LP: example of formulations

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1 LP: eample of formulatos Three classcal decso problems OR: Trasportato problem Product-m problem Producto plag problem Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

2 Trasportato problem The decso problem: cosderg a certa tme perod (e.g., a year), determe the most coveet way to trasport some avalable tems (products) for satsfyg a gve demad the same perod m supplers producg s,...,s m quattes of a product customers requrg r,...,r quattes of that product Assume geeral that the product ca be trasported from ay suppler to ay customers For each ut of product trasported from suppler to customer j the cost s c j Problem: determe the quattes of product trasported betwee each par (,j) suppler-customer such that the demad are satsfed, the avalabltes are respected ad the total trasportato cost s mmzed Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

3 Trasportato problem Graph represetato s r s c j r j s m r Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

4 Trasportato problem Data feasblty codto Formulate TP as LP: m s r j Defe the decso varables Defe the objectve as a fucto of decso varables Defe the set of costrats that mpose that the varables must assume oly feasble values (satsfacto of operatoal codtos ad complacy wth varables meag) j Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

5 Trasportato problem: formulato Varables: Product quatty trasported o each arc (cotuous varables) Objectve fucto: Total trasportato cost Costrats: j R,..., m; j,..., Total quatty suppled by supplers caot eceed avalablty j Total quatty receved by customers must be equal to the demad m Oly postve quattes ca be suppled j 0,.., m; j,..., (3) m j s j r j j j c j j,..., m (),..., (2) Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

6 Trasportato problem: formulato TP formulated as LP problem m s. t. j m j j j m s j c j j,..., m () j rj j,..., (2) 0,..., m; j,..., R,..., m; j,..., (3) Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

7 Trasportato problem: varatos Some possble varatos of TP Mamum trasportato capacty for arcs the cosdered perod Each customer ca be served by a subset of supplers Trasportato through termedate dstrbuto ceters eample s r s s m Dstrbuto ceters r j r Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

8 Product m problem The decso problem: determe the optmal level of producto a referece tme perod for a set of products (actvtes) takg to accout the lmted avalablty of a set of resources the same perod dfferet product types ca be produced m dfferet producto resources (.e., materals) wth mamum avalablty b,...,b m Each product eed a certa quatty of a set of resources: specfcally, each ut of product eed a j uts of resource j For each product the proft for ut produced s c Problem: determe the quattes to produce (the m of products) order to mamze the total proft wthout eceedg the resources avalablty Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

9 Product m problem: formulato Varables: The quattes of products to produce (cotuous varables) R,..., Objectve fucto Total proft from producto Costrats: For each resource, the total used quatty caot eceed the mamum avalablty Oly postve quattes ca be produced aj bj j c,..., m () 0,.., (2) Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

10 Operatos Research Massmo Paolucc DIBRIS Uversty of Geova Product m formulated as a LP problem m j b a t s c j j,..., (2),..., 0 (),...,.. ma R Product m problem: formulato

11 Producto plag problem The decso problem: pla the level of producto for a sgle product type over a tme horzo(e.g., a year) determg the producto for each tme perod (e.g., moth) whch the horzo has bee dvded The product demad s kow for each perod Ivetory s allowed at the ed of a perod to store product for et perods A horzo of N perods (moths) s assumed For each perod are kow: the producto capacty m,...,m N the producto costs c,...,c N the vetory costs r,...,r N the product demad d,...,d N The tal vetory s gve M 0 Problem: determe the product quatty to be produced the cosdered perods to satsfy the demad, mmzg the overall producto ad vetory cost Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

12 Producto plag problem: formulato Varables: The quattes of product plaed to be produced each perod (cot. vars) R,..., N The quattes of product stored at the ed of each perod (cot. vars) s R,..., N Objectve fucto The total cost for producto ad vetory the N perods N ( c r s ) + Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

13 Producto plag problem: formulato Costrats: Product flow coservato for each perod + s d + s,..., N () Producto each perod caot eceed producto capacty m,..., N (2) Startg vetory level s 0 M 0 (3) Producto ad vetory levels caot be egatve 0 s 0,.., N (4) Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

14 Operatos Research Massmo Paolucc DIBRIS Uversty of Geova Plag problem (Sgle Item capactated lot-szg problem) formulated as LP ( ) N s N s M s N m N d s s t s r s c N,..., (4),..., 0 0 (3) (2),..., (),...,.. m R R Producto plag problem: formulato

15 Producto plag problem: formulato Possble varatos: dfferet products (mult-tem producto) Backloggg s allowed Ivetory capactes Mult-stage producto (sem-fshed products ad raw materal Bll of materal - Materal Requremet Plag - MRP) Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

16 LP theory ad soluto methods Ma assumptos for formulatg a problem as LP Proportoalty (varables multpled by costats) Addtvty (sum of varables by costats) Dvsblty (varables assume real values) Certaty (all coeffcets are assumed determstc) Net cocepts: Fudametal deftos Geometrcal aspects Smple method (George Datzg, 947) Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

17 LP: stadard ad caocal forms Two caocal forms for LP problems Caocal form of mamzato ma 0 A b 0 R c T Caocal form of mmzato 0 R T m 0 c A b Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

18 LP: stadard ad caocal forms Ay LP problem ca be epressed stadard form ma 0 A 0 R Where: decso varables vector c objectve fucto coeffcets vector bm rght had sde (rhs) costrat coeffcets vector Am matr of costrats coeffcets A[a j ],,..., j,...,m Assumptos:. b 0 b j 0 j,...,m 2. m< 3. mrak(a) c b T () (2) Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

19 LP: types of solutos Deftos: Costrats () defe as soluto of the LP problem Costrats () ad (2) defe as feasble soluto of LP problem A LP problem ca be: Feasble wth bouded optmal solutos The feasblty rego s a o empty polyhedro X { } R : A b A suffcet codto: f X s closed (Polytope) X Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

LP: types of solutos Deftos: A LP problem ca be: Feasble wthout bouded optmal solutos (ubouded problem) The feasblty rego s a o closed o empty polyhedro (ot

20 LP: types of solutos Deftos: A LP problem ca be: Feasble wthout bouded optmal solutos (ubouded problem) The feasblty rego s a o closed o empty polyhedro (ot suffcet codto) X ad ope X Not feasble The feasblty rego s a empty polyhedro X R such that X X 2 eample X X X2 X Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

21 LP: polyhedra Deftos: A halfspace R the set where a R b R T { R : a b} A polyhedro s defed by the tersecto of a set of halfspaces P { R : A b} A polytope s a bouded polyhedro (for M>0, ) A set X s a cove set gve two pots a, b X ay pot y geerated as y X y λ a + ( λ ) 0 λ s (s a cove combato of a b ) b, X < M X Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

22 LP: polyhedra Graphc represetato of a cove combato of two pots y λ a + ( λ) 0 λ b a X y b Deftos: A halfspace s a cove set The tersecto of cove set s a cove set A polyhedro s a cove set Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

23 LP: polyhedra Defto: y X A pot of a cove set (polyhedro) X s a etreme pot (verte) do ot est a, b X, a b such that y λ a + ( λ) b for some 0 < λ < X Vertces of X Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

24 LP: polyhedra Deftos:,..., Gve a set of pots the cove hull k R cov (,..., k ) { R : λ, λ k k, λ 0 } s the smallest cove set cludg such pots Theorem (property of etreme pots of closed polyhedra) Ay pot X where X s a closed polyhedro (polytope) wth etreme pots (vertces) e,,..., cove combato of such pots E Operatos Research Massmo Paolucc DIBRIS Uversty of Geova E E ca be epressed as λ e λ λ 0

25 LP: polyhedra Eample: cove combato of e e2 e3 e X X e 2 e 3 Operatos Research Massmo Paolucc DIBRIS Uversty of Geova

For all questions, answer choice E) NOTA" means none of the above answers is correct. A) 50,500 B) 500,000 C) 500,500 D) 1,001,000 E) NOTA

For all questions, answer choice E) NOTA means none of the above answers is correct. A) 50,500 B) 500,000 C) 500,500 D) 1,001,000 E) NOTA For all questos, aswer choce " meas oe of the above aswers s correct.. What s the sum of the frst 000 postve tegers? A) 50,500 B) 500,000 C) 500,500 D),00,000. What s the sum of the tegers betwee 00 ad