1 / 29

290 likes | 453 Views

L16 LP part2. Homework Review N design variables, m equations Summary. H15 Prob 1. H15 prob 8.3. ≥. H15 Prob 8.8 and 8.9. H15 prob 8.19. H15 Prob 21. H15 Prob 21. x1=0. x2=0. x3=0. x4=0. Prob 8.21 cont’d. Linear Programming Prob.s. Must convert to standard form LP Problem!.

Download Presentation
## L16 LP part2

**An Image/Link below is provided (as is) to download presentation**
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.
Content is provided to you AS IS for your information and personal use only.
Download presentation by click this link.
While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

**L16 LP part2**• Homework • Review • N design variables, m equations • Summary**Linear Programming Prob.s**Must convert to standard form LP Problem!**Transforming LP to Std Form LP**• If Max, then f(x) = - F(x) • If x is unrestricted, split into x+ and x-, and substitute into f(x) and all gi(x) andrenumber all xi • If bi < 0, then multiply constraint by (-1) • If constraint is ≤, then addslacksi • If constraint is ≥, then subtractsurplussi**Std Form LP Problem**Matrix form All “=“ All “≥0” i.e. non-neg.**Ex 8.4 cont’d**Pivot row Pivot column**Terms**• basic solutions - solutions created by setting (n-m) variables to zero • basic feasible sol’ns - sol’ns @ vertices of feasibility polygon • feasible solution - any solution inS polyhedron • basic variables- dependent variables, not set to zero • non-basic variables - independent variables, set to zero, i.e. not in basis. • basis – identity columns of the coefficient matrixA**Method?**• Set up LP prob in “tableau” • Select variable to leave basis • Select variable to enter basis (replace the one that is leaving) • Use Gauss-Jordan elimination to form identity sub-matrix, (i.e. new basis, identity columns) • Repeat steps 2-4 until opt sol’n is found!**Can we be efficient?**• Are we at the min? • If not which non-basic variable should be brought into basis? • Which basic variable should be removed to make room for the new one coming on? SIMPLEX METHOD!**Simplex Method – Part 1 of 2Single Phase Simplex Method**When the Standard form LP Problem has only ≤ inequalties…. i.e. only slack variables, we can solve using the Single-Phase Simplex Method! If surplus variables exist… we need the Two-Phase Simplex Method –with artificial variables… Sec 8.6 (after Spring Break)**Single-Phase Simplex Method**• Set up LP prob in a SIMPLEX tableau add row for reduced cost, cj’ and column for min-ratio, b/a label the rows (using letters) of each tableau • Check if optimum, all non-basic c’≥0? • Select variable to enter basis(from non-basic) Largest negative reduced cost coefficient/ pivot column • Select variable to leave basis Use min ratio column / pivot row • Use Gauss-Jordan elimination on rows to form new basis, i.e. identity columns • Repeat steps 2-5 until opt solution is found!**Ex 8.7 1 phase Simp Meth**All constraints are “slack” type Therefore, can use single-phase Simplex Method Figure 8.3 Graphical solution to the LP problem Example 8.7. Optimum solution along line C–D. z*=4.**Step 1. Set up Simplex Tableau**Step 2. check if optimum? X1 and x2 are <0! Continue!**Step3 & 4**• 3. Select variable to enter basis(from non-basic) • Largest negative reduced cost coefficient/ pivot column • 4. Select variable to leave basis • Use min ratio column / pivot row**Why use Min Ratio Rule?**We want to add x1 into basis, i.e. no longer is x1=0 How much of x1 can we add? Whoops!!!!**Step 5 Use Gauss-Jordan**form new basis, i.e. identity columns Step 6. Repeat steps 2-5. Step 2. Check if optimal? Since all c’≥0… We have found the optimal solution!**Use Excel to help with arithmetic?**See Excel spreadsheet on website**Summary**• Need to transform into Std LP format Unrestricted, slack, surplus variables, min = - Max • Opt solution is on a vertex • Simplex Method moves efficiently from one feasible combination of basic variables to another. • Use Single-Phase Simplex Method when only “slack” type constraints. • Use Excel to assist w/arithmetic

More Related