An Optimization Procedure In A Production Line Of Sokat Soap Industry, Ikotun, Lagos State. Adamawa State, Nigeria.

Transcription

1 An Optmzaton Pocedue In A Poducton Lne Of Sokat Soap Industy, Ikotun, Lagos State. 1 O.S. Balogun, 2 A.A. Olatun and 3 A.A. Momoh 1 Depatment of Statstcs and Opeatons Reseach, Modbbo Adama Unvesty of Technology, P.M.B. 2076, Yola, Adamawa State, Ngea. 2 Depatment of Statstcs, Unvesty of Ilon, P.M.B. 1515, Ilon, Kwaa state, Ngea. 3 Depatment of Mathematcs, Modbbo Adama Unvesty of Technology, P.M.B. 2076, Yola, Adamawa State, Ngea. Emal: stapsalms@yahoo.com Abstact: Ths eseach wok has been used to demonstate the applcaton of lnea pogammng n optmzng the poft n a poducton lne usng SOKAT Soap Industy, Ikotun, Lagos State. Avalable to us ae the unts of poft of all the poducts as well as the stocks avalable fo the poducton lne. Usng these, the obectve functon s fomulated subect to a numbe of constants based on the pactces. Optmzaton (maxmzaton) was obtaned employng a lnea pogammng technque. It was obseved fom the analyss of the data that the poduct whch contbuted mostly to the poft eaned was the 1kg Salem soap. Howeve, the establshment needed to poduce othes because of the need of the numeous customes (the hotels and othe nsttutons) that needed the othe smalle szes such as the 10kg, 5kg and 2kg soap. [Onyema M.O., Osu L.C., Ofodle S.E. Geochemcal fngepntng of an ol-mpacted ste, Nge Delta: Souce and weatheng pofle of alphatc hydocabons. Reseache 2013;5(10):50-54]. (ISSN: ).. 7 Keywods: Optmzaton, Lnea pogammng, Obectve functon, Constants. 1. Intoducton Company manages ae often faced wth decsons elatng to the use of lmted esouces. These esouces may nclude men, mateals and money. In othe secto, thee ae nsuffcent esouces avalable to do as many thngs as management would wsh. The poblem s based on how to decde on whch esouces would be allocated to obtan the best esult, whch may elate to poft o cost o both. Lnea Pogammng s heavly used n Mco-Economcs and Company Management such as Plannng, Poducton, Tanspotaton, Technology and othe ssues. Although the moden management ssues ae eo changng, most companes would lke to maxmze pofts o mnmze cost wth lmted esouces. Theefoe, many ssues can be chaactezed as Lnea Pogammng Poblems (Svaethnamohan, 2008). A lnea pogammng model can be fomulated and solutons deved to detemne the best couse of acton wthn the constant that exsts. The model conssts of the obectve functon and cetan constants. Fo example, the obectve of SOKAT Soap Industy, Ikotun s to poduce qualty soaps needed by ts customes, subect to the amount of esouces (aw mateals) avalable to poduce the poducts needed by the espectve customes who should also not volate Standad Oganzaton of Ngea (SON). The poblem then s on how to utlze lmted esouces to the best advantage, to maxmze poft and at the sometme selectng the poducts to be poduced out of the numbe of poducts consdeed fo poducton that wll maxmze poft. The eseach s amed at decdng how lmted esouces, aw mateals of SOKAT Soap Industy, Ikotun, Lagos State would be allocated to obtan the maxmum contbuton to poft. It s also amed at knowng how the lmted esouces the ndusty would be allocated to acheve the best poducton mx at mnmum cost and detemnng the poducts that contbute to such poft. The scope of the eseach s to use Lnea Pogammng on some of the soaps poduced SOKAT Soap Industy, Ikotun, Lagos State. The data on whch ths s based ae quantty of aw mateals avalable n stock, cost and sellng pces and theefoe the poft of each poduct. The poft consttutes the obectve functon whle aw mateals avalable n stock ae used as constants. If demands whch must be met ae to be avalable, such can be ncluded n the constants. The data s seconday data collected n the yea 2007 at the SOKAT Soap Industy, Ikotun, Lagos State. The Smplex method, also called Smple technque o Smplex Algothm, was nvented by Geoge Dantzg, an Amecan Mathematcan, n It s the basc wokhose fo solvng Lnea Pogammng Poblems up tll today. Thee have been many efnements to the method, especally to take advantage of compute mplementatons, but the essentals elements ae stll the same as they wee when the method was ntoduced (Chnneck, 2000; Gupta and Ha, 2006). 50

2 The Smplex method s a Pvot Algothm that tansveses the though Feasble Basc Solutons whle Obectve Functon s mpovng. The Smplex method s, n pactce, one of the most effcent algothms but t s theoetcally a fnte algothm only fo nondegeneate poblems (Feng, 1986). To deve solutons fom the LP fomulated usng the Smplex method, the obectve functon and the constants must be standadzed. The chaactestcs of the standad fom ae: 1. All the constants ae expessed n the fom of equatons except the non-negatvty constants whch eman nequaltes The ght-hand-sde of each constant equaton s non-negatve. 3. All the decson vaables ae non-negatve. 4. The Obectve functon s of maxmzaton o mnmzaton type. Befoe attemptng to obtan the soluton of the lnea pogammng poblem, t must be expessed n the standad fom s then expessed n the the table fom o matx fom as gven below: M a x m z e Z C 1 1 S u b e c t t o a b, ( b 0 ), 1, 2, 3,..., m Basc Vaable E y1 0 a11 a12 a13... y2 0 a21 a22 a23... y3 0 a31 a32 a33... b Z E 1 0, 1, 2, 3,..., m In standad fom (Canoncal fom), t s Maxmze Subect to Z C 1 1 a S b, 1, 2, 3,..., m 0, 1, 2,3,..., n S 0, 1, 2,3,..., m Any vecto satsfyng the constants of the Lnea Pogammng Poblems s called Feasble Soluton of the poblem (Fogel, 1996; Schulze, 1998; Chnneck, 2000). 1.1 Algothm to solve lnea pogammng poblem: 1. See that all b s ae postve, f a constant has negatve b multply t by 1 to make b postve. 2. Convet all the nequaltes by the addton of slack o by subtacton of suplus vaable as the case may be. 3. Fnd the statng Basc Feasble Soluton. 4. Constuct the Smplex table as follows: n y1 y2... ym b a1m b1 a2m b2 a3m bm Z Z1 Z2 Z3 Z Z Z0 E E1 E2 E3 E n y1 y2... ym 5. Testng fo optmalty of Basc Feasble Soluton by computng Z E f Z E 0 the soluton s optmal; othewse, we poceed to the next step. 6. To mpove on the Basc Feasble Soluton, we fnd the basc matx. The vaable that coesponds to the most negatve of Z E s the INCOMING VECTOR whle the vaable b that coesponds to the mnmum ato a fo a patcula, and a 0, 1,2,3,..., m s the OUTGOING VECTOR. 7. The key element o the pvot element s detemned by consdeng the ntesecton between the aows that coesponds to both ncomng and outgong vectos. The key element s used to geneate the next table. In the next table, pvot element s eplaced by UNITY, whle all othe elements of the pvot column ae eplaced by zeo. To calculate the new values fo all othe elements n the emanng ows of that fst column, we use the elaton. 8. New ow = Fome element n old ows (ntesecton element n the old ow) 9. (Coespondng element of eplacng ow). 10. Test of ths new Basc Feasble Soluton fo optmalty as (6) t s not optmal; epeat the pocess tll optmal soluton s obtaned. Ths 51

3 was mplemented by the softwae Toa Wndow Veson SIMPLE METHOD FLOWCHART START SET UP LPP IN STANDARD FORMAT INSERT SLACK VARIABLE AND SET UP INITIAL TABLEAU END NO ANY POSITIVE CONTRIBUTION YES OPTIMAL SOLUTION SELECT HIGHEST CONTRIBUTION AND IDENTIFY PIVOT ELEMENT INTERPRET RESULTS MAKE PIVOT ELEMENT INTO UNITY BY ROW OPERATION MAKE ELEMENT IN PIVOT ELEMENT COLUMN TO ZERO SET UP NEW TABLE 52

4 Table 1: Quantty of aw mateals avalable n stock Raw Mateal Maxmum Quantty avalable n Stock(kg) 1. P.K.O Tallow Caustc Soda Chalk Slcate 200 Table 2: Quantty of aw mateals needed to poduce vaous szes of Salem Soap Raw Mateal 10kg 5kg 2kg 1kg 1. P.K.O Tallow Caustc Soda Chalk Slcate Table 3: Aveage Cost and Sellng of a cate of each poduct Szes Aveage Cost pce( ) Aveage Sellng pce( ) Poft( ) 10kg kg kg kg Model Fomulaton Maxmze Z Subect to Fo 0, 1, 2,3, 4 Now, ntoducng the slack vaable to convet nequaltes to equatons, t gves: Maxmze Z Subect to Fo 0, 1, 2,3,...,9 53

5 Whee, 1 = 10kg of Salem soap 2 = 5kg of Salem soap 3 = 2kg of Salem soap 4 = 1kg of Salem soap 2.0 Analyss and Result Optmal Soluton Obectve functon value = 64, Vaables Value The Toa Wndow Veson 2.0 gves: Z = 64, and 4 = Intepetaton of Result Based on the data collected the optmum esults deved fom the model ndcate that one of the poducts should be poduced 1kg of Salem Soap. The poducton quantty should be 3810 unts. Ths wll poduce a maxmum poft of 64, Concluson Based on the analyss caed out n ths eseach and the esult shown, SOKAT Soap Industy, Ikotun should poduce 10kg, 5kg and 2kg of Salem soap but moe of 1kg of Salem soap n ode to satsfy the customes. Also, moe of 1kg of Salem soap should be poduced n ode to attan maxmum poft because t contbute mostly to the poft eaned. Coespondence to: Balogun Oluwafem Samson C/o Depatment of Statstcs and Opeatons Reseach, Modbbo Adama Unvesty of Technology, P.M.B. 2076, Yola, Adamawa State, Ngea. Telephone: E-mal: stapsalms@yahoo.com Refeences 1. Chnneck, J.W. Pactcal Optmzaton, A Gentle ntoducton; Feng, B.R. Lnea Pogammng: An Intoducton; Sage Publcatons Inc., USA, Fogel, M. Opeatons Reseach Poblems Solve; Reseach and Educaton Assocaton; Pscataway, New Jesey, Gupta, P.K. and Ha, D.S. Opeatons Reseach; Raenda Ravnda Pntes Ltd., New Delh, Schulze, M. A. Lnea Pogammng fo Optmzaton; Pespectve Scentfc Instuments Inc., USA, Svaethnamohan, R. Opeatons Reseach; Tata McGaw Hll Publshng Co. Ltd., New Delh, /2/