Outline. Linear Programming (LP): Principles and Concepts. Need for Optimization to Get the Best Solution. Linear Programming

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1 Outline Linear Programming (LP): Principles and oncepts Motivation enoît hachuat Key Geometric Interpretation McMaster University Department of hemical Engineering 3 LP Standard Form he 4G03: Optimization in hemical Engineering For additional details, see Rardin (998), hapter enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 / 9 enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 / 9 Linear Programming Need for Optimization to Get the est Solution plant warehouse customer demand Multiple hoice Question: Why Study Linear Programming? The prof cannot formulate nonlinear models Linear programming is the most frequently used optimization method 3 Linear programming was developed in the 940 s 4 Formulation and solution knowledge will also help when addressing nonlinear models 5 Excellent software is available P P max. 60, W W ,000 units 00,000 units 50,000 units The cost of manufacturing is the same The objective is thus to minimize the tranportation cost (using values in the figure) What is the est Plant? enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 3 / 9 enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 4 / 9

2 Learning Goals Attitudes An optimal solution is much better than an answer Numbers without understanding are useless Learning Goals hallenge Understand both the geometric interpretation (excellent understanding) and the matrix calculations (for the quantitative results) At the same time! Skills Translate a complex problem into a mathematical formulation Solve an optimization problem! ommunicate optimization results in engineering terms Knowledge When/how formulate LP models Analyze the LP solution sensitivity and diagnose weird events a, a,m a,m+ a,n a m, a m,m a m,m+ a m,n It Is Worth the Effort! 0 x n A = b. b m A enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 5 / 9 enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 6 / 9 Linear Programs Definition An optimization model is a linear program (or LP) if it has: continuous variables; a single linear objective function; and 3 ony linear equality or inequality constraints min z = c T x x s.t. A h x = b h A g x b g x min x x max Standard notation for LP: x j = jth decision variable c j = objective function coefficient of x j a i,j = constraint coefficient of x j in the ith main constraint b i = right-hand side (RHS) constant term of main constraint i m = number of main constraints n = number of decision variables Interior, oundary and Extreme Points lassification of the Points in an LP Feasible Region A feasible solution to a linear program is said to be: a boundary point if at least one inequality constraint (that can be strict for some other feasible solutions) is satisfied as equality at the given point an interior point if no such inequalities are active an extreme point (or a corner point or a vertex) if every line segment in the feasible region containing it also has that point as an endpoint Important Remarks About Extreme Points: Extreme point are special boundary points so-named because they stick out (due to convexity of the feasible region of an LP) Every extreme point is determined by a set of constraints that are active only at that solution enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 7 / 9 enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 8 / 9

3 lassifying Feasible Points lass Exercise: lassify the labeled solutions as interior, boundary, and/or extreme points of the following LP feasible region: Extreme Points lass Exercise: List all sets of 3 constraints determining points x (0), x (), x () and x (3) x 3 x (3) A x () x () x (0) H G I J D K E L F x (0) : x () : x () : x (3) : enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 9 / 9 enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 0 / 9 Adjacent Extreme Points and Edges Adjacent Extreme Points Two extreme points of an LP feasible region are adjacent if they are determined by active constraint sets differing in only one element. Edge An edge of an LP feasible region is a -dimensional set of feasible points along a line determined by a collection of active constraints. Adjacent Extreme Points lass Exercise: Which points among x (0), x (), x () and x (3) are adjacent extreme points? Which active constraints do these points have in common? A x 3 x () x (3) Adjacent Extreme Points: Remark: Adjacent extreme points are joined by an edge determined by the active constraints the extreme points have in common H x () x (0) G D E F ommon Active onstraints: I J K L enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 / 9 enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 / 9

4 Optimal Points in LP Fundamental Result Every optimal solution to an LP (that has a non-constant objective function) will be a boundary point of its feasible region Optimal Points in LP Fundamental Result Every optimal solution to an LP (that has a non-constant objective function) will be a boundary point of its feasible region Other Fundamental Properties: lass Exercise: Explain why point cannot be optimal? How about point? If an LP has any extreme point, then one such point must be an optimal solution Why? If an LP has a unique optimal solution, that optimum must occur at an extreme point of its feasible region Why? 3 Every local optimum for an LP is a global optimum Why? enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 3 / 9 enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 3 / 9 Identifying Optimal Points lass Exercise: Indicate which of the labeled points can be optimal or uniquely optimal for some objective function: Naive Algorithm Exploiting the extreme point concept as the foundation for a solution method: Algorithm: alculate the objective function value at all extreme points, and keep whichever is best! Application: onsider an LP comprising 0 variables and 0 inequality constraints. Up to how many extreme points exist for this (rather small) LP problem? enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 4 / 9 enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 5 / 9

5 ack to the Drawing oard! lass Exercise: Develop an algorithm that combines the basic numerical approach with the key geometric insight ack to the Drawing oard! lass Exercise: Develop an algorithm that combines the basic numerical approach with the key geometric insight Starting point asis of the Simplex Algorithm! asic Numerical Optimization: Use local information only, in an iterative scheme Determine a feasible next point that improves the objective value check if further improvement is possible: if yes, continue; else, stop Starting point No further improvement One possible approach: asis of the Simplex Algorithm! onsider only adjacent extreme points for improvement direction Move along the edge that yields the greatest rate of improvement Move until another extreme point has been reached heck if further improvement is possible: if yes, continue; else, stop enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 6 / 9 enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 6 / 9 LP Standard Form Definition: LP in standard form have: only equality main constraints only nonnegative variables objective function and main constraints simplified so that each variable appears at most once (on the left-hand side), and any constant term (possibly zero) appears on the right-hand side min z = c T x x s.t. Ax = b x 0 x j = jth decision variable c j = objective function coefficient of x j a i,j = constraint coefficient of x j in the ith main constraint b i = right-hand side (RHS) constant term of main constraint i m = number of main constraints LP Standard From: Why and How? Why LP in Standard Form? How to Yield LP Standard Form? onvert and inequalities to equalities onvert nonpositive variables to nonnegative 3 onvert unrestricted (URS) variables to nonnegative n = number of decision variables enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 7 / 9 enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 8 / 9

6 LP Standard From: Some Practice! lass Exercise: Place each of the following LP in standard form Identify the m, n, A, b and c of standard matrix representation min 9w + 6w s.t. w + w 0 w 50 w + w = w + w 5 w 0,w 0 ma5( + 8 ) 4x 3 s.t. (0 ) + + 5(9 x 3 ) 0 + x 3, 0,x 3 URS enoît hachuat (McMaster University) LP: Principles and oncepts 4G03 9 / 9