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## Mathematical Program Prototype Example Wyndor Glass (Hillier in addition to Liebermnan)

Hansen, Tyler, High School Sports Reporter has reference to this Academic Journal, PHwiki organized this Journal Explorations in Artificial Intelligence Prof. Carla P. Gomes gomes@cs.cornell.edu Module 6 Intro to Linear Programming Constraint Programming 1- Problem Formulation: A problem is a finite set of constraints involving a finite set of variables. Constraint Satisfaction Problems (CSP) feasibility problem only in addition to — SAT is a particular case of CSP; Constraint Optimization Problems (COP) if in addition the solution is required to maximize an objective function 2- Problem Solution: Domain specific methods General Solution Methods Mathematical Program Optimization problem in which the objective in addition to constraints are given as mathematical functions in addition to functional relationships. Optimize: Z = f(x1, x2, , xn) Subject to: g1(x1, x2, , xn) = , , b1 g2(x1, x2, , xn) = , , b2 gm(x1, x2, , xn) = , , bm Linear Programming problem special type of a mathematical programming problem (all functions are linear)

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Linear Programming (LP) One of the most important scientific advances of the 20th century A variety of applications: Financial planning, Marketing, E-business, Telecommunications, Manufacturing, Transportation Planning, System Design, Health Care Remarkably efficient solution procedures to solve LP models simplex method in addition to interior point methods — Very fast LP solvers (CPLEX from 1981-2001 2,000,000X faster!) Linear Programming Significance. Powerful tool as long as optimal allocation of scarce resources, among a number of competing activities. Powerful model generalizes many classic problems: shortest path, max flow, multicommodity flow, MST, matching, 2-person zero sum games Ranked among most important scientific advances of 20th century. accounts as long as a major proportion of all scientific computation Helps find “good” solutions to NP-hard optimization problems. optimal solutions (branch- in addition to -cut) provably good solutions (r in addition to omized rounding) Linear Programming (LP) Linear all the functions are linear (f in addition to g functions are linear. Ex: f (x1, x2, , xn)= c1x1 + c2 x2 + cn xn Programming does not refer to computer programming but rather planning – planning of activities to obtain an optimal result i.e., it reaches the specified goal best (according to the mathematical model) among all feasible alternatives.

Prototype Example Wyndor Glass (Hillier in addition to Liebermnan) Wyndor Glass Co. Product Mix Problem Wyndor Co. has developed the following new products: An 8-foot glass door with aluminum framing. A 4-foot by 6-foot double-hung, wood-framed window. The company has three plants Plant 1 produces aluminum frames in addition to hardware. Plant 2 produces wood frames. Plant 3 produces glass in addition to assembles the windows in addition to doors.

Wyndor Glass Co. Product Mix Problem Questions: Should they go ahead with launching these two new products If so, what should be the product mix How to as long as mulate this problem as an Linear Programming problem Steps in setting up a LP Determine in addition to label the decision variables. Determine the objective in addition to use the decision variables to write an expression as long as the objective function. Determine the constraints – feasible region. Determine the explicit constraints in addition to write a functional expression as long as each of them. Determine the implicit constraints (e.g., nonnegativity constraints). Algebraic Model as long as Wyndor Glass Co. Let D = the number of doors to produce W = the number of windows to produce Maximize P = 3 D + 5 W subject to D 4 2W 12 3D + 2W 18 in addition to D 0, W 0.

Finding the Optimal Solution Our objective function is: maximize 3D+5W The vector representing the gradient of the objective function is: The line through the origin that contains this vector is: isoprofit line Maximize P = 3 D + 5 W subject to D 4 2W 12 3D + 2W 18 in addition to D 0, W 0. LP: Geometry Geometry. Forms an n-dimensional polyhedron. Convex: if y in addition to z are feasible solutions, then so is ½y + ½z. Extreme point: feasible solution x that can’t be written as ½y + ½z as long as any two distinct feasible solutions y in addition to z. LP: Geometry Extreme Point Theorem. If there exists an optimal solution to st in addition to ard as long as m LP (P), then there exists one that is an extreme point. Only need to consider finitely many possible solutions. Greed. Local optima are global optima.

Graphical Method Draw the constraint boundary line as long as each constraint. Use the origin (or any point not on the line) to determine which side of the line is permitted by the constraint. Find the feasible region by determining where all constraints are satisfied simultaneously. Determine the slope of one objective function line (perpendicular to its gradient vector). All other objective function lines will have the same slope. Move a straight edge with this slope through the feasible region in the direction of improving values of the objective function (direction of the gradient). Stop at the last instant that the straight edge still passes through a point in the feasible region. This line given by the straight edge is the optimal objective function line. A feasible point on the optimal objective function line is an optimal solution. Terminology in addition to Notation Resources m (plants) Activities n (2 products) Wyndor Glass problem optimal product mix — allocation of resources to activities i.e., choose the levels of the activities that achieve best overall measure of per as long as mance Terminology in addition to notation (cont.) Z value of the overall measure of per as long as mance; value of the objective function, xj level of activity j ( as long as j = 1, 2, , n) decision variables cj increase in Z as long as each unit increase in the level of activity j; coefficient of objective function associated with activity j bi amount of resource i that is available ( as long as i=1,2, , m). Right-h in addition to -side of constraint associated with resource i. aij amount of resource i consumed by each unit of activity j. Technological coefficient. (The values of cj, bi, in addition to aij are the input constants as long as the model the parameters of the model. )

xi >= 0 , (i =1,2, ,n) Other as long as ms: Minimize Z (instead of maximizing Z) Some functional constraints have signs >= (rather than <=) Some functional constraints are equalities Some variables have unrestricted sign, i.e., they are not subject to the non-negativity constraints St in addition to ard as long as m of the LP model Terminology of solutions in LP model Solution not necessarily the final answer to the problem!!! Feasible solution solution that satisfies all the constraints Infeasible solution solution as long as which at least one of the constraints is violated Feasible region set of all points that satisfies all the constraints (possible to have a problem without any feasible solutions) Binding constraint the left-h in addition to side in addition to the right-h in addition to side of the constraint are equal, I.e., constraint is satisfied in equality. Otherwise the constraint is nonbinding. Optimal solution feasible solution that has the best value of the objective function. Largest value maximization problems Smallest value minimization problems Multiple optimal solutions No optimal solutions Unbounded Z Corner Point Solutions Corner-point feasible solution special solution that plays a key role when the simplex method searches as long as an optimal solution. Relationship between optimal solutions in addition to CPF solutions: Any LP with feasible solutions in addition to bounded feasible region (1) the problem must possess CPF solutions in addition to at least one optimal solution (2) the best CPF solution must be an optimal solution If the problem has exactly one optimal solution it must be a CFP solution If the problem has multiple optimal solutions, at least two must be CPF solutions Wyndor Glass CPF Edge of Feasible region 0 1 Z=0 Z=30 Let D = the number of doors to produce W = the number of windows to produce Maximize P = 3 D + 5 W 2 Z=36 Z=27 3 No Feasible Solutions Why Maximize P = 3 D + 5 W subject to D 4 2W 12 3D + 2W 18 3 D + 5 W 50 in addition to D 0, W 0. Previous Feasible Region Multiple Optimal Solutions. Why Maximize P = 3 D + 2 W subject to D 4 2W 12 3D + 2W 18 in addition to D 0, W 0. Every point on this line is An optimal solution with P=18

Unbounded Objective Function. Why Maximize P = 3 D + 2 W subject to D 4 in addition to D 0, W 0. (4,2) P=16 (4,4) P=20 (4,8) P=28 (4, ) P= Sensitivity Analysis 0 2 4 6 8 8 6 4 2 Production rate as long as windows Production rate as long as doors Feasible region x=(2, 6) Optimal solution c2>2 10 W D P = 3600 = 300D + 500W P = 3000 = 300D + 500W P = 1500 = 300D + 500W Our objective function is: maximize 3×1+c2x2 How does the optimal solution change as c2 changes Multiple Optimal solution c2=2 P=18 x = (2,6) ; x=(4,3) And any convex combination Optimal solution 0

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