NON LINEAR PROGRAMMING Prof. Stephen Graves

Transcription

1 NON LINEAR PROGRAMMING Prof. Stephen Graves In a lnear program, the constrants are lnear n the decson varables, and so s the objectve functon. In a non lnear program, the constrants and/or the objectve functon can also be non lnear functon of the decson varables. Example: Gasolne Blendng: The qualtes of a blend are determned by the qualtes of the stocks used n the blend. An optmzaton s to determne the volume of each nput stock n each blend so that the objectve functon s optmzed subject to the output blends satsfyng ther qualty specfcatons, stock avalablty constrants, and blend demand constrants. The decson varables are x j denotng the amount of stock n blend j. For the most part the constrants can be wrtten as lnear functons; but some of the qualty constrants are non lnear: Dstllaton Blendng: D jk = b k + c k * ln [ Σ ( S k * VF j ) ] where D jk s the k th dstllaton pont for blend j. S k s the k th dstllaton pont for stock. VF j s the volume fracton of stock n blend j and s equal to x j / (Σ x j ) and b k and c k are constants. 82

2 Octane Blendng: OCT jk = a k { Σ ( b * b * VF j ) - Σ (c * VF j ) 2 } + d k { Σ ( e * VF j ) - Σ (f * VF j ) 2 } 2 + g k Σ { ( h * VF j ) - ( j * k * VF j * VF j )} where OCT jk are the varous octane ndces for blend j. For both D jk and OCT jk the optmzaton problem would have smple upper bounds and lower bounds for each blend and for each qualty ndex. Thus the constrants for the formulaton would nclude: for each stock : x A where A s the avalablty of stock j for each blend j: x R where R s the requrement for blend j j for each combnaton, j, we defne : V j j j j = x j x plus upper bounds and lower bounds on the dstllaton ponts and octane levels for each blend j 83

3 Example: Ste Locaton; gven customer locatons (x, y ), fnd the locaton (X, Y) that mnmzes the weghted dstances from the customer to the central warehouse (or mnmzes the maxmum dstance to an emergency vehcle locaton). The dstance from customer to the warehouse s d and s typcally a non-lnear functon of the decson varables (x, y ), and (X, Y). To wt, we mght have ( ) ( ) 2 2 = + d x X y Y or d = x X + y Y We then have an objectve: Mn N wd subject to constrants on the decson varables. = 1 Example: Determne the producton quanttes for each famly of car (luxury, ntermedate, md sze, compact, subcompact) that maxmzes net revenue subject to producton capacty constrants, fleet fuel mleage constrants. (Haas, SM thess, 1977) Decson varables are q and p, whch denote the quantty and prce for each car famly. We then need to assume a relatonshp between prce and quantty, e.g., lnear supplydemand functon: q = a bp where a and b are postve constants. The objectve of the model s non-lnear, to maxmze profts: Max q p C q p C ( ) = ( ) where C equals the cost per unt for car from famly. We would have lnear capacty constrants: for each resource type j: Rq K where K s the amount of avalable respource of type j, j j j and R s the per unt consumpton of resource j to produce a unt of car. j We also have a non-lnear fleet fuel mleage constrant; the fleet fuel mleage s computed as the harmonc average, and needs to exceed some target, say 30 mpg: q1+ q qn 30mpg where mpg s the mles per gallon for car q1 q2 q n mpg mpg mpg famly. 1 2 n 84

4 Example: Flow n Ppes - In desgnng a network of ppes, say, for a chemcal processng faclty, you mght be gven the network topology (nodes and edges), the desred flow nputs at supply ponts, desred flow outputs at consumpton ponts, and nlet pressures at supply ponts. The decson varables are the sze of ppes (dameter) needed to connect the nodes of the network. The problem s to determne for each edge of the network, the dameter of the ppe and the flow rate on that edge. We defne the varables: q j s the flow rate on edge j, dp j s the pressure drop across edge j, and d j s the dameter of edge j. There are flow balance constrants at each node of the network (.e., flow nto the node = flow out of the node), and constrants on the external flow nputs and outputs. For each edge, the flow rate s a non-lnear functon of the dameter of the ppe and the drop n pressure across the edge, e.g., : q 2 j = c dp j d 5 j where c s a constant, and the drop n pressure across an edge equals the dfference n the node potentals. The objectve would be to mnmze the cost of the ppes, whch depends on the dameters chosen. 85

5 Example: Robot Moton Plannng (taken from LFM thess, Evaluaton of a New Robotc Assembly Workcell Usng Statstcal Expermental Technques and Schedulng Procedures by Erol Erturk, 1991) The problem s to determne the velocty and acceleraton for a new robot assembly system for a gven dsplacement of length d. The objectve s to mnmze tme, subject to a constrant on placement accuracy. We assume the robot accelerates at a constant rate of acceleraton untl t reaches ts peak velocty, then wll travel at ts peak velocty, untl t must decelerate also at a lnear rate. Then the tme (n seconds) to travel a dstance d s: Travel tme = T = v/a + d/v where a s the acceleraton and deceleraton rate (nch/sec 2 ), and v s the peak velocty (nch/sec). The accuracy (n mls) of the placement depends upon the acceleraton rate and the peak velocty and has been found emprcally to be gven by: A = accuracy = v a v a The optmzaton s then to mnmze T, subject to a constrant on accuracy A, as well as upper bounds on acceleraton (a) and velocty (v). 86

6 Example: Desgn parameters for col sprng (from Rajan Ramaswamy s thess, Computer Tools for Prelmnary Parametrc Desgn, Ph. D., LFM 1993) The col sprng s used to provde a clampng force n an ndexng mechansm. Hence, t must delver a specfed force whle satsfyng constrants on compressed length, geometry and materal. The objectve s to fnd the lghtest feasble desgn,. e., mnmze mass. The followng equatons come from Mark s handbook for mechancal engneers: (1) C = Dsprng/Dwre C s sprng coeffcent; Dsprng s sprng dameter (n); and Dwre s the wre dameter (n). (2) Kw = (4*C -1)/(4*C- 4) /C Kw s the Wahl curvature correcton factor. (3) Ksprng = (Dwre 4 * G)/(8 * Dsprng 3 * Ncols) Ksprng s the sprng stffness (lb/n); G s torsonal modulus (Mps); and Ncols s the number of cols. (4) L_Free = (Ncols * Dwre +Tau_Max * 3.14*Dwre 3 )/(8 * Dsprng * Kw * Ksprng) L_Free s free length (n); Tau_Max s the peak allowable shear stress (Mps). (5) F_Act = Ksprng * (L_Free - L_C) F_Act s the force actng (lb); L_C s the compressed length (n). (6) Tau = (Kw * 8 * F_Act * Dsprng) / (3.14 * Dwre 3 ) Tau s the shear stress (Mps). (7) Mass = Rho * Ncols * 3.14 * 3.14 * Dwre 2 * Dsprng/4 Mass s the mass of the sprng (oz); and Rho s the densty of the sprng materal (lb/n 3 ). For ths problem, L_C, G, Rho, Tau_Max are gven constants. We are gven a lower bound on F_Act, namely the desred force; an upper bound on Tau, namely Tau-Max, a lower bound on L_Free, namely L_C, and upper and lower bounds on C, Dsprng and Dwre. The number of cols (Ncols) has to be at least 3. 87

7 ISSUES WITH NON LINEAR PROGRAMS Optmal soluton may occur at extreme pont; may occur on the boundary of feasble regon; may occur n the nteror of feasble regon. Many problems wll have both a global optmum and several local optma; t s very hard to dstngush between local and global optma. 88

8 APPROACHES TO NON LINEAR PROGRAMS Approxmate non lnear functons wth lnear or pece-wse lnear functons, possbly by usng bnary nteger varables. Ths works well f the non lnear functons separate by decson varable. Use monotonc or algebrac transformaton (e. g., log s) to make non lnear functons lnear. Solve for necessary condtons gven by Lagrangean Functon (f constrants are equaltes) or gven by Kuhn Tucker condtons (f constrants are nequaltes). Ths works well f there are very few constrants or f the objectve s quadratc and all of the constrants are lnear (a quadratc program). Use a search algorthm: Fnd a feasble pont (vector) to start X Fnd a feasble drecton of mprovement D Move to new pont along drecton of mprovement, step sze τ: X : = X + τ D Contnue untl some convergence crteron satsfed. 89