1、2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 1Operations Research IThe Simplex Algorithm: Part Two黄红选清华大学工业工程系电话：010-62795308Email： 2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 2Context Degeneracy and the Convergence of the Simplex Al
2、gorithm The Big M Method The Two-Phase Simplex Method Variables That are Unrestricted in Sign2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 3s4=551000010016x1=4400.5000.250.7510000s4-2-430s30.5-1-5x3s2=4s1=16z=240BV8Ratio410-1001601000240001501rhss2s1x2x1zz-value for n
3、ew bfs=z-value of current bfs minus(value of entering variable in new bfs)* (coefficient of entering variable in row 0 of current bfs)3.9 Degeneracy and the ConvergenceFOR Maximization Problem:2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 4s4=5510000100x1=2201.5-0.500
4、1.2510000s4-4-810s3100x3x3=8s1=24z=280BV Ratio820-2002421-200280100501rhss2s1x2x1z3.9 Degeneracy and the Convergence2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 5 (value of entering variable in new bfs)0,(z-value for new bfs)(z-value for current bfs) (value of enteri
5、ng variable in new bfs)=0,(z-value for new bfs)=(z-value for current bfs)z-value for new bfs=z-value of current bfs (value of entering variable in new bfs)* (coefficient of entering variable in row 0 of current bfs)Change of z-values: max problem2010-4-14H.-X. HuangDepartment of Industrial Engineeri
6、ng, Tsinghua University 6Nondegenerate 1mg of vitamin C. Each ounce of orange juice: 0.25oz of sugar, 3mg of vitamin C. The cost of producing orange soda is 2c/oz, and the cost of orange juice is 3c/oz. Orange-flavored soft drink: Each bottle of soft drink has 10oz and must contain at least 20mg of
7、vitamin C and at most 4oz of sugar.2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 16Decision Variables X1: the production (ounces) of orange soda X2: the production (ounces) of orange juice2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 17
8、How to solve the following LP?0,10vitamin,203sugar,4412132min2121212121=+=xxxxxxxxtsxxz0,102034412132min11212122112121=+=+=+=esxxxxexxsxxtsxxz100011020-10310s1=44011/41/20z=0000-3-21ratioBvrhse2s1x2x1znull?null?2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 18Question:
9、 How to find an initial bfs?Add artificial variables! BUT there is no guarantee that the optimal solution to the latter formulation will be the same as the former one.0,102034412132min321121321222112121=+=+=+=aaesxxaxxaexxsxxtsxxz2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua Uni
10、versity 19Make sure that all artificial vars are 0 Modifying the objective function Note that it sometimes happens that in solving the above LP, some of the artificial vars may assume positive values in the optimal solution. - infeasible0,102034412132min32112132122211213221=+=+=+=aaesxxaxxaexxsxxtsM
11、aMaxxz2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 20Flow Chart of Big M methodModify the constraints = rhs of each constraint is nonnegativeFor = or = constraints, add artificial variablesConvert to standard form, add slack or excess variablesLet M denote a very lar
12、ge positive numberMax Problem: - MaMin Problem: + MaEliminate artificial variables from row 0 =WHY?Simplex algorithmCanonical form2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 21CriteriaOptimal criterion:In the optimal solution, all artificial variables = 0Infeasible
13、criterion: At least one artificial variable is positive in the optimal solution, ORThe final tableau indicates that the LP is unbounded and at least one artificial variable is positive.Unbounded criterion:The final tableau indicates that LP is unbounded and all artificial variables in this tableau e
14、quals zero2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 22Example-step130,102034412132min2121212121=+=xxxxxxxxtsxxz0,102034412132min321121321222112121=+=+=+=aaesxxaxxaexxsxxtsxxz0,102034412132min11212122112121=+=+=+=esxxxxexxsxxtsxxz2010-4-14H.-X. HuangDepartment of I
15、ndustrial Engineering, Tsinghua University 23Example-step45322132min MaMaxxz +=10100011000-Ma310-Ma220-10310s1=44011/41/20000-3-21ratioBvrhse2s1x2x1z0,1020344121213212221121=+=+=+xxaxxaexxsxxtsIt is not a initial tableau which can be used !null2010-4-14H.-X. HuangDepartment of Industrial Engineering
16、, Tsinghua University 24Example-step 510a3=10101000110000a3100a220/3*a2=2020-1031016s1=44011/41/20z=30M30M-M0-3 +4M-2 +2M1ratioBvrhse2s1x2x1zIdentity matrix2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 25Example-step 55*a3=10/310/31-1/31/3002/30000a31/3-1/121-4/3Ma220
17、x2=20/320/3-1/3011/3028/5s1= 7/37/31/12105/120z= 20+10/3M20+10/3M-1 +1/3M00-1 +2/3M1ratioBvrhse2s1x2x1z2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 26The final tableaux1=553/2-1/21/20010-1/2-5/83/2-Ma31/21/81/2-Ma2x2=55-1/20100s1= 1/41/4-1/81000z=2525-1/20001ratioBvr
18、hse2s1x2x1z2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 27How to Spot an Infeasible LP?If any artificial variable is positive in the optimal Big M tableau, the original LP has no feasible solution.322132min MaMaxxz +=0,1036344121213212221121=+=+=+xxaxxaexxsxxts2010-4
19、-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 28Example-step 510*a3=10101000110000a3100a212a2=3636-1031016s1=44011/41/20z=46M46M-M0-3 +4M-2 +2M1ratioBvrhse2s1x2x1zIdentity matrix2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 29Example-step 5x2
20、=61010001100-1/43-4Ma3100a2a2=106-100-20s1= 3/23/20101/40z= 30+6M30+6M-M001-2M1Bvrhse2s1x2x1z2010-4-14H.-X. HuangDepartment of Industrial Engineering, Tsinghua University 30Disadvantages of Big M method How to determine the value of Big M ? M is chosen to be at least 100 times larger than the largest coefficient in the original objective function. Such large numbers can cause roundoff errors and other computational difficulties. Return
