Linear Programming Notes for Maple

Linear Programming Notes for Maple 1. Basic Linear Algebra in Maple To use these commands, type with(linalg); first. (1) To define a matrix in Maple: (This example is a 2 × 3 matrix) A:=matrix(2,3,[1,2,3,4,5,6]); A:=matrix([ [1,2,3],[4,5,6] ]); Reminder: In Matlab, it was: A=[1,2,3;4,5,6]; (2) To define a vector in Maple, you can either define an n × 1 or 1 × n matrix, or use the vector command: b:=vector([a,b,c]); b:=matrix(3,1,[a,b,c]); When you use “vector”, Maple will assume it is a row or a column depending on how matrix-vector multiplication is defined. Defining a vector using matrix forces it to be a row/column vector. (3) Scalar Multiplication: In this example, compute γA A:=matrix(2,3,[1,2,3,4,5,6]); gA:=evalm(gamma*A); (4) Add two matrices (must be the same sizes): A:=matrix(2,3,[1,2,3,4,5,6]); B:=matrix(2,3,[a,b,c,d,e,f]); evalm(A+B); (5) Matrix Transpose, Determinant, Inverse, Rank: A:=matrix(2,2,[1,2,b,4]); transpose(A); det(A); inverse(A); rank(A); (6) Matrix-Vector, Matrix-Matrix multiplication: Use &* rather than * b:=vector([3,c]); evalm(A&*b); evalm(b&*A); Compare that with the following (One of these gives an error): b:=matrix(2,1,[3,c]); evalm(A&*b); evalm(b&*A); And Matrix-Matrix multiplication (One of these gives an error): A:=matrix(2,3,[1,2,3,4,5,6]); B:=matrix(3,2,[a,b,c,d,e,f]); evalm(A&*B); evalm(B&*A); (7) To identify a particular element of a matrix, A[i, j] (Note that these are square brackets; in Matlab these would be parentheses). 1 2 2. Solving Systems of Equations There are always three possibilities when solving Ax = b for x: A unique solution, No solution, or an infinite number of solutions (free variables). (1) Maple can give you the explicit equations you’re working with: A:=matrix(3,3,[3,1,1,1,2,1,1,1,7]); b:=vector([10,20,30]); geneqns(A,[x,y,z],b); (2) To solve the system directly: linsolve(A,b); (3) In Linear Algebra, we use Gaussian Elimination with Pivoting to put an augmented matrix into RREF (Row Reduced Echelon Form). Maple can do that for you if you want to see it (sometimes its handy when there is no solution or an infinite number of solutions): C:=augment(A,b); rref(C); (4) Here’s an example of a system with an infinite number of solutions: A:=matrix(2,2,[3,1,6,2]); b:=vector([4,8]); linsolve(A,b); Maple uses t1 , t2 , t3 , . . . to represent the variable for the first column, second column, etc. Compare this to the RREF of the augmented matrix: Ab:=augment(A,b); rref(Ab); (5) Here’s an example of a system with no solution (Again, compare it with the RREF): A:=matrix(2,2,[1,1,1,1]); b:=vector([1,2]); linsolve(A,b); 3. A Specialty plot: INEQUAL EXAMPLE: Plot the region in the plane defined by the following set of inequalities: x1 x2 5x1 + 6x2 x1 x2 ≤ 90 ≤ 60 ≤ 600 ≥0 ≥0 First we’ll define the set of constraints, then we’ll plot them. To use inequal, we first need to call with(plots) with(plots): CS:=[ x[1]<=90, x[2]<=60, 5*x[1]+6*x[2]<=600, x[1]>=0, x[2]>=0 ]; inequal(CS,x[1]=1..120,x[2]=0..120, optionsfeasible=(color=white), optionsexcluded=(color=yellow) ); NOTE: This will only plot linear inequalities and only in the plane. 3 4. Linear Programming in Maple Before using these commands, type with(simplex): (1) We can solve for the intersection points of the boundary in Maple. For example, to solve for the intersection point between constraints 1 and 5 we would write: Eqns1:=convert( { CS[1],CS[2] }, equality ); solve(Eqns1); (2) EXAMPLE: Solve using Maple: Maximize 30x1 + 40x2 so that all the previous constraints in CS are satisfied. We’ll start from scratch so that all relevant commands are right here: with(simplex): CS:=[ x[1]<=90, x[2]<=60, 5*x[1]+6*x[2]<=600]; z:=30*x[1]+40*x[2]; Sol:=maximize(z, { seq( CS[i],i=1..3)},NONNEGATIVE); assign(Sol); z; The last line will give you what the maximum value is. Using NONNEGATIVE will save some typing- this is in place of the last set of inequalities. (3) EXAMPLE: We can get Maple to tell us what it is computing by using infolevel[simplex]. In this example, we have an unbounded problem so there is no solution. Without using infolevel, Maple returns an empty solution so we would be unsure as to why it is empty. CS:=[x[1]<=20,x[2]>=10]; Z:=x[1]+x[2]; Sol:=maximize(Z, {seq(CS[i],i=1..2)}); infolevel[simplex]:=1; Sol:=maximize(Z, {seq(CS[i],i=1..2)}); 5. Exercises: Use Maple to solve the following problems: (1) Solve the system: 3a − b + 5c = 2 2a − b − c = 1 b − 3c = 5 (2) Solve the equation Ax = b  1 0 1 A= 0 1 0 5 6 0 for x, if  0 0 1 0 , 0 1 (3) Maximize x1 + x2 subject to: 2x1 + x2 x1 + 2x2 x1 x2 Also plot the feasible region. ≤ 29 ≤ 25 ≥2 ≥5   90 b =  60  600 4 (4) Maximize x1 + x2 + x3 + x4 + x5 subject to: x1 + x2 x3 + x4 x2 + x3 + 2x4 + 5x5 xi ≤ 100 ≤ 70 ≤ 250 ≥0 1≤i≤5 (5) Minimize 2x1 + 4x2 subject to: x1 + 5x2 4x1 + 2x2 x1 + x2 xi And plot the feasible region. ≤ 100 ≥ 20 = 10 ≥0 1≤i≤2