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CO-350-1031-Midterm_exam

CO-350-1031-Midterm_exam - Surname L First Name M—...

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Unformatted text preview: Surname: L First Name: M— Name: (Signature) L Id.#: _————————— CO 350 — Linear Optimization Midterm Examination Professor D.H. Younger February 26, 2003 Qéection 1: 2:30 MWF D Section 2: 10:30 MWF INSTRUCTIONS: 1. Write your name and Id.# in the blanks above. Indicate the Section in Which you are registered. 2. If you require additional space for your solution, please use the back of the previous page. Indicate clearly Where your additional work is located. 3. No calculators are permitted. Mark Awarded CO 350 1 1 (a) Solve the following linear program by the ordinary simplex method, using the 2—phase method if needed. You may use the entering rule of your choice. minimize £152 —’ 533 subject to 331 + 3562 + x3 = 2 — .732 + 373 S _1 331 a $2 a $3 2 0 mm. --w + x3 @V fi£, X‘ +})(1 4. X3 —’— glad: wank“; (401.046! XUXAI ‘5‘ x7;0 .‘ {M (0 warms! Avx‘vlknj LP '5' X5‘ Y5 +Yq +Y5: I .2“ “Q (MW @45be £65)": 9={ l, 5% pNI/l’;(5,"l> m. :1- DJ“: 4' 3 X1“ X5 4Yfi+5:' FINA, : (l, 3) (Gang awiiaiiu 012‘) x 2(3xa4)’; ‘3' 3\ / “A ”“3 Wl~7<r§ 930 C) 4 a v )‘ 5N./ iii—«1:, ,mw-,-...._..m__wm, HMW’KL’O: g #g, He gamut/«r (78/ Q70 1 warm er team, ‘a [10] / Z ‘P O CO 350 2 1 (b) Solve this same linear program, reproduced below, by the Direct 2-phase Method. This method does not use artificial variables in phase 1. minimize $2 _ $3 Subject to $1 ’ $2 7 $3 0. ”“2" vi. *Y3 5. . ~YJ\ TYg4XL‘ :-| Mid/’5 dud ' SILLFL WK A-¥§+' (51’0“? Cr) 0~l w 9.. -OC> CO 350 3 [15] 2 (a) Carry out one full pivot of the solution by the Revised Simplex Method for the linear program maximize CTZL‘ subject to A3: = b, :L' 2 0, with data as follows: cT=[~11—1-1—11] 100106 9 A=31—4002;b=2. 102012 6 Start with the basis B = [2, 4, 5]T. Stop When you reach the new basis 3' and its solution X31, unless the solution terminates during this first pivot. O ‘ 0 2 4 2 ~ X;"b.’ ,. ‘ '— .. I w :- Mm d « Mm l , g I l 5 kg - at?" L I )4! me5 W bay?) ‘3 {0:}. '1 6% g 3 4 XQX: [7- %(\)‘ j? V 9 X1; £3 g y 716 )(5:6 “13(i)’ [10] CO 350 4 2 (b) Consider the linear program which has the following tableau: B: [4, 2, 5]T J As you see, the objective function row is missing. (i) Suppose that the objective function is z: —x1+x2—x3—z4—x5+$6. Find the objective function row for the above tableau. (ii) We suppose. that B0 = [4, 5, 6]T is the reference basis. Show that the above tableau is, or is not, lexifcographically feasible. If it is not feasible, stop. If it is feasible, find the pivot to a new basis that gives the greatest lexicographic improvement (but do not carry out that pivot). [If you were unable to answer part (i), use 10 [101001Iz—0] as the objective function row for this part. l7 llanflloflf'kafg 1;;th , CO 350 5 3. A tableau corresponding to basis B may be written as 2 ijj = z — 17 jeN _ Z (—lijflij = bi, 2 E B, jEN ZEj 2 0, jEN. [10] (a) Let (r, k) be a pivot from B to new basis Bl. Derive a formula for 0;, the reduced cost of 337, in the new tableau, in terms of the coefficients of the present tableau. (gun-,1" bLQflni a. K A» axd’l’ "f E) Y; 2 Z' a... _' .— ‘V X" I , Ki )0 On“ (um-l“ rmm {5; hX( + ar;¥+*, “+5‘rKXK‘1- ' "*3ij; =3 bf A150; LYQ LRGJJ ‘lkf" 0‘4 r ORCC‘FN-t, fifiz‘gim r21»; {5 ‘_ .— (ixlg +(&)(‘4r V'"yCKX,( 4, ..4 CK). =Z‘U ALL" (NAN-S, GU’ Mk) (cu lc LuaMSC “an?“ +23; «”4" ‘4 XK+”‘+»..»£—'-fl_ ‘xkf—EL an: an fi'm a’K TM) 9M5 ‘”£ “(a)"; 0}??ka tench)» “v”. C. ..~ Cicaf‘y' ‘ .1 2-11 ‘ (Jr (“5‘ ... _ “‘4 <) X (a '5K‘ 0. + 4' OX“. 1 an: ’ < 0r, ’- ’— ’ ‘ - A , £5315. fl‘w—n-wns ‘Hw'l 7‘1} X0 )‘qu (ovrfspf: C, MW be ; (’, - ( ’04 C V .11: ‘ ‘ {K (K [10] (b) Suppose that pivot (72k) satisfies the requirements of the simplex algorithm. Prove that the variable acr that leaves B to form basis Bl cannot be the entering variable in the pivot that takes B, to the next new basis. A; above we Cam seek-f Cr‘: a—fm 0\n< 1 EV hum“, Ex 5 Q _ We aka law/v 74:14 E, ‘S 7 a [h k6 quolwc ‘ ' itch Hun-l 6:70f' I I5 Q N M”"+FMAJ‘“ k (9 H‘... a: , .1 .cK,a,.(m, .— w, «.c/ ,a an: SlellX aljtVl-Mm/ we ”.AYJF CLWO$€ an I" 46 (”la fiucA+lq+ Crl > a. l’L—wrrl {vi ‘3 13. ”l. G 1 (“awe—l- bc Ewelvml “LP“ 61‘0”“? 4/601, Xr (an/wt M 0m WW) Vania Mo Tu JrLe proi' blink) @143 41% twelve“) 174/4}, l20l CO 350 6 4. Three students, Xavier, Yee and Zinka, have entered a linear programming contest, for which they have been given one common computer terminal. They are assigned 6, 8, and 10 linear programs to solve, respectively. Using the terminal, they solve problems at the rates 2, 4, and 3 problems per hour, respectively. When they solve problems by hand they do it at the rates 1, 2, and 3 problems per hour. Each person solves only his or her own problems and each must solve at least one problem by machine. The objective is for the last of these problems to be solved as soon as possible. Formulate this as a linear program. Include all constraints. Le‘l X; p XX X5 At ‘l'LQ «mam 'l O“: M Xa‘ljul YCé, 2mlc‘ spl’rd dwbtj («bums Niflfc lfirl‘j . 9 mgggwww We Iva/1+ lo Within-{2c 6X. +3X; 4» NY; 3'"? \‘OMMi’v‘t 'l‘W WW.“ All") [1&3 " I’M/HAMS NMCMIB” 9x. w» strict {ix-1M; 43 MA ML? 2. each I‘n) é 39L! 64"“1 an ‘13 Wfi¢hfi 23‘:* LIXJ'JEY} > Z Far satay?) M's fiua N7», 2%. 4')».Kg 1’ 3x3 ye) Ag; “(A Fm“ “”4" “W" "“ “M ‘w‘ wife"; as X}. was Lave) LP 1‘? 1 1L éhrl +llx‘+ 3X3 >67 QOlQl“; lodaM/(Q'? 4U" Q 2‘, +231 +?Y3 2’59 )9, x,“ x} are? Xsh/f/B’) \ fl CO 350 Use this page for trial calculations. ...
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