MIT 15 066J - NON LINEAR PROGRAMMING (8 pages)

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NON LINEAR PROGRAMMING



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NON LINEAR PROGRAMMING

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Pages:
8
School:
Massachusetts Institute of Technology
Course:
15 066j - System Optimization and Analysis for Manufacturing
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NON LINEAR PROGRAMMING Prof Stephen Graves In a linear program the constraints are linear in the decision variables and so is the objective function In a non linear program the constraints and or the objective function can also be non linear function of the decision variables Example Gasoline Blending The qualities of a blend are determined by the qualities of the stocks used in the blend An optimization is to determine the volume of each input stock in each blend so that the objective function is optimized subject to the output blends satisfying their quality specifications stock availability constraints and blend demand constraints The decision variables are xij denoting the amount of stock i in blend j For the most part the constraints can be written as linear functions but some of the quality constraints are non linear Distillation Blending Djk bk ck ln i Sik VFij where Djk is the k th distillation point for blend j Sik is the k th distillation point for stock i VFij is the volume fraction of stock i in blend j and is equal to xij i xij and bk and ck are constants 15 066J 82 Summer 2003 Octane Blending OCTjk ak i bi bi VFij i dk gk i i ei VFij hi VFij c i VFij 2 i f i VFij 2 2 ji ki VFij VFij where OCTjk are the various octane indices for blend j For both Djk and OCTjk the optimization problem would have simple upper bounds and lower bounds for each blend and for each quality index Thus the constraints for the formulation would include for each stock i x ij Ai where Ai is the availability of stock i j for each blend j x ij R j where R j is the requirement for blend j i for each combination i j we define Vij xij x ij i plus upper bounds and lower bounds on the distillation points and octane levels for each blend 15 066J 83 Summer 2003 Example Site Location given customer locations xi yi find the location X Y that minimizes the weighted distances from the customer to the central warehouse or minimizes the maximum distance to an emergency vehicle location The distance from customer i to the warehouse is di and is typically a non linear function of the decision variables xi yi and X Y To wit we might have di xi X yi Y 2 2 or di xi X yi Y N We then have an objective Min wd i 1 i i subject to constraints on the decision variables Example Determine the production quantities for each family of car luxury intermediate mid size compact subcompact that maximizes net revenue subject to production capacity constraints fleet fuel mileage constraints Haas SM thesis 1977 Decision variables are qi and pi which denote the quantity and price for each car family We then need to assume a relationship between price and quantity e g linear supplydemand function qi ai bi pi where ai and bi are positive constants The objective of the model is non linear to maximize profits Max qi pi Ci qi pi Ci where Ci equals the cost per unit for car from i i family i We would have linear capacity constraints for each resource type j R q ij i K j where K j is the amount of available respource of type j i and Rij is the per unit consumption of resource j to produce a unit of car i We also have a non linear fleet fuel mileage constraint the fleet fuel mileage is computed as the harmonic average and needs to exceed some target say 30 mpg q1 q2 qn 30mpg where mpgi is the miles per gallon for car qn q1 q 2 mpg1 mpg 2 mpg n family i 15 066J 84 Summer 2003 Example Flow in Pipes In designing a network of pipes say for a chemical processing facility you might be given the network topology nodes and edges the desired flow inputs at supply points desired flow outputs at consumption points and inlet pressures at supply points The decision variables are the size of pipes diameter needed to connect the nodes of the network The problem is to determine for each edge of the network the diameter of the pipe and the flow rate on that edge We define the variables qj is the flow rate on edge j dpj is the pressure drop across edge j and dj is the diameter of edge j There are flow balance constraints at each node of the network i e flow into the node flow out of the node and constraints on the external flow inputs and outputs For each edge the flow rate is a non linear function of the diameter of the pipe and the drop in pressure across the edge e g qj2 c dpj dj5 where c is a constant and the drop in pressure across an edge equals the difference in the node potentials The objective would be to minimize the cost of the pipes which depends on the diameters chosen 15 066J 85 Summer 2003 Example Robot Motion Planning taken from LFM thesis Evaluation of a New Robotic Assembly Workcell Using Statistical Experimental Techniques and Scheduling Procedures by Erol Erturk 1991 The problem is to determine the velocity and acceleration for a new robot assembly system for a given displacement of length d The objective is to minimize time subject to a constraint on placement accuracy We assume the robot accelerates at a constant rate of acceleration until it reaches its peak velocity then will travel at its peak velocity until it must decelerate also at a linear rate Then the time in seconds to travel a distance d is Travel time T v a d v where a is the acceleration and deceleration rate inch sec2 and v is the peak velocity inch sec The accuracy in mils of the placement depends upon the acceleration rate and the peak velocity and has been found empirically to be given by A accuracy 0 022v 0 0079a 0 0002v a The optimization is then to minimize T subject to a constraint on accuracy A as well as upper bounds on acceleration a and velocity v 15 066J 86 Summer 2003 Example Design parameters for coil spring from Rajan Ramaswamy s thesis Computer Tools for Preliminary Parametric Design Ph D LFM 1993 The coil spring is used to provide a clamping force in an indexing mechanism Hence it must deliver a specified force while satisfying constraints on compressed length geometry and material The objective is to find the lightest feasible design i e minimize mass The following equations come from Mark s handbook for mechanical engineers 1 C Dspring Dwire C is spring coefficient Dspring is spring diameter in and Dwire is the wire diameter in 2 Kw 4 C 1 4 C 4 0 615 C Kw is the Wahl curvature correction factor 3 Kspring Dwire4 G 8 Dspring3 Ncoils Kspring is the spring stiffness lb in G is torsional modulus Mpsi and Ncoils is the number of coils 4 L …


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