Supply Chain Management Chapter 5: Application of ILP. Unified optimization methodology. Beun de Haas

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1 Supply Cha Maagemet Chapter 5: Ufed Optmzato Methodology for Operatoal Plag Problem What to do whe ILP take too much computato tme? Applcato of ILP Tmetable Dutch Ralway (NS) Bu ad drver chedulg at Coeo, GVU, Qbuzz Producto plag at Hoogove Advaced bledg at Shell Plat locato at Iterbrew Cuttg platc at Eo Mobl Crew plag at Amerca Arle Emergg area: Healthcare Lk: Cutomer of AIMMS Ufed optmzato methodology Beu de Haa = Decompoto approach = Colum geerato Fater tha traghtforward ILP, ofte better tha heurtc Poblty to fd very good (but ot eceary optmal) oluto wth qualty meaure May ucceful applcato SCM More advaced Day Moday Tueday Wededay Thurday Frday Job 2 3 Workg tme 6 hour 8 hour 4 hour 8 hour 4 hour Durato Reveue 2 hour 5 3 hour 6 2 hour 4 Day Mo, wed, fr Mo, tue, thu Wed, thu, fr etc......

2 The Beuhaa problem Advaced ILP formulato Beu de Haa a depedet etrepreeur. Clet cotact hm for mall job. For each job j gve: the reward (c j ); the tme t take (a j ); the day o whch they ca be doe Plag perod: day,..., T. Beu ha Q t tme o day t (t =,..., T). Goal. Chooe ad pla the work to ear a much a poble. Formulato wth day pla. A day pla for day t a et of job that Beu ca do o day t. S t the et of feable day pla for day t The reward of day pla j equal to C jt Ue a bary varable: jt = f day pla j from S t the choe day pla for day t (ad 0 otherwe). ILP wth day pla Job at mot oce Dadvatage: olvg ILP may take a log tme Oe pla per day Soluto: rela tegralty cotrat, LP-relaato. Dadvatage: There are o may poble day pla Soluto: Coder oly teretg day pla Colum geerato. Decompoto!!!!!

3 Colum geerato for LP Job at mot oce Oe pla per day. Start wth a mall et of day pla 2. Solve LP-relaato. 3. Fd out f there a ew daypla that ca mprove the oluto (= prcg) 4. No optmum foud 5. Ye add pla to model ad go to 2. Prcg= Lagragea ubproblem Prcg (2) Fdg out f there are day pla to mprove oluto Recall: varable ca mprove oluto f ad oly f reduced cot are potve Prcg problem: Fd day pla wth mamal reduced cot If mamum > 0, add day pla Otherwe top So, fd a `optmal daypla for each day t Y = f job elected, 0 otherwe Reduced cot Day pla ha to ft = c y = π = a y Q t y λ t

4 Prcg problem for gve t Fdg tegral oluto ma ubject to y = a = y (c {0,} Q π t )y Solve LP-relaato = upper boud The: Roudg Solve ILP wth reultg et of colum Brach-ad-prce Decompoto Decompoto Mater IP: Varable: 0/ (or teger) for electo of feable ub pla I prcple huge umber of varable Solve Mater LP: Oly clude retrcted collecto of varable,.e. oly teretg varable Mater LP Shadow prce Feable Subpla Lagragea Subproblem (prcg problem) Soluto procedure:. Start wth feable oluto from Mater LP 2. Solve LP ad fd hadow prce 3. Coder ub problem to fd teretg varable (wth egatve reduced cot for mmzato problem) 4. Go to 2. f teretg colum (wth egatve reduced cot for mmzato problem) were foud Mater IP Feable Pla

5 Local delvery problem Chemcal are traported by truck from depot to 3 cutomer What hould be the route of the truck uch that the cot are mmal? Depot Local delvery problem Mater problem: I every cutomer vted? m.t. = r c a r r r r {0,} for all for all r Deco varable r dcate electo of route r c r : cot of route r a jr = f cutomer route r Local delvery problem (2) Gate agmet at Schphol Subproblem: Route for dvdual truck. Solvg the ub problem: Mamze π c cutomer route route. ILP: 80 flght 2. colum geerato: 560 flght = oe day at Schphol correpod wth mmze reduced cot c π route cutomer route

6 Gate agmet: Wat het probleem? Twee fae aapak We hebbe ee verzamelg vluchte: Aakomt- e vertrektjd Type vlegtug Herkomt e betemmg Evetuele voorkeure va de maatchappj Grodafhadelaar E we hebbe ee verzamelg gate: Mogeljke rego' (Schege/EU/No-EU) Mogeljke vlegtugtype (grootte) Mogeljke grodafhadelare Gateplae: verzamelg vluchte op ee gate. Zoek voor elke groep va `geljke gate ee eve groot aatal gateplae. 2. Koppel de gateplae aa fyeke gate. Gezocht: ee optmale plag Kotefucte Gate agmet at Schphol fae Robuute oploge Varate: (tuetjde) 2 Of oortgeljke fucte Evetueel correcte voor: Vluchte zelfde maatchappj Vluchte zelfde grodafhadelaar Betrouwbaarhed maatchappj Mater problem: Varable are pla for oe gate Each flght o eactly oe gate Flght aged to gate of correct type Other cotrat.. Mamze robute Subproblem: Feable pla for oe gate

7 Kolom geerate fae Rekereultate Model m.t. c a geeft electe va pla aa c : kote va pla a = al vlucht pla = overge beperkge {0,} Ee dagje Schphol: Looptjd LP: ecode Looptjd ILP: ecode Producto chedulg 6 type of tre, plag for 8 week demad are gve lmted mache capacty cot: et up cot ad vetory cot Mater problem: overall mache capacty Producto Schedule Tre.. Producto Schedule Tre 6 Cuttg tock Iro bar of gve legth Order for mall bar of dfferet legth How to cut gve order from mmal umber of bar? Decompoto: Mater LP: producto of all ordered format Lagragea ubproblem: patter for dvdual bar

8 Cutomer order Legth b 9 cm 27 cm 3 cm 35 cm 42 cm 48 cm 60 cm Number m Legte taaf L: meter ILP wth patter Dadvatage: olvg ILP may take a log tme Soluto: rela tegralty cotrat, LP-relaato. Dadvatage: There are o may poble patter Soluto: Coder oly teretg patter: Colum geerato. Colum geerato for LP. Start wth a mall et of patter 2. Solve LP-relaato. 3. Fd out f there a ew patter that ca mprove the oluto 4. No optmum foud 5. Ye add patter to model ad go to 2.