INDR 262 Optimization Models and Mathematical Programming LINEAR PROGRAMMING MODELS


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1 LINEAR PROGRAMMING MODELS Commo termiology for liear programmig:  liear programmig models ivolve. resources deoted by i, there are m resources. activities deoted by, there are acitivities. performace measure deoted by z A LP Model: max z = c x = = a x b i =,...,m i i z : value of overall performace measure x : level of activity (=,,) c : performace measure coefficiet for activity b : amout of resource i available (i=,,m) a i : amout of resource i cosumed by each uit of activity Decisio Variables: x Parameters: c,a i,b Stadard Form of the LP Model A Liear programmig problem ca be expressed i the followig stadard form: max z= c x + c 2 x c x a x + a 2 x a x b a 2 x + a 22 x a 2 x b a m x + a m2 x a m x b m x 0 x x 0 Obective fuctios: overall performace measure c x + c 2 x c x Costraits: a i x +a i2 x a i x b i i=,,m (Fuctioal costraits) x 0 =,, (Noegativity costraits)
2 Variatios i LP Model A LP model ca have the followig variatios:. Obective Fuctio: miimizatio or maximizatio problem. 2. Directio of costraits a i x +a i2 x a i x b i i=,,m less tha or equal to a i x +a i2 x a i x b i i=,,m greater tha or equal to a i x +a i2 x a i x = b i i=,,m equality 3. Noegativity costraits  x Termiology for solutios of the LP Model Solutio: ay specificatio of values for the decisio variables, x, is called a solutio Ifeasible Solutio: a solutio for which at least oe costrait is violated. Feasible Solutio: a solutio for which all of the costraits are satisfied The best CPF = optimal solutio Corer  Poit Feasible (CPF) solutio: a solutio that lies at the corer of the feasible regio. Optimal Solutio: a feasible solutio that has the most favorable value of the obective fuctio. maximizatio largest z miimizatio smallest z Multiple Optimal Solutios: ifiite umber of solutios with the most favorable value of the obective fuctio 2
3 Assumptios of Liear Programmig. Proportioality:  cotributio of each activity to the obective fuctio, z, is proportioal to its level. c x  cotributio of each activity to each fuctioal costrait is proportioal to its level. a i x 2. Additivity:  every fuctio is the sum of the idividual cotributio of the respective activities. z = c x = = a x b i =,...,m i i 3. Divisibility:  decisio variables are allowed to have ay real values that satisfy the fuctioal ad oegativity costraits. 4.Certaity:  the parameter values are assumed to be kow costats. Examples of LP  Radiatio Therapy Desig  Regioal Plaig  Cotrollig Air Pollutio  Reclaimig Solid Water  Persoel Schedulig  Distributio Network  Product Mix  Plaig read pp
4 Cosider the Wydor Glass Co. problem: SOLVING LP PROBLEMS max z= 3x + 5x 2 x 4 2x 2 2 3x + 2x 2 8 x 0 x A x2 x = 4 B C 2x2 = 2 D E 5 6 3x + 2x2 = 8 x CPFs (Corerpoit Feasible Solutios): itersectio poits of the system of equatios that defie the feasible regio. Defiitio: Adacet CPF solutios For ay liear programmig problem with decisio variables, two CPF solutios are adacet to each other if they share  costrait boudaries. The two adacet CPF solutios are coected by a lie segmet that lies o these same shared costrait boudaries. Such a lie is referred to as a edge o the feasible regio. Poit Coordiates (x, x 2 ) A (0,0) B (0,6) C (2,6) 4
5 Poit Coordiates (x, x 2 ) Its Adacet CPF solutios A (0,0) B (0,6) B (0,6) A (0,0) C (2,6) C (2,6) B (0,6) C (2,6) A (0,0) Optimality Test: If a CPF solutio has o adacet CPF solutios that are better, the it must be a optimal solutio. Solutio Algorithm:. Iitialize: choose a iitial CPF solutio. 2. Optimality Test: evaluate the performace measure at the curret solutio. If its value is larger tha all of its adacet CPF solutios; the curret solutio is optimal. Otherwise go to step Iteratio: Move to a better adacet CPF solutio. Go to step 2. Example: Wydor Glass Co. Iitializatio: select poit A (0,0) as the iitial CPF solutio. Optimality Test: z = 0. There may be better solutios. Iteratio: move to B. Why? z = 3 x + 5 x 2 Movig alog x 2 would make z larger. Optimality Test: z = 30. It is better tha Ai but is it the best? Iteratio: Move to C. Why? There is o poit else but C to move. Optimality Test: z = 36. It is better tha B, but is it the best? Iteratio: move to D. Why? Optimality Test: z = 27. Optimal Solutio correspods to CPF solutio C. Some observatios o the solutio algorithm: Simplex method focuses solely o CPF solutios. Simplex method is a iteratio algorithm. Wheever possible, the iitializatio of the simplex method chooses the origi as the iitial CPF solutio. Give a CPF solutio, it is much quicker to gather iformatio about its adacet CPF solutios tha its oadacet CPF solutios. After the curret CPF solutio is idetified, the simplex method examies each of the edges of the feasible regio. that emaate from this CPF solutio. The most promisig edge is selected ad this edge is chose for the ext iteratio. A positive rate of improvemet i Z implies that the adacet CPF solutio is better; a egative rate of improvemet i Z implies that the adacet CPF solutio is worse. The optimality test cosists of simply of checkig whether ay of the edges give a positive rate improvemet i Z. If oe do, the the curret CPF solutio is optimal. 5
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