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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
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m. :1 DJ“:
4' 3
X1“ X5 4Yﬁ+5:' FINA, : (l, 3) (Gang awiiaiiu 012‘) x 2(3xa4)’; ‘3' 3\ /
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O CO 350 2 1 (b) Solve this same linear program, reproduced below, by the Direct 2phase Method. This
method does not use artiﬁcial variables in phase 1. minimize $2 _ $3
Subject to $1 ’ $2 7 $3 0.
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5. .
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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—11—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
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)(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“ (uml“ rmm {5;
hX( + ar;¥+*, “+5‘rKXK‘1 ' "*3ij; =3 bf A150; LYQ LRGJJ ‘lkf" 0‘4 r ORCC‘FNt, ﬁﬁz‘gim r21»; {5 ‘_ .— (ixlg +(&)(‘4r V'"yCKX,( 4, ..4 CK). =Z‘U ALL" (NANS, 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' ‘ .1 211 ‘
(Jr (“5‘ ... _ “‘4 <) X
(a '5K‘ 0. + 4' OX“. 1 an: ’ < 0r, ’ ’—
’ ‘  A , £5315.
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[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 seekf 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 Hunl 6:70f' I I5 Q N M”"+FMAJ‘“ k (9 H‘... a: , .1
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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.
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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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