Prelim2_Solutions

# Prelim2_Solutions - 1‘(20 points Solve the following...

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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 x - ‘ 7 z 0 \ ’6\ t T 2; \f1\{L B \L3\Lgrﬂ%§ (:\ 01/A¥B[1 ’\ . 5— n3 - Us 3» ‘ i i i, i“ w(“\$75 o emit :\n N I J (J- A“? 503 c" « s. 0 . = 2, l \ s §> __ “7‘9 W’ A315 ‘95: [\ £313 [5L] z 3%? is as isms 2». as L \ r L . v 3 a The, 347 as} la): lea} (kc:th ‘iﬁtv‘t'ho t>o 53% Femic ' W ‘ W- Hal 3 etikﬂﬂee {3:31 igigkésix » ‘ A.‘ t *“zc ed :5] \[i J Ellis k_ Vt”: ~ 3; i f ~ , Q \l 30 \ 033 \r j 3] 5 _ 2 ‘x 1 \ " z \ ’ 2:} a: saw with t ale Ll «L Lil “t M w, l 93 x" : j7>\1:txl,l)lq20‘ll(‘l) :1 ,"EP’UD j: * Em; . _\ T f p q . k A i . a y } [elk 51\$le 7: CB \ K _ 2 [ 1 i a I ‘r 4 a 9 N ”\ 9 :9 i ’3 R (Z) . Consider-a 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 gce-isk L a h . z ‘3” s OS ' S\~*SD‘ : J J:\Il‘3_ (LA 33:5; S>\‘\~,SL 9M e o w» :sé: > 0, S}; Minna, 3‘ (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 WQDKQSK; So Wis. 96% 92AM Obj now}, One ConﬁWk La is {awed}! gt», (1mK R Qév afﬁw‘ﬂ _SO\U.}H<3I\ k4; s 11K \993 Weet‘LAAeS. \Q «UL ngfmaedji bx {affix}; 1*7:¥L’~‘~é Vargxkgg‘ on COAVM‘Q/MN: ‘5: ALL owe, (Realm I 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 UK\ A‘ V1 “"Cik W-\ ’3< V3 ’5 ~QL3 no; 4e v\ ‘ a; “CM UKL “Cl-Z“; 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)? t , _, , :5 ii \ N (We absuﬁbwﬂ fwd/“L 0§ be“ V3) “\5 L 3 _ Z s;(w*\4\g) 4r JZ:\ (vi ~Tj ‘i=\ . O 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: my“ L (we OK\C\\ p in {Mogul}; (Cw 3d \\ rdesbeilg {most —~ Um i3 <« e “Jeri eat-s (3 «Twain»; R90 Spar/3&3 '?~Co§i: k“ in «Mix ...
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Prelim2_Solutions - 1‘(20 points Solve the following...

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