Linear Progamming- the Simple method with greater-than-or-equal-to or equality minimization problem Quantitative deciion making technique /5/6
Tableau form- dealing with greaterthan-or-equal-to contraint Ma 5. t. 8 where 5 5 5 Warehoue capacity 5 Minimum total production number of number of Aembly time Portable diplay unit of the Dekpro unit of the UltraPortable /5/6
Tableau form- dealing with greaterthan-or-equal-to contraint 5 8 5 a where 5 Portable diplay Warehoue capacity a Aembly time 5 a Minimum total are the lack variable production i the urplu variable and i the artificial variable for the contraint. /5/6
Initial baic feaible olution If a 5 5 then /5/6
/5/6 5 Tableau form for the initial imple tableau production Minimum total 5 Warehoue capacity 5 8 Portable diplay Aembly time 5 5.. 5 M a a t Ma Ma
Initial imple tableau a Bai C b 5 -M 5 5 8 5 a -M - 5 z j -M -M M -M -5M c i -z j 5+M +M -M /5/6 6
Improving the olution- determination of the variable entering to the baic olution a Bai C b 5 -M 5 5 8 5 a -M - 5 z j -M -M M -M -5M c i -z j 5+M +M -M /5/6 7
Improving the olution- determination of the variable leaving the baic olution a ratio Bai C b 5 -M 5 5 5-8 5 7.5 a -M - 5 5 z j -M -M M -M -5M c i -z j 5+M +M -M /5/6 8
Firt iteration of the imple tableau a Bai C b 5 -M - 75-8 -8 5-5 z j 5 5-5 5 5 c i -z j - 5 -M-5 /5/6 9
Simple tableau at the end of phae I Bai C b 5 75-8 5-5 z j 5 5-5 5 c i -z j - 5 /5/6
Firt iteration of phae II Bai C b 5.5 -.75 7.5 -.75.5.5 5.65.5 7.5 Z j 5.5 6.5 875 c i -z j 8.75 -.65 /5/6
Second iteration of phae II Bai C b 5. -. -.. 8..8 7 5 -.. z j 5.8 5. 98 c i -z j -.8-5. /5/6
Equality contraint When an equality contraint occur in a linear programming problem an artificial variable i alo needed to adopt to obtain tableau form and an initial baic feaible olution. /5/6
Eliminating negative right-hand ide value In circumtance where the value on the right-hand ide of the contraint are negative an equivalent contraint with a nonnegative right-hand ide value can be developed by multiplying both ide of the contraint by -. For a greater-than-or-equal-to / equality contraint multiplying by - create an equivalent le-than-or-equal-to contraint /5/6
Summary of creating tableau form If the original formulation of the linear programming problem contain one or more contraint with negative right-hand ide value multiply each of thee contraint by (-). For <= contraint add a lack variable to obtain an equality contraint. The coefficient of the lack variable in the objective function i aigned a value of zero. /5/6 5
Summary of the tep to create tableau form For >= contraint ubtract a urplu variable to obtain an equality contraint and then add an artificial variable to obtain the tableau form. The coefficient of the urplu variable in the objective function i aigned a value of zero. The coefficient of the artificial variable in the objective function i aigned a value of M. /5/6 6
Summary of the tep to create tableau form For equality contraint add an artificial variable to obtain the tableau form. The coefficient of the artificial variable in the objective function i aigned a value of M. The artificial variable become one of the baic variable in the initial baic feaible olution. /5/6 7
/5/6 8 An eample converting into tableau form 5 5.6667 6 6.5.t. 6 Ma
/5/6 9 An eample converting into tableau form 5 5.6667 6 6.5.t. 6 Ma
/5/6 An eample converting into tableau form 5 5.6667 6 6.5.t. 6 a a a a Ma Ma Ma
Initial imple tableau formulated a a Bai C b 6 -M -M a -M.5-6 6.6667 a -M 5-5 z j -M -.5M -M -6M M -M -M -M c i -z j 6+M +.5M +M +6M -M /5/6
Solving a minimization problem- the M&D Chemical eample Min where ubject to number of number of 5 Demand for product A litre of litre of 5 Total production 6 Proceing time product A product B /5/6
LP model for the M&D Chemical eample Ma - ubject to 5 Demand for product A 5 6 Total production Proceing time /5/6
/5/6 Tableau form for the M&D Chemical eample 6 5 5 ubject to Ma - a a a a Ma Ma
Initial imple tableau for minimization problem a a Bai C b - - -M -M a -M - 5 a -M - 5 6 z j -M -M M M -M -M -75M c i -z j -+M -+M -M -M /5/6 5
Firt iteration of the imple tableau for minimization problem a Bai C b - - -M - - 5 a -M - 5 5 z j - -M -M M -M -5-5M c i -z j -+M -+M -M /5/6 6
Final imple tableau for minimization problem Bai C b - - - 5 - - - 5 z j - - -8 c i -z j - - /5/6 7
Special cae- infeaibility Ma 5. t. 8 where 5 5 5 Warehoue capacity 5 Aembly time Portable diplay Minimum total production number of unit of the Dekpro number of unit of the UltraPortable /5/6 8
Special cae- infeaibility cae tudyfinal tableau- after two iteration a Bai C b 5 -M. -. -.. 8 5 -.. a -M -. -.5-8 z j 5.8+.M 5.+.8M M -M 98-8M c i -z j -.8-.M -5.-.8M -M /5/6 9
/5/6 Special cae- unbounded problem 5.. a a t Ma Ma
Special cae- unbounded problem after iteration Bai C b - 5 z j - c i -z j /5/6
Special cae- alternative optimal olution Ma. t. 8 where 5 5 5 5 Warehoue capacity Aembly time Portable diplay number of unit of the Dekpro number of unit of the UltraPortable /5/6
Special cae- alternative optimal olution- final imple tableau Bai C b 5 5 -.67 8. 66.67. -.67 6.67 z j 5 5 c j -z j - /5/6
Special cae- alternative optimal olution- after introducing Bai C b 5 5. -. -.. 8 -.. z j 5 5 c j -z j - /5/6