Lecture 1: Linear Programming Models. Readings: Chapter 1; Chapter 2, Sections 1&2


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1 Lecture 1: Linear Programming Models Readings: Chapter 1; Chapter 2, Sections 1&2 1
2 Optimization Problems Managers, planners, scientists, etc., are repeatedly faced with complex and dynamic systems which they have to manage or control in order to realize certain goals. These systems have the following properties in common: decision variables representing the options and operating levels the decisionmaker can control in order to drive the behavior of the system; constraints limiting the range of control the decisionmaker has on the decision variables; an objective that measures how well the system is operating with regard to the goals of the decision maker. 2
3 Mathematical Programs Mathematical programs are simply the presentation of an optimization problem in a mathematically precise form. The general description of a mathematical program is of the form where (MP ) max min x X f(x) x is the set of decision variables, which can be represented by numerical values, sets, functions, etc. X represents the feasible region describing the constraints on the decision variables. Feasible regions are usually described by giving equalities and inequalities involving functions of the decision variables. Any point x X is called a feasible solution to the problem. f represents the objective function, a function of the variable values that one wishes to maximize or minimize. 3
4 Solving Mathematical Programs The goal of solving a mathematical program (MP ) is to find an optimal solution to the problem, that is, an assignment x of values to the decision variables x in such a way that (i) x is feasible to (MP ); (ii) x has the best objective function value for (MP ) in the sense that any other feasible solution x to (MP ) has f(x) f(x ) (for a max problem) f(x) f(x ) (for a min problem) Questions Associated with Solving Mathematical Programs: What are properties of the feasible region X, and its description, that help us in finding feasible and optimal solutions to (MP )? What conditions can we give to show that a given solution x is optimal to (MP )? What computational methods are available to find feasible and optimal solutions to (MP )? What is the relationship between the parameters describing (M P ) and the associated optimal solutions? How sensitive are the solutions to the accuracy of the description of (MP )? 4
5 Woody s Problem: A Resource Allocation Model Woody s Furniture Company makes chairs and tables. Chairs are made entirely out of pine, and require 8 linear feet of pine per chair. Tables are made of pine and mahogany, with tables requiring 12 linear feet of pine and 15 linear feet of mahogany. Chairs require 3 hours of labor to produce and tables require 6 hours of labor. Chairs provide $35 profit and tables provide $60 profit. Woody has 120 linear feet of pine and 60 linear feet of mahogany delivered each day, and has a work force of 6 carpenters each of whom put in an 8hour day. How should Woody use his resources to provide the largest daily profits? The mathematical program: x 1 = chairs per day, x 2 = tables per day, max z = 35x x 2 profits 8x x pine 15x 2 60 mahogany 3x 1 + 6x 2 48 labor x 1 0 x 2 0 nonnegativity constraints 5
6 Linear Programming Models Each of the models above are examples of linear programs. Linear programs are characterized by the following properties: optimization problem/mathematical program: The problem involves finding the best value for a given objective, subject to a given set of constraints. deterministic: The behavior of the model is completely determined, that is, there is no probability contributing to the objective or constraints. proportionality: Each variable contributes in direct proportion to its value. additivity: The variables in the objective and each constraint contribute the sum of the contributions of each variable. divisibility: The variables can take on continuous values subject to the constraints. Any function satisfying the proportionality and additivity properties is called a linear function, and will always have the form f(x 1,..., x n ) = a 1 x a n x n ( + d). 6
7 Examples of Situations violating the LP Assumptions nonproportionality: fixed charges or volume discounting ( $2000 to set the printer up, and $.15 per each copy made, $100 for the first 50 copies, $1 for each copy thereafter ) nonadditivity: distances from a point (x, y) in the plane to given point (a, b): (x a)2 + (y b) 2 (this is nonproportional as well). Volumes and square footage (in terms of dimension) are also nonadditive (but are proportional). nondivisibility: integer/binary programs ( How many aircraft to be made?, Should a new plant be built or not?, must be made in lots of 5 ) 7
8 max min The Mathematical Description z = c 1x 1 + c 2 x c n x n a i1 x 1 + a i2 x a in x n = b i i = 1,..., m x j 0 unrestricted, ( 0) j = 1,..., n x j are the variables of the problem, and are allowed to take on any set of real values that satisfy the constraints. c j, b i, and a ij are parameters of the problem, and (along with dimensions n, m, and objective/row/variable types) provide the precise description of a particular instance of the LP model that you wish to solve. c j are the objective/cost/profit coefficients. b i are the righthandside/resource/demand coefficients. a ij are the proportionality/production/activity coefficients or coefficients of variation. 8
9 LP forms There are several specific LP forms that will useful for various purposes. The first is the standard (equality form) max LP: max z = c 1 x 1 + c 2 x c n x n a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1. + a 22 x a 2n x n =. b 2 a m1 x 1 + a m2 x a mn x n = b m x 1 0 x x n 0. Also useful is the inequality max LP: max z = c 1 x 1 + c 2 x c n x n a 11 x 1 + a 12 x a 1n x n b 1 a 21 x 1. + a 22 x a 2n x n. b 2 a m1 x 1 + a m2 x a mn x n b m 9
10 The Canonical Max and Min Problems There are also two symmetric related LPs, the canonical max and min LPs, that play an important role in duality theory: Canonical Max LP: max z = c 1 x 1 + c 2 x c n x n a 11 x 1 + a 12 x a 1n x n b 1 a 21 x 1. + a 22 x a 2n x n. b 2 a m1 x 1 + a m2 x a mn x n b m x 1 0 x x n 0. Canonical Min LP: min z = c 1 x 1 + c 2 x c n x n a 11 x 1 + a 12 x a 1n x n b 1 a 21 x 1. + a 22 x a 2n x n. b 2 a m1 x 1 + a m2 x a mn x n b m x 1 0 x x n 0. 10
11 Definitions and Issues feasible solution: A specific set of values of the variables that satisfy the constraints of that instance. feasible region: The set of all feasible solutions for that instance. optimal solution: Any feasible solution whose objective function value is the greatest (for a max problem) or smallest (for a min problem) of all points in the feasible region. Some issues: Does a particular instance of an LP have any feasible point? Does the instance have an optimal solution? How can we describe the feasible region? What properties does it have? How will we know that a solution to a particular instance is in fact an optimal solution for that instance? Can we give an algorithm to find feasible/optimal solutions for any instance of an LP, or to determine that none exist for that instance? How efficient is that algorithm, particularly for solving large LP instances? 11
12 A Graphical View of LPs We can gain insight into the dynamics of LPs by looking at 2variable problems. Consider Woody s problem written in inequality max form: max z = 35x x 2 8x x x x 1 + 6x 2 48 x 1 0 x 2 0 Draw a picture of this LP, by performing the following steps: 1. Take each of the constraints (including the nonnegativity constraints), and draw the line describing the associated equality. 2. Determine which side of each equality line satisfies the inequality constraint. The feasible region is set of points which lie on the correct side of all of the equality lines. 3. For particular objective function value z 0, the isocost line (or level set) at value z 0 is described by the equality line c 1 x 1 + c 2 x 2 = z Slide this isocost line parallel and in the direction of improving objective value until the last point at which it still intersects the feasible region. 5. Any feasible point in this final intersection is an optimal solution. 12
13 Example max z = 35x x 2 (C1) 8x x (C2) 15x 2 60 (C3) 3x 1 + 6x 2 48 x 1 0 x x C2 C3 0 1 C1 x
14 Special subsets of the feasible region interior: set of points having neighborhoods that lie entirely in the feasible region. Equivalently, an interior point is one that satisfies all inequalities strictly. boundary: set of feasible points not in the interior, that is, which satisfy one or more inequalities at equality extreme point/corner point/vertex: boundary point lying on n equalities, where n = the dimension of the problem. LP types optimal solution: one can find a feasible solution for which no other solution has a better objective function value. infeasible LP: the constraints of the problem are such that the feasible region is empty, in other words, there is no feasible solution. unbounded LP: (maximization problem) the objective and constraints of the problem are such that there is no upper bound on the objective function values of the feasible points. That is, for every feasible solution there exists another feasible solution with larger objective function value. (A symmetric definition applies for minimization problems.) 14
15 Three Types of Linear Programs Optimal Solution max z = 35x x 2 (C1) 8x x (C2) 15x 2 60 (C3) 3x 1 + 6x 2 48 x 1 0 x x C2 C3 0 1 C1 x
16 Infeasible LP max z = 35x x 2 (C1) 8x x (C2) 15x 2 60 (C3) 3x 1 + 6x 2 48 x 1 0 x x C C3 C1 x
17 Unbounded LP max z = 35x x 2 (C1) 8x x (C2) 20x x 2 60 (C3) 3x 1 6x 2 48 x 1 0 x
18 The Graphical Method in 3Dimensions Consider the Woody s Problem with a third item, say desks, represented by variable x 3, requiring 16 l.f. of pine, 20 l.f. of mahogany, 9 hrs. of labor, and giving a profit of $75: max z = 35x x x 3 8x x x x x x 1 + 6x 2 + 9x 3 48 x 1 0 x 2 0 x 3 0 This also has a graphical representation, now in 3dimensional Euclidean space: 4 x x x
19 The level set is now a flat sheet perpendicular to the objective function vector (35,60,75). Adding this to the picture, we get 10 5 desks 0 tables chairs We can slide this level set to the optimal solution (12,2,0), although this is difficult to conceptualize even in 3 dimensions, and becomes completely unwieldy in higher dimensions. 19
20 Transforming a General LP into Equality Form minimization problems: Replace with min z = c 1 x 1 + c 2 x c n x n max z = z = c 1 x 1 c 2 x 2... c n x n constraints: Add a nonnegative slack variable indicating the difference between the LHS value and the b i value. Specifically, replace with a i1 x 1 + a i2 x a in x n b i a i1 x 1 + a i2 x a in x n + x n+i = b i, x n+i 0 constraints: Subtract the slack variable in that row. Specifically, replace with a i1 x 1 + a i2 x a in x n b i a i1 x 1 + a i2 x a in x n x n+i = b i, x n+i 0 unrestricted variables: Replace unrestricted x i by x i = x + i x i, x + i 0, x i 0 negative variables: Replace x i 0 by x i = x i, x i 0. 20
21 Examples Woody s problem can be put into standard max form by simply adding slack variables to each of the inequalities max z = 35x x 2 8x x 2 + x 3 = x 2 + x 4 = 60 3x 1 + 6x 2 + x 5 = 48 x 1 0 x 2 0 x 3 0 x 4 0 x 5 0 Now consider the following LP: min z = 35x x 2 8x x 2 = x x x 1 6x 2 48 x 1 unrest. x 2 0 Here we subtract a slack in Row 2, add a slack in Row 3, negate the objective function, replace x 1 with x 1 = x + 1 x 1 and replace x 2 by x 2 to get standard max LP max z = 35x x x 2 8x x 1 12x 2 = x x 1 15x 2 x 4 = 60 2x + 1 2x 1 + 6x 2 + x 5 = 48 x x 1 0 x 2 0 x 4 0 x
22 A Statistics Problem You are given a set of sample points ( x 1, ỹ 1 ), ( x 2, ỹ 2 ),..., ( x n, ỹ n ), which are suppose to satisfy a linear equation of the form Y = ax + b. Due to experimental error, they do not simultaneously satisfy the same equation. You want to find the best equation to fit to these sample points. Y (3,10) n=9 (9,7) (3,7) (9,6) (14,3) (6,2) (18,1) X (12,1) (16,2) Y=aX+b 22
23 Criteria for good fit : Each point should be as close to the line Y = ax + b as possible. We will measure this by the vertical distance from ( x i, ỹ i ) to the line Y = ax + b, that is, d i = ỹ i (a x i + b) (Note that this could be positive or negative.) We can measure goodness of fit using the d i in several ways L 1 measure: z = sum of the absolute values of the d i s Chebyshev measure: z = max of the absolute values of the d i s leastsquares measure: z = sum of the squares of the d i s Goal: To minimize the appropriate measure over all possible choices of a and b. 23
24 Formulating the LP for the L 1 Measure The optimization problem has as variables the two numbers a and b describing the linear equation. Once we have these numbers we can compute the d i s by adding the n equations d i = ỹ i (a x i + b), i = 1,..., n. Note that while the x s and ỹ s are parameters, the d i s are now auxiliary variables. The optimization problem can be stated mathematically as: min z = d i + d d n d i = ỹ i (a x i + b), i = 1,..., n a, b, d i unrestricted Although the constraints are linear, the objective function is not. 24
25 Dealing with Unrestricted Variables Each unrestricted variable is now separated into its positive and negative parts: a = a + a b = b + b, and d i = d + i d i i = 1,..., n Added bonus: The absolute value of d i can be written d i = d + i + d i. Thus the objective function can be rewritten min z = d d d + n + d n 25
26 Formulating the LP for the Chebyshev Measure The objective here is min z = max{ d i, d 2,..., d n } = max{d d 1,..., d + n + d n } Again, this is not linear. To formulate it as a linear program, we introduce still another auxiliary variable M, restricting it by the inequalities M d + i + d i, i = 1,..., n or, in equality form, d + i + d i + s i = M, i = 1,..., n where the s i s are the slack variables. Minimizing z = M subject to the above constraints will insure that M will in fact take the value of the maximum of the d i s. 26
27 Formulating the LP for the Leastsquares Measure The objective here can be written min z = d d d 2 n = [ỹ 1 (a x 1 + b)] 2 + [ỹ 2 (a x 2 + b)] [ỹ n (a x n + b)] 2 with no restrictions on the variables a and b. How can we solve this problem? 27
28 Transforming a General LP into Canonical Form We can also transform LPs into canonical max form. The transformations for variable types are the same, and the transformations for constraints are as follows: constraints: Simply negate the constraint. Equality constraints: Replace the constraint a i1 x 1 + a i2 x a in x n = b i with the two constraints and a i1 x 1 + a i2 x a in x n b i a i1 x 1 a i2 x 2... a in x n b i Note that no slack variables are necessary for this transformation. Example: The general form LP given above has equivalent canonical max form max z = 35x x x 2 8x x 1 12x x + 1 8x x x x x x + 1 2x 1 + 6x 2 48 x x 1 0 x
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