng ths problem by the smplex method (Tble 16) the totl costs of brck storge nd trnsportton would mount to only 6000 thousnd PLN. Tble 16. The optml progrm of brck delvery wthout pror greement, drwn up by the smplex method Wrehouses А * W W * W * В * 40* 70* 30* 50* Although n ths optml soluton the levels of brck delvery from W 1 nd W 3 re the sme, ther costs re dfferent. The totl cost of trnsportton from the wrehouse number 3 wll ncrese by 250 thousnd PLN, nd from the wrehouse number 1 on the contrry wll decrese by 300 thousnds PLN. At the sme tme verge of the costs trnsportton thousnd peces of brcks from the wrehouse W 1 to ll the consumers [( ):4] re hgher thn the delvery costs from W 3 [( ):4] by 1.7 tmes nd n ths optml soluton by tmes! It s therefore evdent tht the soluton of ths problem of the smplex method or potentl method would be uncceptble to consumers brck wrehouse W 1, but becuse of ths optml pln wll never be relzed. Conclusons Whle summrzng the presentton of the method of structurl optmzton nd exmples of Zeszyty Nukowe 34(106) 63

6 Sergey A. Krgnov ts mplementton, the smplcty nd verstlty of the method should be hghlghted. As the exmples presented n the pper hve demonstrted, the method of potentls nd the smplex method cnnot be lwys successfully used. Becuse the trnsportton of products s connected wth costs whch re pd by both consumers nd dstrbutors, then the dstrbuton method should stsfy the needs nd requrements of both prtes nvolved n the trnscton. The pln, drwn up usng the crteron (8) does not only stsfy the needs of the prtes of contrct, but lso better fts the fnncl condtons of the prtes. The method of structurl optmzton cn become n mportnt rgument whle doptng mutully dvntgeous condtons n trnsportton trnsctons, but lso the bss for determnng socl snctons for volton of the rules of the optml delvery of products n fvor of ether supplers or customers (the effects of monopoly of ether producers or consumers). References 1. JĘDRZEJCZYK Z., KUKUŁA K., SKRZYPEK J., WALKOSZ A.: Bdn opercyne w przykłdch zdnch. Redktor nukowy Krol Kukuł. Wydwnctwo Nukowe PWN, Wrszw Scentfc Journls 34(106)