4 LINEAR PROGRAMMING (LP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
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1 4 LINEAR PROGRAMMING (LP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
2 Mathematical programming (optimization) problem: min f (x) s.t. x X R n set of feasible solutions with linear objective function f and X={ x R n : g i (x) 0, i {1,, m} } with linear g i for i = 1,, m E. Amaldi Fondamenti di R.O. Politecnico di Milano 2
3 General form: min z = c 1 x c n x n Matrix notation: min z = c T x Ax b x 0 a 11 x a 1n x n b 1 (=) ( ) a m1 x a mn x n b m (=) ( ) x 1,, x n 0 min x 1 c 1 c n x n a 11 a 1n x 1 b 1 a m1 a mn x n b m E. Amaldi Fondamenti di R.O. Politecnico di Milano0 3 xn x 1
4 X = { x R n : A x b, x 0 } region of the feasible solutions x* X is an optimal solution if c T x* c T x x X A variety of decision making problems can be formulated as Linear Programs (LPs) They often involve the optimal allocation of a given set of scarce resources Origins: LP models (L. Kantorovich 1939) simplex algorithm (G. Dantzig 1947) E. Amaldi Fondamenti di R.O. Politecnico di Milano 4
5 Given n Diet problem aliments j = 1,, n m basic substances i = 1,, m a ij b i c j amount of i th substance contained in a unit of j th aliment daily requirement of substance i cost of a unit of aliment j Determine a diet of minimal total cost that meets the daily requirements. E. Amaldi Fondamenti di R.O. Politecnico di Milano 5
6 Decision variables: x j = amount of j th aliment in the diet, with j = 1,..., n n m in j=1 n j=1 c j x j a ij x j b i i=1,...,m x j 0 j=1,...,n E. Amaldi Fondamenti di R.O. Politecnico di Milano 6
7 Transportation problem (single product) m production plants i = 1,, m n clients j = 1,, n c ij p i transportation cost for one unit of product from plant i to client j maximum supply at plant i d j q ij demand of client j maximum amount transportable from i to j Determine a transportation plan that minimizes costs while respecting plant capacities and client demands. E. Amaldi Fondamenti di R.O. Politecnico di Milano 7
8 m n m in i=1 j=1 n j=1 m i=1 Assumption: c ij x ij x ij p i x ij d j m i=1 n p i j=1 Decision variables: x ij = amount transported from i to j i = 1,, m j = 1,, n d j plant capacity client demand 0 x ij q ij i, j transportation capacity E. Amaldi Fondamenti di R.O. Politecnico di Milano 8
9 Production planning problem n products (j = 1,, n) that compete for m scarce resources (i = 1,, m) c j profit (price cost) per unit of j th product a ij amount of i the resource needed per unit of j th product b i maximum available amount of i the resource Determine a production plan that maximizes the total profit given the available amount of resources. E. Amaldi Fondamenti di R.O. Politecnico di Milano 9
10 Decision variables: x j = amount of j th product, with j = 1,..., n n m ax j=1 n j=1 c j x j a ij x j b i i=1,...,m x j 0 j=1,...,n E. Amaldi Fondamenti di R.O. Politecnico di Milano 10
11 Assumptions of LP models 1) Linearity (proportionality and additivity) of the objective function and constraints Proportionality: contribution of each variable = constant variable Do not consider large scale savings! Additivity: contribution of all variables = sum of single contributions Profits of competing products are not independent! E. Amaldi Fondamenti di R.O. Politecnico di Milano 11
12 2) Divisibility The variables can take fractional (real) values. Parameters: We assume that the model parameters are constant and can be estimated with a sufficient degree of accuracy. Several scenarios sensitivity analysis E. Amaldi Fondamenti di R.O. Politecnico di Milano 12
13 4.1 Equivalent forms General form : min z = c T x (max) A 1 x b 1 A 2 x b 2 A 3 x = b 3 x j 0 for j J {1,, n} x j unrestricted for j {1,, n} \ J Standard form: min z = c T x Ax = b x 0 inequality constraints equality constraints only equality constraints all variables are non negative E. Amaldi Fondamenti di R.O. Politecnico di Milano 13
14 Both forms are equivalent Simple transformation rules allow to pass from one form to the other form. NB: the transformation may involve changes in the (number of) variables and constraints E. Amaldi Fondamenti di R.O. Politecnico di Milano 14
15 max c T x = min c T x Transformation rules a T x b a T x + s = b s 0 slack variable a T x b a T x s = b s 0 surplus variable E. Amaldi Fondamenti di R.O. Politecnico di Milano 15
16 x j unrestricted in sign {x j =x j x j } x j 0 x j 0 By substituting x j with x j+ x j, we delete x j from the problem. Alternatively the expression of x j can be derived from an equation and substituted everywhere (deleting the equation and the variable). E. Amaldi Fondamenti di R.O. Politecnico di Milano 16
17 Example max 2x 1 3x 2 s.t. 4x 1 7x 2 5 6x 1 2x 2 4 x 1 0, x 2 unrestricted x 2 = x 3 x 4 x 3, x 4 0 max 2x 1 3x 3 + 3x 4 s.t. 4x 1 7x 3 + 7x 4 5 6x 1 2x 3 + 2x 4 4 x 1, x 3, x 4 0 E. Amaldi Fondamenti di R.O. Politecnico di Milano 17
18 introducing slack and surplus variables x 5 and x 6 + change the objective function sign min 2x 1 + 3x 3 3x 4 s.t 4x 1 7x 3 + 7x 4 + x 5 x 6 = 5 6x 1 2x 3 + 2x 4 + x 5 x 6 = 4 x 1, x 3, x 4, x 5, x 6 0 E. Amaldi Fondamenti di R.O. Politecnico di Milano 18
19 4.2 Geometry of LP Example Capital budgeting Capital and two possible investments A and B with, respectively, 4% and 6% expected return. Determine a portfolio that maximizes the total expected return while respecting the diversification constraints: at most 75% of the capital is invested in A at most 50% of the capital is invested in B E. Amaldi Fondamenti di R.O. Politecnico di Milano 19
20 Model x A = amount invested in A x B = amount invested in B max s.t. z = 0,04 x A + 0,06 x B x A + x B x A x B x A, x B 0 E. Amaldi Fondamenti di R.O. Politecnico di Milano 20
21 4.2.1 Graphical resolution x B x A Feasible region x B x A x A + x B x A, x B 0 K euro E. Amaldi Fondamenti di R.O. Politecnico di Milano 21
22 8 z = K euro x B 10 2 z = 500 z = 240 z grows The level curves are straight lines 0,04 x A + 0,06 x B = z x A Optimal solution: x * A = x * B constant z * = The objective function gradient: f (x) = indicates the direction of fastest growth of f 0,04 0,06 constant E. Amaldi Fondamenti di R.O. Politecnico di Milano 22
23 4.2.2 Vertices of the feasible region Each constraint (with or ) defines an affine half space in the variable space half plane in R 2 a hyperplane b = 0 {x R n : a T x b} a 0 {x R n : a T x = b} which defines the half space E. Amaldi Fondamenti di R.O. Politecnico di Milano 23
24 Def.: The feasible region X is a polyhedron P P can be empty and unbounded half spaces # finite Def.: A subset S R n is convex if y 1, y 2 S contains the (whole) segment connecting y 1 and y 2. x. y 2 y 1 y 1 y 2 convex [y 1, y 2 ] = { x R n : x = α y 1 + (1 α) y 2 with α [0, 1] } segment all convex combinations of y 1 and y 2 E. Amaldi Fondamenti di R.O. Politecnico di Milano 24
25 Property: A polyhedron P is a convex set of R n Indeed, any half space is convex y 1 y 2 and the intersection of a finite number of convex sets is also convex. E. Amaldi Fondamenti di R.O. Politecnico di Milano 25
26 not a vertex! x 2 (0,3,1) (0,3,0) H 1 (0,1,3)... H 2 x 3 (0,0,0) (1,3,0) (0,0,3) (2,2,0) (1,0,3). (2,0,2) (2,0,0) facet edge H 3 vertex x ~ Polyhedron P x 1 ~ x is a vertex of P if y 1, y 2 P, y 1 y 2 and α (0, 1) s.t. ~ x = α y 1 + (1 α) y 2 Def.: The vertices of P are the points of P that cannot be expressed as convex combinations of two other (distinct) points of P. E. Amaldi Fondamenti di R.O. Politecnico di Milano 26
27 Property: A polyhedron P ={ x R n : Ax = b, x 0 } ø has a finite number ( 1) of vertices. Def.: Given a polyhderon P, a vector d R n with d 0 is an unbounded feasible direction of P if for any point x 0 P the ray { x R n : x = x 0 + λ d, λ 0 } is contained in P.. P x 0 d d. 0 E. Amaldi Fondamenti di R.O. Politecnico di Milano 27
28 Representation of polyhedra: Every point x of a polyhedron P can be expressed as the convex combination of its vertices plus (possibly) an unbounded feasible direction d of P : x = α 1 x α k x k + d where x 1,..., x k are the vertices of P and the multipliers α i 0 satisfy α α k = 1. P x [x 1, x 2 ], d = d x x 1 x 2 E. Amaldi Fondamenti di R.O. Politecnico di Milano 28
29 Consequence: (Weyl Minkowski) Every point x of a bounded polyhedron P (polytope) can be expressed as the convex combination of its vertices. x 4. (d = 0) x 1.. x 2 P.. x 3 x = α 1 x α 4 x 4 with α i 0 and α α 4 = 1 E. Amaldi Fondamenti di R.O. Politecnico di Milano 29
30 Fundamental theorem of LP: If the polyhedron P ={ x R n : Ax = b, x 0 } of the feasible solutions of the LP min{ c T x : x P } is not empty, either there exists (at least) one optimal vertex or the value of the objective function is unbounded on P. Proof Case 1: P has an unbouded direction d such that c T d < 0 P is unbounded and the value z = c T x along the direction d E. Amaldi Fondamenti di R.O. Politecnico di Milano 30
31 Case 2: P has no unbounded direction d such that c T d < 0 Every point of P can be expressed as: x = α i x i d i=1 where x 1,..., x k are the vertices of P, α i 0 with α α k =1, and d = 0 or d is an unbounded feasible direction. For every x P, d = 0 or c T d 0 and hence... since α i 0 i and α α k =1. k E. Amaldi Fondamenti di R.O. Politecnico di Milano 31
32 Although the variables can take fractional values, LPs can be viewed as combinatorial problems: we just need to examine the vertices of the polyhedron of the feasible solutions! finite number The graphical approach is only applicable for n 3 algorithm? E. Amaldi Fondamenti di R.O. Politecnico di Milano 32
33 An interior point x P cannot be an optimal solution: c = direction of fastest growth of z (constant gradient) x an improving direction In an optimal vertex all feasible directions (respecting feasibility for a sufficiently small step) are not improving: improving directions feasible direction E. Amaldi Fondamenti di R.O. Politecnico di Milano 33
34 4.2.3 Four types of LPs Unique optimal solution x 2. c x 1 Multiple (infinite) number of optimal solutions c E. Amaldi Fondamenti di R.O. Politecnico di Milano 34
35 Unbuounded LP c unbounded polyhedron and unlimited objective function value Infeasible LP empty polyhedron (no feasible solution) E. Amaldi Fondamenti di R.O. Politecnico di Milano 35
4 Linear Programming (LP) E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 1
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