Math Models of OR: The Revised Simplex Method

Transcription

1 Math Models of OR: The Revised Simplex Method John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY USA September 2018 Mitchell The Revised Simplex Method 1 / 25

2 Motivation Outline 1 Motivation 2 Pivot matrices 3 The revised simplex method 4 A symbolic representation of the pivot matrix and the tableau Mitchell The Revised Simplex Method 2 / 25

3 Motivation Reducing the computation in simplex At each iteration, we move from a basic feasible solution to a neighboring basic feasible solution. When we perform a simplex pivot, we update every entry in the tableau. Do we need to know all these entries in order to find the next BFS? We need to know: the pivot column, so we have to find a variable that has a negative cost. If the current tableau is in optimal form, we need to check that by looking at every cost c j. the pivot row, so we need to compute the minimum ratio. This requires knowledge of the entries in b and the pivot column. Note: once we ve used the minimum ratio to select the pivot row, we only need to know two entries in that row in order to update x: the value on the right hand side, and the value in the pivot column. the basic sequence, so we know which variable is leaving the basis. Mitchell The Revised Simplex Method 3 / 25

4 Motivation Reducing the computation in simplex At each iteration, we move from a basic feasible solution to a neighboring basic feasible solution. When we perform a simplex pivot, we update every entry in the tableau. Do we need to know all these entries in order to find the next BFS? We need to know: the pivot column, so we have to find a variable that has a negative cost. If the current tableau is in optimal form, we need to check that by looking at every cost c j. the pivot row, so we need to compute the minimum ratio. This requires knowledge of the entries in b and the pivot column. Note: once we ve used the minimum ratio to select the pivot row, we only need to know two entries in that row in order to update x: the value on the right hand side, and the value in the pivot column. the basic sequence, so we know which variable is leaving the basis. Mitchell The Revised Simplex Method 3 / 25

5 Motivation Reducing the computation in simplex At each iteration, we move from a basic feasible solution to a neighboring basic feasible solution. When we perform a simplex pivot, we update every entry in the tableau. Do we need to know all these entries in order to find the next BFS? We need to know: the pivot column, so we have to find a variable that has a negative cost. If the current tableau is in optimal form, we need to check that by looking at every cost c j. the pivot row, so we need to compute the minimum ratio. This requires knowledge of the entries in b and the pivot column. Note: once we ve used the minimum ratio to select the pivot row, we only need to know two entries in that row in order to update x: the value on the right hand side, and the value in the pivot column. the basic sequence, so we know which variable is leaving the basis. Mitchell The Revised Simplex Method 3 / 25

6 Motivation Example For example, consider the tableau, with the circled pivot entry: ratio x 1 x 2 x 3 x 4 x basic sequence S = (3, 4). To determine the pivot position, we need to know the objective function costs, the right hand side b, and the entries in the pivot column: ratio x 1 x 2 x 3 x 4 x basic sequence S = (3, 4). Mitchell The Revised Simplex Method 4 / 25

7 Motivation Updating x ratio x 1 x 2 x 3 x 4 x basic sequence S = (3, 4). We can tell from these entries that x 1 will replace x 4 in the basic sequence, since x 4 is the basic variable corresponding to the pivot row. We can also calculate the updated entries in b, so we can calculate the new solution x based just on this part of the tableau: R 0 + 3R 2, R 1 R 2 x 1 x 2 x 3 x 4 x S = (3, 1). Updated BFS: x 3 = 2 and x 1 = 3, x 2 = x 4 = x 5 = 0 and value 9. Mitchell The Revised Simplex Method 5 / 25

8 Motivation What about the next pivot? In order to perform the next pivot, we need to be able to calculate the updated objective function terms efficiently. A note on terminology: the updated objective function entries c j are the reduced costs. Thus, the entries in the top row of a simplex tableau are known as the reduced costs of the corresponding variables. Mitchell The Revised Simplex Method 6 / 25

9 Pivot matrices Outline 1 Motivation 2 Pivot matrices 3 The revised simplex method 4 A symbolic representation of the pivot matrix and the tableau Mitchell The Revised Simplex Method 7 / 25

10 Pivot matrices The revised simplex golden rule Pivoting corresponds to premultiplying the tableau by a pivot matrix. We find the pivot matrix using the simplex golden rule: Do unto the identity as you would do unto the tableau. For our example problem, this gives a pivot matrix: R R 2, R 1 R 2 Q 1 = Mitchell The Revised Simplex Method 8 / 25

11 Using the pivot matrix Pivot matrices Let s call the original tableau M 0. If we premultiply M 0 by Q 1, we get a new tableau: Q 1 M 0 = = =: M 1 Mitchell The Revised Simplex Method 9 / 25

12 Pivot matrices Comparing with the simplex pivot The resulting tableau M 1 is identical to the one obtained by performing the pivot on the original tableau: ratio x 1 x 2 x 3 x 4 x R 0 + 3R 2, R 1 R 2 x 1 x 2 x 3 x 4 x Mitchell The Revised Simplex Method 10 / 25

13 Pivot matrices Why do the two approaches agree? Let s look at just one entry, a 15. In the pivot, we first divide a 25 by the pivot entry (which is just a 21 = 1 in this case), and then we subtract a 11 multiplied by a 25 from a 15. Algebraically, we can write this update to a 15 as a 15 a 15 a 11 a 25 /a 21 = 2 1 4/1 = 2 ratio x 1 x 2 x 3 x 4 x Mitchell The Revised Simplex Method 11 / 25

14 Pivot matrices Expressing Q 1 in terms of the pivot column Q 1 = 1 0 c 1 /a a 11 /a /a 21 The updated x 5 column of the product Q 1 M 0 is equal to the matrix-vector product of Q 1 with the x 5 column of M 0 : 1 0 c 1 /a a 11 /a /a 21 c 5 a 15 a 25 = a 15 a 25 a 11 /a 21 This is exactly the same formula as calculated for the pivot update. All the other entries work similarly. Mitchell The Revised Simplex Method 12 / 25

15 Pivot matrices The next iteration We can now perform another iteration on the updated tableau M 1, M 1 = x 1 x 2 x 3 x 4 x The new pivot column would be the x 2 column and the pivot would: (i) divide row 1 by 2, 1 2 R 1, (ii) add 6 copies of row 1 to row 0, R 0 + 6R 1, (iii) add 1 copy of row 1 to row 2, R 2 + R 1. Performing these operations on the identity matrix gives the pivot matrix Q 2 : R then R 0 + 6R 1, R 2 + R 1 Q 2 = Mitchell The Revised Simplex Method 13 / 25

16 Pivot matrices Premultiply by the pivot matrix If we premultiply M 1 by Q 2, we get a new tableau: Q 2 M 1 = = =: M 2 Mitchell The Revised Simplex Method 14 / 25

17 The revised simplex method Outline 1 Motivation 2 Pivot matrices 3 The revised simplex method 4 A symbolic representation of the pivot matrix and the tableau Mitchell The Revised Simplex Method 15 / 25

18 The revised simplex method The revised simplex method The revised simplex method carries out exactly the pivots of the usual simplex method, but uses pivot matrices to calculate required entries, and is selective about which entries get calculated. Note that since we never pivot on the objective function row, the first column of the pivot matrix is always the first column of the identity matrix. We give an example of solving a linear optimization problem with the revised simplex method in a separate handout. Mitchell The Revised Simplex Method 16 / 25

19 A symbolic representation of the pivot matrix and the tableau Outline 1 Motivation 2 Pivot matrices 3 The revised simplex method 4 A symbolic representation of the pivot matrix and the tableau Mitchell The Revised Simplex Method 17 / 25

20 A symbolic representation of the pivot matrix and the tableau Expressing the pivot matrix symbolically We have an initial tableau which we can write as M 0 = d ct b A After a few pivots, we end up with a new canonical form tableau ˆM = ˆd ĉ T ˆb  The sequence of pivots can be represented by a single pivot matrix P. So we have ˆM = PM 0. Mitchell The Revised Simplex Method 18 / 25

21 A symbolic representation of the pivot matrix and the tableau Treating the basic and nonbasic columns separately Let us assume without loss of generality that the basic variables consist of the first m columns of ˆM, by reordering the variables if necessary. So we can write ˆM = ˆd 0 T ĉ T N ˆb I ˆN The original tableau can be similarly written M 0 = d ct B c T N b B N Mitchell The Revised Simplex Method 19 / 25

22 A symbolic representation of the pivot matrix and the tableau Rewriting the pivot matrix We also choose to write the pivot matrix as [ ] 1 y T P = 0 G for some vector y IR m and matrix G IR m m. Mitchell The Revised Simplex Method 20 / 25

23 A symbolic representation of the pivot matrix and the tableau Calculating the pivot The equation ˆM = PM 0 can then be written ˆM = ˆd 0 T ĉ T N ˆb I ˆN = PM 0 = [ 1 y T 0 G ] d c T B cn T b B N = d y T b cb T y T B cn T y T N Gb GB GN Mitchell The Revised Simplex Method 21 / 25

24 A symbolic representation of the pivot matrix and the tableau Finding G We have ˆd 0 T ĉn T ˆb I ˆN Comparing the entries in the portion of the constraint matrix corresponding to the basic variables shows that I = GB, so G = B 1, the inverse of the matrix B. = d y T b cb T y T B cn T y T N Gb GB GN The objective function coefficients for the basic variables imply Thus we get [ 1 c T P = B B 1 0 B 1 ] 0 T = cb T y T B, so y T = cb T B 1. We will interpret y and the other terms in ˆM when we discuss duality. Mitchell The Revised Simplex Method 22 / 25

25 A symbolic representation of the pivot matrix and the tableau Finding G We have ˆd 0 T ĉn T ˆb I ˆN Comparing the entries in the portion of the constraint matrix corresponding to the basic variables shows that I = GB, so G = B 1, the inverse of the matrix B. = d y T b cb T y T B cn T y T N Gb GB GN The objective function coefficients for the basic variables imply Thus we get [ 1 c T P = B B 1 0 B 1 ] 0 T = cb T y T B, so y T = cb T B 1. We will interpret y and the other terms in ˆM when we discuss duality. Mitchell The Revised Simplex Method 22 / 25

26 A symbolic representation of the pivot matrix and the tableau Finding G We have ˆd 0 T ĉn T ˆb I ˆN Comparing the entries in the portion of the constraint matrix corresponding to the basic variables shows that I = GB, so G = B 1, the inverse of the matrix B. = d y T b cb T y T B cn T y T N Gb GB GN The objective function coefficients for the basic variables imply Thus we get [ 1 c T P = B B 1 0 B 1 ] 0 T = cb T y T B, so y T = cb T B 1. We will interpret y and the other terms in ˆM when we discuss duality. Mitchell The Revised Simplex Method 22 / 25

27 A symbolic representation of the pivot matrix and the tableau The symbolic version of P for the example We have M 2 = x 1 x 2 x 3 x 4 x S = (2, 1) M 0 = x 1 x 2 x 3 x 4 x Since the basic sequence is S = (2, 1), the matrix B contains the second column followed by the first: [ ] 1 1 B = so B 1 = 1 [ ] Mitchell The Revised Simplex Method 23 / 25

28 A symbolic representation of the pivot matrix and the tableau The symbolic version of P for the example (continued) Similarly, Then Thus, c T B B 1 = 1 2 P = c B = [ c2 c 1 ] = [ 3 3 [ 3 3 ] [ [ 1 c T B B 1 0 B 1 ] = ] ] = [ 3 0 ] Mitchell The Revised Simplex Method 24 / 25

29 A symbolic representation of the pivot matrix and the tableau Checking P We can calculate PM 0 = = = M 2 Mitchell The Revised Simplex Method 25 / 25