Sectin 6-2: Simplex Methd: Maximizatin with Prblem Cnstraints f the Frm ~ Nte: This methd was develped by Gerge B. Dantzig in 1947 while n assignment t the U.S. Department f the Air Frce. Definitin: Standard Maximizatin Prblem in Standard Frm A linear prgramming prblem is said t be a standard maximizatin prblem in standard frm if its mathematical mdel is f the fllwing frm: Maximize the bjective functin P = c1x] + C2X2 +... + cnxn subject t prblem cnstraints f the frm a]x] + a2x2 +... + anxn ~ b, b;:::: 0 with nnnegative cnstraints XI' X2', xn ;::::0. In Ql f the Sectin 5.3 lecture we used a gemetric methd t slve the fllwing prblem:* Maximize P = 9x] + 3x2, subject t 3x] + x2 ~ 30 {3X] + x],x2 7x2 ;::::0 ~ 84 We saw that the maximum slutin ccurred at the cmer pint (7,9) and (10,0) and, therefre, at all pints n the bundary line fr the regin cnnecting (7,9) and (10,0). ]S (0, 12) 10 5 s Cmer Pint (0,]12 (7,9) (10,0) (0,0) P=9x] +3x2 36 90 90 t- t (3,3) ~-f-h-i-h (0, 0)1... 5 (l0, 0) 15 * In the Sectin 5.3 lecture we used X and y instead f Xl and X2. Nte that the inequality y ~ -tx + 12 fq1, Sectin 5.3, culd be written as X2 ~ -tx] + 12 r 3xl+7x2 ~ 84. Similarly, y ~ -3x +30 fq1, Sectin 5.3, culd be written as X2 ~ -3x] + 30 r 3x] +x2 ~30. Sectin 6.2, p. 1
Q1. Use the Simplex Methd t find the ptimal slutin t the prblem Maximize P = 9x1 + 3x2, subject t 3x1 + x2 ~ 30 {3XI + xi' 7x2 2:: ~ 084 In the Simplex Methd we cnvert the cnstraints frm inequalities t equalities by intrducing slack variables. A) Intrduce slack variables Sj and Sz t cnvert the first tw inequalities t equalities. ~X,+ 7)<.,).-+.5, =8Y 3 '1-, -I- )(.}. "f 5d-.:: 30 The initial system fr the linear prgramming prblem is a set f equatins, cnsisting f the prblem cnstraints written as equatins with slack variables, tgether with the bjective functin rewritten in the frm -c\x\ -C2X2 -.. '-cnxn + P = O. B) Write the initial system fr this prblem. j,xl;' 7x~-tS, = 5'<f 3)( I + X 6 +Sd-. =- 30 d" +- P = 0... 11'1-3)C.}.. X,.) XaJ 5,./ SlI P.2 0 Definitin: Basic Variables and Nnbasic Variables:Basic Feasible Slutins Given a system f linear equatins assciated with a linear prgramming prblem: The variables are divided int tw (mutually exclusive) grups as fllws: Basic variables are selected arbitrarily with the ne restrictin that there be as many basic variables as there are equatins. P is always ne f the basic variables. The remaining variables are called nn basic variables. The nnbasic variables are always set equal t O. If the values f the basic variables are 2: 0, the crrespnding slutin is said t be a basic feasible slutin. We are interested in finding basic feasible slutins fr the linear prgramming prblem. These will crrespnd t cmer pints f the graph f the feasible slutin regin. We will d this using what is called a simplex tableau. After setting up the initial simplex tableau the tableau is mdified by selecting basic and nnbasic variables and perfrming pivt peratins until the ptimal (maximum value f the bjective functin) slutin is btained. The initial simplex tableau is an augmented matrix f the initial system fr the linear prgramming prblem (as cnstructed in part b) abve). Sectin 6.2, p. 2
C) What is the initial simplex tableau fr the abve linear prgramming prblem?. 'X, X.l 5,.5~ P.3-7~3:l~(:17 f 0 0 ) B''11 Prcedure: Selecting Basic and Nnbasic Variables fr the Simplex Methd Given a simplex tableau, Step 1. Determine the number f basic variables and the number f nnbasic variables. These numbers d nt change during the simplex prcess. Step 2. Selecting basic variables. A variable can be selected as a basic variable nly if it crrespnds t a clumn in the tableau that has exactly ne nnzer element (usually 1) and the nnzer element in the clumn is nt in the same rw as the nnzer element in the clumn f anther basic variable (This prcedure always selects P as a basic variable.) Step 3. Selecting nnbasic variables. After the basic variables are selected in step 2, the remaining variables are selected as the nnbasic variables. (The tableau clumns under the nnbasic variables usually cntain mre than ne nnzer element.) ~,f h~ v~..::: /U> {~ «f~-.c-c.. D) Select the basic and nnbasic variables fr the initial simplex tableau. Find the initial ~ basic feasible slutin (by setting the nnbasic variables equal t 0 and slving fr ~ ~asic <::J (5) Sj variables). S2 P 13~ ~- b~ v~ v~r..~j~ S); aa.(.. 5~1 X" PX ~. SJVf' ~ = ~ ~ S2 3 1 0 1 0 30 _ " _. _ '0 isl3 P -9-37 01 0 01 841 0 ~/ SJ -S'<{J ~",,-3~ P=c>,. (N" I '1 J We will find the ther basic feasible solutins V1 by perfrming pivt peratins n the simplex tableau. Prcedure: Selecting the Pivt Element Step 1. Lcate the mst negative indicatr in the bttm rw f the tableau t the left f the P clumn (the negative number with the largest abslute value). The clumn cntaining this element is the pivt clumn. If there is a tie fr the mst negative, chse either. Step 2. Divide each psitive element in the pivt clumn abve the dashed line int the crrespnding element in the last clumn. The pivt rw is the rw crrespnding t the smallest qutient btained. If there is a tie fr the smallest qutient, chse either. If the pivt clumn abve the dashed line has n psitive elements, there is n slutin and we stp. Step 3. The pivt (r pivt element) is the element in the intersectin f the pivt clumn and pivt rw. Nte: The pivt element is always psitive and is never in the bttm rw. The entering variable is the variable at the tp f the pivt clumn. The exiting variable is at the left f the pivt rw. Sectin 6.2, p. 3
E) Lcate the pivt element in the initial simplex tableau. Determine which variables are the exiting and entering variables. x2 Sj S2 P XI ~ 7 1 0 0 84 f'ff3:: ~y 5.,,).~ 1 0 1 0 30 JD 1-.3 :::: 10-3 0 0 1 0 )(\=/0) )(d-.':::::c» 5, =5'<-(j S.;l. =~ JO,..,. 1D Jn~~ (I? When there are n mre negative indicatrs in the bttm rw f the simplex tableau the ptimal slutin has been fund. Prcedure: Stpping the Simplex Methd If it is nt pssible t select a pivt clumn, the simplex methd stps and we cnclude that the ptimal slutin has been fund. If the pivt clumn has been selected and it is nt pssible t select a pivt rw, the simplex methd stps and we cnclude that there is n ptimal slutin. Sectin 6.2, p. 4
Q2 (#44A, page 314). Manufacturing: resurce allcatin. A cmpany manufactures threespeed, five-speed, and ten-speed bicycles. Each bicycle passes thrugh three departments fabricatin, painting & plating, and final assembly. The data is shwn in the table belw. 100 Ten- Five- A speed 70 45 35 vailable 120 130 Da per 43 Maximum 580 Labr-Hurs Labr-Hurs per Bicycle Hw many bicycles f each type shuld the cmpany manufacture maximize its prfit? What is the maximum prfit? L.d- )({ : rw 13-4-p-u~ )(~.:. II II s-- ~ )(,,3:: Il., Jp-~ 1J~rrtL7'~~J./? k h, per day in rder t WI2- ~ -Iv ~ P=cx,+70X,)..i-/~6X3 s~ -Iv E X,+t.(x~ +-5~ ~ (ilo b-x,+3 x~4- S-X3= (~ If X (-+-.3 K.;. + s~!::. (~ :r~~~~ XI}X,J..JXJ.20 3;«;- 4X.l1-SX3 + S, S-X] -.. 3x,;}.--l- S~ +.$.). '-I X I -t- 3 X..)..,.. ~ + s~ 1.;10 (.30 ( --- - I «0 0 D I.= I~ ::::/36 = I~t> - 80 't,-70 Y.J.-I()O~ + P -= 0,.A X I :.)( J. ;:: 'X -3 :::. ()) S 1 -= I J.O/ S,. ::::. (3 0~~ 51. -=/ ~/ Pc C), Sectin 6.2, p. 5
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