4 Linear Programming (LP) E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 1

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1 4 Linear Programming (LP) E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 1

2 Definition: A Linear Programming (LP) problem is an optimization problem: where min f () s.t. X n the objective function f : X is linear, the feasible region X={ n : g i () r i 0, i {1,, m} } with r i =,, and g i : n linear functions i = 1,, m. E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 2

3 Definition: * n is an optimal solution of the LP min f () s.t. X n if f (*) f () X. A wide variety of decision-making problems can be formulated as linear programs (LPs). They often involve the optimal allocation of a given set of limited resources to different activities. E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 3

4 General form: min z = c c n n a a 1n n b 1 (=) ( ) a m a mn n b (=) m ( ) 1,, n 0 Matri notation: min z = c T A b 0 min c 1 c n 1 n a 11 a 1n 1 b 1 a m1 a mn n b m E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano n

5 Historical sketch 1825/26: Fourier presents a method to solve systems of linear inequalities and discusses LPs with 2-3 variables. 1939: Kantorovitch lays the foundations of LP (Nobel prize, 1975) 1947: Dantzig independently proposes LP and invents the Simple algorithm. E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 5

6 Eample 1: Diet problem Given n aliments j = 1,, n m a ij b i c j nutrients (basic substances) i = 1,, m amount of i-th nutrient contained in one unit of the j-th aliment daily requirement of the i-th nutrient cost of a unit of j-th aliment, determine a diet that minimizes the total cost while satisfying all the daily requirements. E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 6

7 Decision variables: j = amount of j-th aliment in the diet, with j = 1,..., n min n j1 n j1 c a j ij j j j b 0 i i 1,..., m j 1,..., n E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 7

8 Eample 2: Transportation problem (single product) Given m production plants i = 1,, m n clients j = 1,, n c ij p i d j unit transportation cost from plant i to client j maimum supply (production capacity) of plant i demand of client j q ij maimum amount transportable from plant i to client j, determine a transportation plan that minimizes the total costs while respecting plant capacities and client demands. E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 8

9 Assumption: m i1 n p i d j1 j Decision variables: ij = amount of product transported from i to j, with i = 1,, m and j = 1,, n. min m n i1 j1 c ij ij n j1 ij p i i = 1,, m (plant capacity) m i1 ij d j j = 1,, n (client demand) 0 ij q ij i, j (transportation capacity) E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 9

10 Eample 3: Production planning problem Given n products (j = 1,, n) which compete for resources m resources (i = 1,, m) c j a ij b i profit (selling price cost) per unit of j-th product amount of i-th resource needed to produce one unit of j-th product maimum available amount of i-th resource, determine a production plan that maimizes the total profit given the available resources. E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 10

11 E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 11 j = amount of j-th product, with j = 1,..., n n j m i b a c j n j i j ij n j j j 1,..., 0 1,..., ma 1 1 Decision variables:

12 Assumptions of LP models 1) Linearity (proportionality and additivity) of the objective function and constraints. Proportionality: contribution of each variable = constant variable Drawback: does not account for economies of scale. Additivity: contribution of all variables = sum of single contributions Drawback: competing products profits are not independent. E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 12

13 2) Divisibility The variables can take fractional (rational) values. 3) Parameters are considered as constants which can be estimated with a sufficient degree of accuracy. Uncertainty in the parameters may require more comple mathematical programs. In Linear Programming, if we have different scenarios we adopt sensitivity analysis (see end of Chapter 4). E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 13

14 4.1 Equivalent forms General form: min (ma) z = c T A 1 b 1 A 2 b inequality constraints 2 A 3 = b 3 equality constraints j 0 for j J {1,, n} j free for j {1,, n}\j Definition: Standard form min z = c T A = b 0 only equality constraints and all variables non negative. E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 14

15 The two forms are equivalent. Simple transformation rules allow to pass form one form to the other form. Warning: the transformation may involve adding/deleting variables and/or constraints. E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 15

16 Transformation rules mac T = - min -c T a T b a T + s = b s 0 slack variable a T b a T -s = b s 0 surplus variable E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 16

17 E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 17 After substituting j with j + j,wedelete j from the problem. 0 0 j j j j j j unrestricted in sign

18 Eample General from: ma s.t , 2 unrestricted Step 1: 2 = 3 4 3, 4 0 ma s.t , 3, 4 0 E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 18

19 Step 2: introduce slack and surplus variables 5 and 6 min = = 4 1, 3, 4, 5, 6 0 Step 3: change the objective function sign E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 19

20 Other straightforward transformations a b -a -b a b -a -b a= b a b a b a b -a -b E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 20

21 4.2 Geometry of Linear Programming Eample Capital budgeting Capital of and two possible investments A and B with, respectively, 4% and 6% epected return. Determine a portfolio that maimizes the total epected 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 -- Foundations of Operations Research -- Politecnico di Milano 21

22 Model: A = amount invested in A B = amount invested in B ma s.t. z=0,04 A + 0,06 B A + B A B A, B 0 E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 22

23 4.2.1 Graphical solution B A Feasible region B A A + B A, B 0 K euro E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 23

24 10 z=500 8 z=360 6 z= K euro B 2 z= 0 f () z ma increase f () Definition:A level curve of value z of a function f is the set of points in n where f is constant and takes value z. The level curves of a LP are lines: 0,04 A + 0,06 B = z constant A Optimal solution: * A * B = z * =500 f () = 0,04 0,06 is the direction at of fastest increase of f. E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 24

25 4.2.2 Vertices of the feasible region Consider a LP with inequality constraints (easier to visualize). Definitions: H = { n : a T = b } is a hyperplane and H - = { n : a T b } is an affine half-space. Half-plane in 2 Each inequality constraint (a T b) defines an affine half-space in the variable space. a H - ={ n : a T b} a 0 b = 0 E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 25

26 Definition: The feasible region X is a polyhedron P half-spaces # finite P can be empty or unbounded Definition: A subset X n is conve if for each pair y 1, y 2 X, X contains the whole segment connecting y 1 and y 2.. y 2 y 1 y 1 y 2 conve [y 1,y 2 ] = { n : = y 1 + (1 - ) y 2 with [0, 1] } segment all the conve combinations of y 1 e y 2 E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 26

27 Property: A polyhedron P is a conve set of n. Indeed: any half-space is clearly conve y 1 y 2 and the intersection of a finite number of conve sets is also a conve set. E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 27

28 not a verte! 2 (0,3,1) (0,3,0) 3 H 1 (0,1,3)... H 2 (0,0,3) (1,0,3) (2,0,2) Polyedron P (0,0,0) (2,2,0) (1,3,0). (2,0,0) facet edge verte ~ H 3 1 Algebraically: ~ is a verte of P if y 1, y 2 P, y 1 y 2 and (0, 1) s.t. ~ = y 1 + (1 - ) y 2 Definition: A verte of P is a point of P which cannot be epressed as a conve combination of two other distinct points of P. E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 28

29 Property: A non-empty polyhedron P ={ n : A = b, 0 } (in standard form) or P ={ n : A b, 0 } (in canonical form) has a finite number ( 1) of vertices. Definition: Given a polyhedron P, a vector d n with d 0 is an unbounded feasible direction of P if, for every point 0 P, the ray { n : = 0 + d, 0 } is contained in P.. P 0 d d. 0 E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 29

30 Theorem (representation of polyhedra -- Weyl-Minkowski): Every point of a polyhedron P can be epressed as a conve combination of its vertices 1,..., k plus (if needed) an unbounded feasible direction d of P : = k k + d where the multipliers i 0 satisfy k = 1. [ 1, 2 ], d = d. P E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 30

31 Definition:A polytope is a bounded polyhedron, that is, it has the only unbounded feasible direction d = 0. Consequence: Every point of a polytope P can be epressed as a conve combination of its vertices. 4. = (d = 0) P.. 3 with i 0 and = 1 E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 31

32 Fundamental theorem of Linear Programming: Consider a LP min{ c T : P } where P n is a non-empty polyhedron of the feasible solutions (in standard or canonical form). Then either there eists (at least) one optimal verte or the value of the objective function is unbounded below on P. Proof Case 1: P has an unbounded feasible direction d such that c T d < 0 P is unbounded and the values z = c T - along the direction d E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 32

33 E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 33 Case 2: P has no unbounded feasible direction d such that c T d < 0 i T k i T k i i T i k i i i T T c d c c d c c min For any P, we have d = 0 or c T d 0 and hence Any point of P can be epressed as: where 1,..., k are the vertices of P, i 0 with k =1, and d = 0, or d is a unbounded feasible direction. d k i i i 1 since i 0 i and k =1. 0

34 Geometrically An interior point P cannot be an optimal solution: c = direction of fastest increase in z (constant gradient) always an improving direction In an optimal verte all feasible directions (respecting feasibility for a sufficiently small step) are worsening directions: -c improving directions vertices feasible directions E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 34

35 The theorem implies that, although the variables can take fractional values, Linear Programs can be viewed as combinatorial problems: we only need to eamine the vertices of the polyhedron of the feasible solutions! Finite but often eponential number Graphical method only applicabile for n 3. E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 35

36 4.2.3 Four types of Linear Programs Observation: Since min c T, better solutions found by moving along -c. A unique optimal solution 2. c 1 Multiple (infinitely many) optimal solutions c E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 36

37 Unbounded LP c Unbounded polyhedron and unlimited objective function value Infeasible LP Empty polyhedron (no feasible solution) E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 37