LINEAR PROGRAMMING. Introduction. Prototype example. Formulation of the LP problem. Excel Solution. Graphical solution

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1 Itroductio LINEAR PROGRAMMING Developmet of liear programmig was amog the most importat scietific advaces of mid 0th cet. Most commo type of applicatios: allocate limited resources to competig activities i a optimal way. Liear programmig uses a mathematical model. Liear because it requires liear fuctios. Programmig as syoymous of plaig. Very efficiet solutio procedure: simplex method. João Miguel da Costa Sousa / Alexadra Moutiho 3 Prototype example Wydor Glass Co. produces glass products, icludig widows ad glass doors. Plat produces alumium frames, Plat produces wood frames ad Plat 3 produces glass ad assembles. Two ew products: Glass door with alumium framig (Plat ad 3). New wood framed glass widow (Plat ad 3). As the products compete for Plat 3, the most profitable mix of the two products is eeded. João Miguel da Costa Sousa / Alexadra Moutiho 3 Formulatio of the LP problem x = umber of batches of doors produced per week x = umber of batches of widows produced per week Z = total profit per week (i thousads) Wydor Glass Co. Product Mix Problem Doors Widows Profit Per Batch $3.000 $5.000 (Batches of 0) Hours available Hours Used Per Batch Produced per week Plat 0 4 Plat 0 Plat maximize Z= 3x+ 5x : x 4 x 3x + x 8 ad x 0, x 0. João Miguel da Costa Sousa / Alexadra Moutiho 33 Graphical solutio Z= 3x + 5x 3 x = x + Z ( y= mx+ b) 5 5 Slope of the lie is 3/5 Trial ad error: Z = 0, 0, 36 Feasible regio: satisfies all costraits maximize Z= 3x + 5x : x 4 x 3x + x 8 ad x 0, x 0. Excel Solutio Wydor Glass Co. Product Mix Problem Doors Widows Rage Name Cells Profit Per Batch $3.000 $5.000 BatchesProduced C:D HoursAvailable G7:G9 Hours Hours Hours Used Per Batch Produced Used Available HoursUsed E7:E9 4 HoursUsedPerBatchProduced C7:D9 Plat 0 <= Plat 0 <= ProfitPerBatch C4:D4 Plat <= 8 TotalProfit G Doors Widows Total Profit 6 $ Batches Produced João Miguel da Costa Sousa / Alexadra Moutiho 34 João Miguel da Costa Sousa / Alexadra Moutiho 35

2 Matlab solutio Geeralizig the example All solvers are desiged to miimize the obective fuctio i the optimizatio toolbox. To maximize f(x), miimize f(x). solutio João Miguel da Costa Sousa / Alexadra Moutiho 36 Prototype example Geeral problem Productio capacities of plats Resources 3 plats m resources Productio of products Activities products activities Productio rate of product, x Level of activity, x Profit Z Overall measure of performace Z The most commo type of applicatio of LP ivolves allocatig resources to activities. João Miguel da Costa Sousa / Alexadra Moutiho 37 Liear programmig model Stadard form of the model Z = value of overall performace measure. x = level of activity, for =,,,. Decisio variables c = parameters of Z related to x. b = amout of resource i that is available for allocatio of activities, for i =,,, m. Parameters a i = amout of resource i cosumed by each uity of activity. Select the values for x, x,, x to maximize: : Z= c x + c x + + c x a x + a x + + a x b a x + a x + + a x b a x + a x + + a x b ad x 0, x 0,, x 0. m m m m João Miguel da Costa Sousa / Alexadra Moutiho 38 João Miguel da Costa Sousa / Alexadra Moutiho 39 Allocatio of resources to activities Termiology for solutios Resource Usage per Uit of Activity Activity Amout of Resource Resource available a a a b a a a b m a m a m a m b m Cotributio to Z per uit of activity c c c Solutio: ay specificatio of values for the decisio variables (x, x,, x ). Feasible solutio: solutio for which all the costraits are satisfied. Ifeasible solutio: solutio for which at least oe costrait is violated. Feasible regio: collectio of all feasible solutios. No feasible solutio: o solutio satisfies all costraits. João Miguel da Costa Sousa / Alexadra Moutiho 40 João Miguel da Costa Sousa / Alexadra Moutiho 4

3 Termiology for solutios Examples of solutios Optimal solutio: feasible solutio with the most favorable value of the obective fuctio. Most favorable value: largest value if maximizig or smallest value if miimizig. No optimal solutios: () o feasible solutios or, () ubouded Z (see ext slide). More tha oe solutio: see ext slide. More tha oe solutio Ubouded cost fuctio João Miguel da Costa Sousa / Alexadra Moutiho 4 João Miguel da Costa Sousa / Alexadra Moutiho 43 Corer poit feasible solutio Solutio that lies at a corer of feasible regio. Best CPF solutio must be a optimal solutio.. Assumptios of liear programmig Proportioality: the cotributio of each activity to the value of the obective fuctio Z ad left had of each fuctioal costrait is proportioal to the level of activity x. Violatios: ) Setup costs. João Miguel da Costa Sousa / Alexadra Moutiho 44 João Miguel da Costa Sousa / Alexadra Moutiho 45 Violatios of proportioality Violatios of proportioality ) Icrease of margial retur due to ecoomy of scale (loger productio rus, quatity discouts, learig curve effect, etc.). 3) Decrease of margial retur due to e.g. marketig costs: to sell more the advertisemet grows expoetially. João Miguel da Costa Sousa / Alexadra Moutiho 46 João Miguel da Costa Sousa / Alexadra Moutiho 47 3

4 Assumptios of liear programmig Additivity: every fuctio i a liear programmig model is the sum of the idividual cotributios. Divisibility: decisio variables ca have ay values. If values are limited to iteger values, the problem eeds a iteger programmig model. Certaity: parameters of the model c, a i ad b i are assumed to be kow costats. I real applicatios parameters have always some degree of ucertaity ad sesitivity aalysis must be coducted (idetify sesitive parameters). João Miguel da Costa Sousa / Alexadra Moutiho 48 The assumptios i perspective Mathematical model is idealized represetatio of real problem. Approximatios ad simplifyig assumptios are required for the model to be tractable. A reasoably high h correlatio lti betwee predictio of model ad reality is required. It is commo i real applicatios of LP that almost oe of the 4 assumptios holds completely. It is importat to aalyze how large the disparities are, ad maybe use alterative models. Alexadra Moutiho 49 Coclusios Liear programmig models ca be solved usig: Spreadsheet as Excel Solver for relatively simple problems Problems with thousads (or eve millios!) of decisio variables ad/or fuctioal costraits must use modelig laguages such as CPLEX/MDL ad LINGO/LINDO. Matlab Optimizatio toolbox (GUI orcode). Liear programmig solves may problems. However, whe oe or more assumptios are violated seriously, other programmig techiques are eeded. SIMPLEX METHOD João Miguel da Costa Sousa / Alexadra Moutiho 50 Example revisited Settig up the simplex method Wydor Glass Co. Origial form Maximize Z= 3x + 5x x 4 x 3x + x 8 ad x 0, x 0. Augmeted form Maximize Z= 3x + 5x () x + x = 4 3 () x + x = (3) 3x + x + x = 8 ad x 0, =,,3,4, Covert fuctioal iequality costraits to equivalet equality costraits usig slack variables João Miguel da Costa Sousa / Alexadra Moutiho 5 João Miguel da Costa Sousa / Alexadra Moutiho 53 4

5 Settig up the simplex method Slack variable is: Zero: solutio lies o costrait boudary; Positive: solutio lies o feasible of costrait; Negative: solutio lies o ifeasible of costrait Example p problem has 5 variables ad 3 oredudat equatios, ad so (5 3) degrees of freedom. Two variables have a arbitrary value. Simplex uses the zero value for these obasic variables. The solutio for the other three variables (basic variables) is a basic solutio. Basic solutios have thus a umber of properties. João Miguel da Costa Sousa / Alexadra Moutiho 54 Properties of a basic solutio Each variable is basic or obasic. Number of basic variables is equal to umber of fuctioal costraits (equatios). The obasic variables are set to zero. Values of basic variables are obtaied solvig system of equatios (fuctioal costraits i augmeted form). Set of basic variables are called the basis. If basic variables satisfy the oegativity costraits, basic solutio is a BF solutio. João Miguel da Costa Sousa / Alexadra Moutiho 55 Fial form Cor the obective fuctio as the ew costrait equatio: Maximize Z (0) Z 3x 5x = 0 () x + x = 4 () x + x = (3) 3x + x + x = 8 ad x 0, =,, Tabular form (simplex tableau) (a) Algebraic form Basic variable Eq. (b) Tabular Form Coefficiet of: Z x x x 3 x 4 x 5 (0) Z 3x 5x = 0 Z (0) () x + x 3 = 4 x 3 () () x + x 4 = x 4 () (3) 3x + x + x 5 = 8 x 5 (3) João Miguel da Costa Sousa / Alexadra Moutiho 56 João Miguel da Costa Sousa / Alexadra Moutiho 57 Simplex method i tabular form Iitializatio. Itroduce slack variables. Select decisio variables to be iitial obasic variables (=0) ad slack variables to be iitial basic variables. Iitial solutio of example: (0, 0, 4,, 8) Optimality test. Curret BF solutio is optimal iff every coefficiet i row (0) is oegative. If yes, stop; otherwise, iterate to obtai the ext BF solutio. Example: coefficiets of x ad x are 3 ad 5. Simplex method i tabular form Iteratio. Step : Determie the eterig basic variable, which is the obasic variable with the largest egative coefficiet. Basic variable Eq. pivot colum Coefficiet of: Z x x x 3 x 4 x 5 eterig basic variable Z (0) x 3 () x 4 () x 5 (3) João Miguel da Costa Sousa / Alexadra Moutiho 58 João Miguel da Costa Sousa / Alexadra Moutiho 59 5

6 Iteratio Step : Determie the leavig basic variable, by applyig miimum ratio test: Basic variable Eq. Coefficiet of: Z x x x 3 x 4 x 5 Z (0) pivot row x 3 () x 4 () / = 6 miimum x 5 (3) / = 9 pivot umber (PN) leavig basic variable replace leavig BV for eterig BV i the BV colum of the ext simplex tableau João Miguel da Costa Sousa / Alexadra Moutiho 60 Iteratio Step 3: solve for the ew BF solutio by usig elemetary row operatios. Iteratio Basic Coefficiet of: Eq. variable Z x x x 3 x 4 x 5 Z (0) row with 0 x 3 () egative coef. x 4 () x 5 (3) row with Z (0) positive coef. x 3 () (PN) x () x 5 (3) João Miguel da Costa Sousa / Alexadra Moutiho 6 Iteratio : Steps ad Complete set of simplex tableau Iteratio Basic variable Optimality test: solutio still ot optimal oe more iteratio Eq. Coefficiet of: Z x x x 3 x 4 x 5 Z (0) x 3 () / = 4 x () x 5 (3) /3 = miimum Ratio João Miguel da Costa Sousa / Alexadra Moutiho 6 Iteratio Basic variable Eq. Coefficiet of: Z x x x 3 x 4 x 5 Z (0) x 3 () x 4 () x 5 (3) Z (0) x 3 () x () x 5 (3) Z (0) x 3 () /3 /3 x () x (3) /3 /3 João Miguel da Costa Sousa / Alexadra Moutiho 63 Tie breakig i simplex method Tie for eterig basic variable: arbitrary choice; Tie for leavig basic variable: arbitrary choice; No leavig basic variable: ibl ubouded Z, i.e., error i model or computatio; Multiple optimal solutios: simplex fids oe optimal BF solutio; fid the others by cotiuig the algorithm choosig a obasic variable with a zero coefficiet as the eterig BV. Adaptig to other model forms Equality costraits There is o obvious iitial BF solutio, so artificial variables ad the Big M method are used. Negative g right had s: Multiply iequality by ; Fuctioal costraits i form Surplus ad artificial variables / big M method are used. Miimizatio Maximize Z. João Miguel da Costa Sousa / Alexadra Moutiho 64 João Miguel da Costa Sousa / Alexadra Moutiho 65 6

7 Artificial variables ad Big M method The real problem The artificial problem Maximize Z= 3x + 5x x 4 x 3x + x = 8 ad x 0, x 0. Algebraically elimiate x 5 Defie x = 8 3x x. 5 Maximize Z= 3x + 5 x Mx, 5 () x 4 () x (3) 3x + x 8 João Miguel da Costa Sousa / Alexadra Moutiho 66 (so 3x + x + x = 8) 5 ad x 0, x 0, x 0. 5 from equatio (0) for proper form. Postoptimality aalysis Reoptimizatio: whe the problem chages slightly, ew solutio is derived from the fial simplex tableau. Shadow price for resource i (deoted y i* ) measures the margial value of this resource, i.e., how much Z ca be icreased by (slightly) icreasig the amout of this resource b i. Example: recall b i values of Wydor Glass problem. The fial tableau gives y * = 0, y * =.5, y 3* =. João Miguel da Costa Sousa / Alexadra Moutiho 67 Graphical check Shadow prices Icreasig b i uit, the ew profit is 37.5 João Miguel da Costa Sousa / Alexadra Moutiho 68 Icreasig b i uit icreases the profit i $500. Should this be doe? Depeds o the margial profitability of other products usig that hour time. Shadow prices are part of the sesitivity aalysis. Costrait o resource is ot bidig the optimal solutio. There is a surplus of this resource. Resources as this are free goods. Costraits o resources ad 3 are bidig costraits. They have positive shadow prices ad o surplus. Resources are scarce goods. João Miguel da Costa Sousa / Alexadra Moutiho 69 Sesitivity aalysis Sesitivity aalysis of parameters c i Idetify sesitive parameters (those that caot be chaged without chagig optimal solutio). Parameters b i ca be aalyzed usig shadow prices. Parameters c i for variables ca be aalyzed graphically. Parameters a i ca also be aalyzed graphically. For problems with a large umber of variables usually oly b i ad c i are aalyzed, because a i are determied by the techology beig used. Graphical aalysis of Wydor example: c ca have values i [0, 7.5] without chagig optimal solutio. c ca have ay value greater tha without chagig optimal solutio. João Miguel da Costa Sousa / Alexadra Moutiho 70 João Miguel da Costa Sousa / Alexadra Moutiho 7 7

8 Theory of simplex method Recall the stadard form: Maximize Z= c x + c x + + c x a x + a x + + a x b a x + a x + + a x b a x + a x + + a x b ad x 0, x 0,, x 0. m m m m Augmeted form Maximize Z= c x + + c x + c x + + c x m a x + a x + + a x + x = b + a x + a x + + a x + x = b + a x + a x + + a x + x = b m m m + m m ad x 0, =,,,. João Miguel da Costa Sousa / Alexadra Moutiho 7 João Miguel da Costa Sousa / Alexadra Moutiho 73 Matrix form (stadard form) Maximize Z = cx Ax b ad x 0 c = [ c, c,, c ], x b a a a x = b = a = a a x, b, A. x bm am am am João Miguel da Costa Sousa / Alexadra Moutiho 74 Matrix augmeted form Itroduce the colum vector of slack variables: x+ x + x = s x + m The costraits are (with I (m m) ad 0 ( (+m)) ): [, ] x = ad x AI b 0 xs xs João Miguel da Costa Sousa / Alexadra Moutiho 75 Solvig for a BF solutio Oe iteratio of simplex: The obasic variables are set to zero. The m basic variables are i a colum vector deoted as x B. The basic matrix B is obtaied from [A, I] by elimiatig the colums with coefficiets of obasic variables. The problem becomes: Bx B = b. The solutio for basic variables ad the value of the obective fuctio for this basic solutio are: x B b c x c B b B = ad Z = B B = B Revised simplex method. Iitializatio (as origial): Itroduce slack variables x s. Decisio variables are iitial obasic variables.. Iteratio: Step (as origial): Determie eterig basic variable (obasic variable with the largest egative coefficiet). Step : Determie leavig basic variable. As origial, but computatios are simplified. Step 3: Determie ew BF solutio settig x B = B b. 3. Optimality test: solutio is optimal if every coefficiet i Z is 0. Computatios are simplified. João Miguel da Costa Sousa / Alexadra Moutiho 76 João Miguel da Costa Sousa / Alexadra Moutiho 77 8

9 Fudametal isight Fudametal isight Iitial tableau Row 0: Other rows: Combied: Fial tableau Row 0: Other rows: Combied: t= [ c 0 0] T=[ A I b] t c00 = T A I b t* = t+ y* T= [ y* A c y* y* b], y* = cb B T* S* T [ S* A S* S* b], S* B = = = t* y* A cy* y* b = T* S* A S* S* b Optimal performace idex: Z*=y*b Optimal solutio: x B =S*b, x B =0 Shadow prices: y* After ay iteratio, the coefficiets of the slack variables i each equatio immediately reveal how that equatio has bee obtaied from the iitial equatios. João Miguel da Costa Sousa / Alexadra Moutiho 78 João Miguel da Costa Sousa / Alexadra Moutiho 79 Duality theory DUALITY THEORY AND SENSITIVITY ANALYSIS Every liear programmig problem (primal) has a dual problem. Most problem parameters are estimates, others (as resource amouts) represet maagerial decisios. The choice of parameter values is made based o a sesitivity aalysis. Iterpretatio ad implemetatio of sesitivity aalysis are key uses of duality theory. João Miguel da Costa Sousa / Alexadra Moutiho 8 Primal ad Dual Problems Example: Wydor Glass Co. Primal problem (stadard form) Maximize i i = Z = = c x ax b, for i=,,, m ad x 0, for =,,,. Dual problem Miimize W m m = i= yb i i y a c, for =,,, i i i= ad y 0, for i=,,, m. i Primal problem = [ ] x Maximize Z 3 5, x 0 4 x 0 x 3 8 x 0 ad. x 0 Dual problem 4 Miimize Z= y y y3, 8 0 y y y [ ] ad y y y3 [ 000. ] João Miguel da Costa Sousa / Alexadra Moutiho 8 João Miguel da Costa Sousa / Alexadra Moutiho 83 9

10 Primal dual table for LP Example: Wydor Glass Co. Primal problem Coeff ficiet of: Coefficiet of: x x x y a a a b y a a a b y m a m a m a m b m c c c Coef fficiets of obect ive fuctio (M iimize) Dual problem x x y 0 4 y 0 y Coefficiets of obective fuctio (Maximize) João Miguel da Costa Sousa / Alexadra Moutiho 84 João Miguel da Costa Sousa / Alexadra Moutiho 85 Duality theory Primal dual relatioships Geeral relatios betwee primal ad dual problems:. The parameters for a (fuctioal) costrait i either problem are the coefficiets of a variable i the other problem.. The coefficiets i the obective fuctio of either problem are the right had s for the other problem. Weak duality property: If x is a feasible solutio for the primal problem ad y is a feasible solutio for the dual problem, the cx yb Strog duality property: If x * is a optimal solutio for the primal problem ad y * is a optimal solutio for the dual problem, the cx * = y * b João Miguel da Costa Sousa / Alexadra Moutiho 86 João Miguel da Costa Sousa / Alexadra Moutiho 87 Primal dual relatioships Primal dual relatioships Complemetary solutios property: At each iteratio, the simplex method idetifies a CPF solutio x for the primal problem ad a complemetary solutio y for the dual problem, where cx = yb If x is ot optimal for the primal problem, the y is ot feasible for the dual problem. Complemetary optimal solutios property: At the fial iteratio, the simplex method idetifies a optimal solutio x* for the primal problem ad a complemetary optimal solutio y* for the dual problem, where cx* = y*b The y i * are the shadow prices for the primal problem. João Miguel da Costa Sousa / Alexadra Moutiho 88 João Miguel da Costa Sousa / Alexadra Moutiho 89 0

11 Primal dual relatioships Symmetry property: For ay primal problem ad its dual problem, all relatioships betwee them must be symmetric because the dual of this dual problem is this primal problem. Duality theorem: The followig are the oly possible relatioships betwee the primal ad dual problems:. If oe problem has feasible solutios ad a bouded obective fuctio, the so does the other problem, so both the weak ad strog duality properties are applicable. Primal dual relatioships. If oe problem has feasible solutios ad a ubouded obective fuctio, the the other problem has o feasible solutio. 3. If oe problem has o feasible solutios, the the other problem has either o feasible solutios lti or a ubouded d obective fuctio. João Miguel da Costa Sousa / Alexadra Moutiho 90 João Miguel da Costa Sousa / Alexadra Moutiho 9 Primal dual relatioships Primal dual relatioships Relatioships betwee complemetary basic solutios Primal basic solutio Complemetary dual basic solutio Both basic solutios Primal feasible? Dual feasible? Suboptimal Superoptimal Yes No Optimal Optimal Yes Yes Superoptimal Suboptimal No Yes Neither feasible or superoptimal Neither feasible or superoptimal No No João Miguel da Costa Sousa / Alexadra Moutiho 9 João Miguel da Costa Sousa / Alexadra Moutiho 93 Applicatios of duality theory If the umber of fuctioal costraits m is bigger tha the umber of variables, applyig the SM directly to the dual problem will achieve a substatial reductio i computatioal effort. Evaluatig a proposed solutio for the primal problem. i) if cx=yb, x ad y must be optimal eve without applyig the SM; ii) if cx<yb, yb provides a upper boud o the optimal value of Z. Ecoomic iterpretatio João Miguel da Costa Sousa / Alexadra Moutiho 94 Duality theory i sesitivity aalysis Sesitivity aalysis ivestigates the effect i the optimal solutio of chagig parameters a i, b i, c. It ca be easier to study these effects i the dual problem: Optimal solutio of the dual problem are the shadow prices of the primal problem. Chages i coefficiets of a obasic variable. Oly chages oe costrait i the dual problem. If this costrait is satisfied the solutio is still optimal. Itroducig a ew variable. Oly itroduces a ew costrait i the dual problem. If the dual problem is still feasible, solutio is still optimal. João Miguel da Costa Sousa / Alexadra Moutiho 95

12 The essece of sesitivity aalysis Simplex tableau with parameter chages Oe assumptio of LP is that all the parameters of the model (a i, b i ad c ) are kow costats. Actually: The parameters values used are ust estimates based o a predictio of future coditios; The data may represet deliberate overestimates or uderestimates to protect the iterests of the estimators. A optimal solutio is optimal oly with respect to the specific model beig used to represet the real problem sesitivity aalysis is crucial! What chages i the simplex tableau if chages are made i the model parameters, amely b b, c c, A A? New iitial tableau Revised fial tableau Coefficiet of: Eq. Z Origial variables Slack variables (0) c 0 0 (,,,m) 0 A I b (0) z* c= y* A c y* (,,,m) 0 A* = S* A S* Z* = y* b b* = S* b João Miguel da Costa Sousa / Alexadra Moutiho 96 João Miguel da Costa Sousa / Alexadra Moutiho 97 Applyig sesitivity aalysis Applyig sesitivity aalysis Chages i b i : Allowable rage: rage of values for which the curret optimal BF solutio remais feasible (fid rage of b i such that b* = S* b 0, assumig this is the oly chage i the model). The shadow price for b i remais valid if b i stays withi this iterval. The 00% rule for simultaeous chages i RHS: the shadow prices remai valid as log as the chages are ot too large. If the sum of the % chages does ot exceed 00%, the shadow prices will defiitely still be valid. Chages i coefficiets of obasic variable: Allowable rage to stay optimal: rage of values over which the curret optimal solutio remais optimal (fid c y* A, assumig this is the oly chage i the model). The 00% % rule l f for simultaeous i chages h i i obective fuctio coefficiets: if the sum of the % chages does ot exceed 00%, the origial optimal solutio will defiitely still be valid. Itroductio of a ew variable: Same as above. João Miguel da Costa Sousa / Alexadra Moutiho 98 João Miguel da Costa Sousa / Alexadra Moutiho 99 Other simplex algorithms Dual simplex method Modificatio useful for sesitivity aalysis, based o the duality theory. Parametric liear programmig Et Extesio for systematic ti sesitivity aalysis. Vary oe or more parameters cotiuously over some iterval(s) to see whe the optimal solutio chages. Upper boud techique A simplex versio for dealig with decisio variables havig upper bouds: x u, where u is the maximum feasible value of x. João Miguel da Costa Sousa / Alexadra Moutiho 00