o R M s graphical solution algebraic method ellipsoid algorithm Karmarkar 97 George B Dantzig 979 Khachian Karmarkar 98 Karmarkar interior-point algorithm
o R 08 gallon 000 000 00 60 g 0g X : : X : :
Ms 09 Max Z 000X 000X st X X 00 X X 60 X 0 X X 0 X X X X 00 X X 00 hyperplane half plane half space X X X 00 X
o R 00 X X X 60 A(0, 0) B(, 0) C(., ) X 0 D(, 0) X X X X X Z Z Z extreme point X X Z 00000 D E
Ms 0 C Z X X C X X 00 X X 60 000 Z 000 000 000 Z iso-profit line Z X X (0, ) E(0, 0) A(0, 0) Z 00000 Z 000 B(, 0) C(., ) : D(, 0) X
o R 0 Z 00000 iso-cost line Z X X convex set canonical form capability requirement X X 00 X X X X 00 X X
Ms 0 s 0 X X s 00 s slack variable X X s 0 binding constraint s 0 X X s nonbinding constraint X X X X s 0 00 X X s 0 60 X s 0 0 s 0 s 0 s 0 0g X X 00
o R 0 s 0 X X s 00 s surplus variable negative slack variable artificial variable a 0 X X s a 00 Min X X st X X 0 X X 8 X X 0 s s a 0 Min X X 0s 0s 0a st X X s 0 X X s a 8 X X s s a 0
Ms 0 unit matrix X X s 0 X X s a 8 column vector linear combination 0 0 0 X [ ] X [ ] s [ ] s [ ] a [ ] [ ] 0 8 0 0 X X s s a [ ] [ ] [ 0 ] [ ] [ ] 0 [ 8 ] 0 [ 8 ] X 0 X 0 s 0 s 0 a 8 X X 0 s 6 s 0 a 0 X X s 0 s 0 a 0 X 0 X s 0 s 0 a linearly independent 0 m m
o R 06 [ ] 0 [ ] [ ] 0 [ ] [ ] m m weighted sum 0 [ ] [ ] [ ] m X X m m m v v m X v X m v m 0 X X m 0 v v m 0 linearly dependent
Ms 07 [ ] [ ] [ ] [ ] [ ] [ ] X [ ] X [ ] X 0 0 0 X 0 [ ] [ ][ ][ ] [ ][ ][ ] 0 basis 0 0 m m 0 m basic variable 0 nonbasic variable basic feasible solution m
o R 08 big- Mbig-M method two Phase method dual simplex method Max Z X X st X X 0 X X 6 X X 0 X (0, ) (/, ) (, 0) X
Ms 09 Max Z X X st X X s 0 X X s 6 X X s s 0 initial basic feasible solution 0 0 iterative method X 0 X 0 Z 0 Z X X Î s 0 X X X s 6 X X X 6 0 6 Z X [ X s ] 6 X s X X s X 8 Î s 6 X [ X s ] X s X [ 8] Z 6 [ s s ] s 8 s s X [ s s ] s s X s s 8 8 8 8 8 8
o R 00 0 0 0 0 / iteration Z Z 0 0 Z 0 0 0 X X X X 0 0 0 0 6 / 8 0 0 s s 0 X X 0 entering variable 0 leaving variable 0 X X X X 0 X X X 0 X
Ms 0 s s minimum ratio test 0 s X s X 6 s X s X 6 X 0 s 0/ 6/ X s s 0 X X pivot row elementary row operation X X X / S / X X / S / / /8 s s 0
o R 0 6 PC IE 0008 pp 7 7 unbounded 000 6000 X 6000 6 Max Z 000X 6000X st X X 00 X X 60 X 0 X X 0
Ms 0 X B(, 0) C(., ) X X 6000 Z 000 extreme face BC B 0 C Z 000
o R 0 00 00 7 Max Z X X st X 00 X X 00 X X 0 X : X : Z 000000 Z 0000000 Z 000000
Ms 0 Z unbounded optimal solution 0 0 000 6 8 Max Z X X st 0X 0X 000 X 6 X X 0 X X 0 X : X :
o R 06 X X 6 X X 0 0X 0X 000 X infeasible solution
Ms 07 6 0 Max Z X X st 0X 0X 000 X 0 X X 0 X X 0 9 X Z 0 Z 00 Z A(0, 7) B(60, 60) X
o R 08 A0 7 B60 60 X X 0 7 Z 0X 0X 000 0X 0X 000 0X 0X 000 unrestricted X X X X X X X 0 Max Z X X st X X X X X 0 X X X X X X 0
Ms 09 Max Z X X X st X X X X X X X X X 0 0 X [ ] X [ ] X [ ] s [ ] s [ ] [ ] 0 [ ] [ ] [ ] [ ] [ ] 0 0 X X X 0 X 0 X 0 X 0 X 0 X 0 X X 0 X 0 Z Max Z X X Í Min Z X X
o R 00 [ ] [ ] [ ] 0 [ ] [ ] [ ] 6 0 [ ] [ ] [ ] [ 0 ] 0 0 [ ] [ ] [ ] [ 0 ] 6 0 0 X : : m X : : m Max X X st X 00 X 00 0X 0X 000
Ms 0 X X 0 0 00 0 0 0 X : : X X 00 X 00 0X 0X 000 X
o R 0 X X X (8, ) Z 000 Z 790 Z 00 X X 00 X X : : Min 0X 0X st X X X X 00 X X 0 X X 0 X X X X 00 X 8 X Z 790 Max X X st X X 0
Ms 0 X X 0 X X 0 Max X X st X X s a 0 X X s 0 X X s s a 0 s s a. X X X Max X X X st X X X 0 X X X 0 X X X 0 Max X X st X X 8 X X X 0 X X X Max X X X
o R 0 st X X X 8 X X X X X X 0 Max X X st X X X X X X 0 0 0 0 0 Z X X s X X X Î s X X X 0 6 Z X [ X s ] 6 X s Î s X [ X s ] X s X X X s X [ 6 ] : 0 6 Z 6 [ s s ] s s
Ms 0 X s s 6 X [ s s ] s s s Max X X st X X 0 X X 8 X X 0 0 0 0 0 Z X X Î s 0 X X X 0 s 8 X X X 8 0 0 [ ] 0 Z [ X s ] X X s 0 X X s X 0 0 Î s X s X [ ] : 0 8 Z [ s s ] s s 9
o R 06 0 X [ s s ] s s X s s 6 6