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Unformatted text preview: 1‘ (20 points) Solve the following linear programming problem by the revised simplex method,
using the smallest subscript rule for entering and leaving variables. Begin with the basis
[3, 4]. Give computational details at all iterations (except for the inverses of basis matri—
ces). State which basic feasible solutions you encounter are degenerate, and which pivots
you perform are degenerate. If you run out of space, continue on the back of the previous” page.
maximize 5:131 + 5232 + 2333 + x4
subject to 2:51 + 332 + 334 = 4
I1 + £62 + .233 = 2
(I31 , 332 3 £133 , $4 2 X "K
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4 a 9 N ”\ 9 :9 i ’3 R (Z) . Considera standard transportation problem with two origins, three destinations, no upper
bounds an the flows from any origin to any destination, and total supply 51 + 52 equal
to total demand d1 + d2 + d3. All supplies and demands are positive, and the cost for shipping from origin 1' to destination j is cij. (a) (4 points) Write a linear programming problem i andard equality prrn equivalent
to the transportation problem. You should have M 7— 3 z "' \\ I \I
.23
24 l k : Sr». K :\ 2— (b) (4 points) Verify that shipping an amount sidj/ (81 + 82) from origin 71 to destination j, for each 2', j, yields a feasible solution. 3 3 a 2: 3v ~ 6' S“ i t L a ‘ 2 a5; \‘J ‘2 5‘5 j:\ J :pSK/y ‘ \ \‘L
1"\ §\**< SL gceisk L a h . z ‘3” s OS ' S\~*SD‘ : J J:\Il‘3_
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(c) (6 points) Show that\there is an optimal solution in which two destinations each
receive a shipment from only one origin (the origin can be different for the two destinations). Partial credit for showing the same for one destination. in e») tame tweet s 5&9 H are “i
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Swww ﬁn,“ \osttk o5i‘an‘ ems“ no) Vatéth\@\ an Comvi‘pgﬁc i w) “(\VAT WKQVQQ cme Qiijkrv. (d) (6 points) State the dual problem to the problem in (3). Use variables ui, 2' z 1.2.
and 121,] = 1.23. man S\ m\ ~\ SLMKL’ ‘K 6&3“ ‘4': figs/1“: (£3 V3 w l + V'\ , 3 “Cu
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ML ~¥ \/3 l? "‘ (:13 (e) (6 points) Show that, if (ui. 2' 2 1, 2. '03, j = 1, 2,3), is a feasible solution to the dual?
so is (111' =: ui + 6, i = 1,2, {Jj == 123 — 6, j 2 1,2,3), for any (positive or negative)
number 6. How do the objective values of these two solutions compare? Ea..th SQVL3 0K 0’3 Semi w: 4: \/~ , Ewe? a A A
‘  ‘33 (when)?
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x J Q’ICLIVJ) ’ (f) (4 points) Show that the feasible region of the dual has no extreme points. Lek WM ‘G‘xg.’ v3 Mail Le. aha SAM, Mel cinema 5 :> o . Tkeux
(v3\ 3 V5) : .3: (Kai\g‘gcofj ~ﬂ) ~\ :Cﬁmk gleAeg’) i Slwvéx‘ﬂ “Ct Ls wk cm {asva 3. (a) ('5 points) State What is meant by a certiﬁcate of optimality for a feasible solution
to a linear programming problem in standard equality form. week «it a as a 3 i. x “l ’ A
ﬁaojLLA/W 1K ‘ v? as Snwm kg #0 For the remaining parts, consider the linear program maximize 6171 + 5132 + .733 + 5334 3331 + 3:2 "l’ 554 =3 4
(C2 + .733 ‘l’ 211:4 = 3
9:1 a $2 7 $3 a $4 2 O (b) (7 points) State the dual problem (D) of problem (P) and solve it graphically. (You
can use the graph paper on the preceding page.) » 39» (c) (3 points) What is the optimal solution of (D’), which is (D) with the last two constraints deleted? (d) (4 points) What is the standard form problem (P’) of which (D’) is the dual? 05}; “\‘w rewweeiﬁyﬁ §> “m.— 641 Q) s 00E» OWL “\vdh/m ‘L 2;”: 3 ‘
K ‘r' K .‘ . \
$0 \(\\ It. 3 ' 55’ ‘K C. 5 (f) (7 points) Find the optimal solution of (P) and prove that it is indeed optimal. 4. On the preceding page is an AMPL ﬁle: which is the same as the multi—commodity model
multino‘d except that the set of destinations DEST has been replaced with the index
set from 1 to .ndests and the set of products by the index set from 1 to nprods. The
reason is that the destination ndests refers to a dummy destination, with demand for
each commodity equal to the difference between the total amount of that commodity
available to be shipped and the total demand for the commodity at the real destinations. For each of the parts below, show how this model ﬁle (and / or possibly a data file) should
be modiﬁed to model the stated conditions. (a) (7 points) For each origin and each commodity, the amount actually shipped out
should be at least half of the amount available for shipping. saga ’W 9:95? it in ON“ p La unpack}; (b) (7 points) There is a particular product prodl and a constraint that, at each origin,
the amount of that product actually shipped out should be at least 10% of the total
amount of all products actually shipped out from that origin. sﬁgece "(‘0 Pas? $0M, it it. owe»)
m rimmed rm [L4, WM] >=~ 04 ¥:SM—N\ ﬁat v I CM 3 Fokme Frod\i ‘7 0) <3” ancakg MAL pram. kg ﬁg MK gkv \ (C) (6 points) Suppose that the origins are factories that produce the products that are
actually shipped out, and the production cost of one unit of product p at origin
{factory} 21 is pCOSti1,p]. The supply amounts ior each iactory and each product
are in fact production capacity limits. The goal is to minimize the total co bined
production and transportation cost. f‘tki‘aem {cost {QRV‘L 1., ‘A 7: m’kmﬂ “ 72%..Csu’o:
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 Fall '08
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