YORK UNIVERSITY. Faculty of Science and Engineering Faculty of Liberal Arts and Professional Studies MATH A Test #3.

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1 YORK UNIVERSITY Faculty of Science and Engineering Faculty of Liberal Arts and Professional Studies MATH A Test #3 July 25, 212 Surname (print): Given Name: Student No: Signature: INSTRUCTIONS: 1. Please write your name, student number and final answers in ink. 2. This is a closed-book test, duration- 75 minutes. 3. No calculator is permitted. 4. There are six questions on eight pages. Answer all the questions. 5. Show all work necessary to justify each answer you give. Clearly indicate each time you use the back of a page for your work. 6. Remain seated until we collect all the test papers. 7. Do the easiest questions first. GOOD LUCK! Question I Points I Scored I Total: 6

2 Name: Student No: 1. ( pts) (a) For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For this question, each correct answer is worth 1.5 point and each incorrect answer is worth -.5 (negative half!) point. If the number of incorrect answers is more than three times greater than the number of correct ones, then the total mark will be zero. If you don't know the answer, don't write anything. For this question only, you do NOT need to explain your answer or show your work. I Statement I TRUE/FALSE I Sensitivity analysis allows to determine how much can the RHS of a constraint be changed without changing of the optimality of a basic feasible solution. r-r'ru.e_ Sensitivity analysis is used to determine the range of values for an OFC of a basic variable, within which the optimal objective value remains unchanged. -f;i?sz._ Faiee- A weak duality may be used for getting an upper bound for optimal objective value of the primal (max) LP. If the dual LP is unbounded, then the primal LP is infeasible. In a balanced transportation problem with m sources and n destinations the number of basic variables is m + n. An assignment problem is a balanced transportation problem. 'Tvu.c 'T'rUL Fdce_ 1

3 Name: (b) Using the LINDO output bellow, answer the following questions: Student No: i. If the RHS of the second constraint is changed from 1 to 7, (and the first RHS is still 5) what is the new optimal objective value?.i.l r/je.ww.m- t't... fu R/{ $ of Jk "'-cf COtt <::A-vcut.;{. ANSWER: 4 _Lt 3 > UM-J ffj... du.af fv"e-'cl o.p1t-e.?.ta.of 1:- 'YCM--wi..:s -. i>. Hu; V\ 2 opt-.+ 3 (--1. )-=- 1'3,4. ii. If the OFC of X2 is changed from 3 to 5, what is the new optimal objective value? -ft._.t.. OF C!. o.f X t. s t:a-eve.m.e.cf ANSWER: J.."f. b 5-3 -=-(t).,wi.-t:e-l c<; t\... o re_. So) tk opfc.:l-1.<o. :t<j t.'> U-6 t o.ffe.clecp.f:l nht<- w...ai_ o s'e.c>h 'l,..e... V opt- ( ) If zopf-=- 1.i>.S + (5""-3)X -=- g.g ;-..t Lt.4 MIN X1+3X2+2X3 SUBJECT TO 2) 3Xl-2X2+X3>5 3) Xl+X2+X3 >1 4) -2Xl+3X2-X3>2 END OBJECTIVE FUNCTION VALUE 1) , g -+. -:::- e2. :f. b. VARIABLE Xl X2 X3 ROW 2) 3) 4) VALUE SLACK OR SURPLUS 3... REDUCED COST...6 DUAL PRICES (;-:- 4 RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE X X X3 2. INFINITY.6 RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE INFINITY 3 1. INFINITY

4 Name: Student No: 2. ( pts) Consider the LP: max z = 4xl subject to 8x1 6x1 +x2 +3x2 +2x2 +2x3 +x3 +x3 X1 7 X2, X3 < 2 < 8 2: and its optimal tableau f Z X1 X2 X3 S1 S2 RH S ] (a) Write the dual LP and determine its optimal solution. <fk.e_ ka.f_ L P t 1Mi (A_ w- -= c1 'cj 1 + g ra 2. S. {;. g1 + 6"'2.. Lt 3ce -r 62.-'cf 'c!2.. '>- 1., l<j 2.. f V'OiM._..tk_ t tj...eficjv.-l. rov-= \J<.:3)1 > ) [: 1= f-{ut) 'ijud. 1hevvt {U_ ettt:a..r &v.- LP [<t:pt 'i:f J = SI{ - = cj._ 1 L ] = c o J OM. J< Us opf--=- z_o pf 4. e.--f.m-.._, w-ot>f- = L( > tj (>f--.i, a... of 1 :!1-. 3

5 Name: Student No: (b) Determine the range of values for the objective function coefficient of x 3 for which the current basis remains optimal. Hint: Recall that row has the form z + (cbvb-l N- CNBV )xnbv = CBV s- 1 b. su.. & V "= \_'X 3,.&2-1? fu_ fa_r ANSWER: Cg 6 [, oe> J..e,- B -=- [ 1 o 1 1 -= r 3 ] - _ [L 1 1 j ) A L-.1, ) c N'B v - ( _J. Le.f eb '";:::.. [c!> o1 ) '1(!?:. E \'R..fk e_u_x-v S -, - \e..\m.(llu)) ott" CB\f rj- CNr1\f c2::::> r o] r 3 A ' - L-4 " o l > o C?> [ (':> o J lo 1\ \_-.t -J.. -1) j...\ _... ( c - 4 '3 3 - e. 3. J. J g C 3-4 > 3 c 3-1 &x' C. ll:e-1 cfw X.... C 3 E [ J eo) (c) Determine the range of values for the objective function coefficient of x 1 for which the current basis remains optimal. 1h Cu/frewi & rev tv\ ot.f,;l-uap c:: 1 ] trro'i>ljej.f:tt.e.j 1:.-e_ cure.j{"? ANSWERo C, C C, 6. 1_ i ) lj,-= 1 Ye.IM.4.-t. <{ ' so 1 M.u_Y'f CV/ALg u -+- (ju >- c -="> J. T- CA. (!"6 <1\ <12 :.- ".. '1c.,{;(-s:o.o, G] &. cu.v-v 4 s r L-Vl ort- t4a-e. (d) Were we add the constraint x1 + x2 + X3 ::; 5, would the current basis (i.e., x 3, s 2 ) remain optimal? Explain. 1fu_.rewt t's'" 'J'- ANSWER: "d -2. of> 4 'N< fit "Jc,opf = Xf =D x;p(--.l. 1/,u:s t- VeM- W"> f'e..m-t t-el Lb--ke.vL o..jj CU.:wt '"JC 1 -t- )(2- -t X.. 1 2).. C!.,tv/vJ -t..- Ve-u<eu- 4

6 Name: Student No: 3. (4 + 4 pts) Given the LP for minimizing the total transportation cost of a certain commodity: subject to min z = 9xu + 2x12 + Sx21 + lox x31 + 8x32 Xu + X12 X31 xu + X21 + X31 X12 + X22 xu, X12, x2l, X22, X3l, (a) Write down the corresponding transportation tableau. + X32 + X32 X32 > 1 = 2 = = s, )\ SL.$ 1J 4- '):_2.\ li_ 4o w _w ')c._ 3\ (b) Use the minimum-cost method to find a BFS for the problem. ANSWER: s" 1_2_ AKf M:fo $2_ rj.1). / s> 4 S.o 7 BfS t.5 x -=- 1- ;c. L \ -=- 2 a ) ) x,.,n ')C..? o l - ovj ) ;.2._ O.eR 1-k ofu..e e.. U&Y<;. OAd -=-cd. 4

7 Name: Student No: 4. (6 pts) Use Vogel's Method to find an initial BFS for the transportation tableau: I) _.., () 5+ ':)-+eio...;...j_-=-6-==- O+':)+lD -tcflo, <;;) Pv- t.'s (96t' p :t' l ') -::::... ") ':t 4.-= ')c -=- o 1 7C :3. 2 -=- x tr 2- -:::. 5' ) x 4 -:=... 3 JD G.RJ_ k s: Cl+".e... N f3, \f <k =-o. c - J...J -;;::. (,[;) ro 1/-o o { ol@-- 5. ( pts) Consider the transportation tableau b in part (a) with a basic feasible solution. A.3 (a) Calculate the reduced costs for all nonbasic variables. Is the BFS optimal? Explain. -+ }) + 55' -t- 4 -=-- { b '$"-:=- 4 + '1 + ANSWER: () -4: w + w' ) TP L<;. cu;(( "#=-'DNBVs -=- t1tt k.. -(l+l+-t-) = l{g- +=-g. kt u. =- caf_j..cvfe_ 15- J &-J.J 7 U.4, '15".;, Lt2-, u-lf a..._j u?:> u-,.-:::: t- u-..2-= :> 'l5:- 5" L re..ctnv"k ' l,..( k >- ul.(=--\ - /J ' D 1 D2 D3 D4 ')W'IM-c.A. (t:;l._. u-o 51 - ()...2.-"='4 u \::_ \2. \ \.::_ e \.2 /X) _ ol\ =-n l ' L{. + 1.r.- e... =o t.. J LJ 6

8 Name: Student No: fit.' (b) Is there an alternative optimal solution? If no, explain why. If yes, perform one pivot on the tableau from part (a) to find it and write your solution in the empty tableau below. ct, "'-' c...ta-el c 3i =- u3 + 15', - c13 =- ANSWER: 'J ==- G :, = Ct.M f:t._e_y.e. t-6 :. (3,1)- (3/t)- (,4) -(:L,3) --(4)3) - (.ti}1) Ju-- 3,) TP /A tz4 --"b-e. PerVtA- CL., u-e... olcu.-k- x23 t- o.- c'la_- tfsc_ia} Va. t-'- oph. ke.ur- ti il!u- Sl S2 S3 S4 Dl D2 {5 \.2. jl(:)\!_ \22. \g_ 5" 6 25 D3 D4 \.2_ \...::_.3 1' \-2. 5o\!_ z.tff--= :t\15' " + S..3 -t-13..w +!J g.!),/-* 6'.5.) -==- 1 Q., 15-, (c) What two conditions are necessary and sufficient to make a feasible solution a basic feasible solution? 4) rft,._,_. of c:_ vo..,h fk_ JIU..+- k. -1 == 4 -rl(- 1 -=- ;, J {1./ tbef& Ck c_ va.-.-,. 7

9 Name: Student No: 6. (7 pts) A company is taking bids on four construction jobs. Three people have placed bids on the jobs. Their bids (in thousand dollars) are given in the table Jobl Job2 Job3 Job4 Personl Person2 Person3 51 * * 45 where * indicates that the person did not bid on the given job. Person 1 can do only one job but persons 2 and 3 can each do as many as two jobs. Use the Hungarian method to determine the minimum cost assignment of persons to jobs. u_ pw--.s c2_. 3 C:.Q..u... e.o.o-l ANSWER: Jo 'f fo iura d dy, 1 CA>k.<;...,fev e.ad, oj M two dt-"ffe -ed -d?ci -/e1{jolen- '"-fr rr-y ll-l "- -fiu._ Uff Ct.>6.f.}r";; 1-tr 1}- -. ot z p Pot Pot' P3 5' 5 51 '-16 4g, 4g 'J9'9 P3' So 4(; Lc.t 4o o \ttu.. '"'-- ; Lt 'f '-fj_ Lro '-r'r ggg 44 9q9 45" its ,-r, ; r"\ tr\ -...:: v '-../ \.../ (\) 1 95i, c <b (D gsg ( St z <D 4 cb qt g, p J._ '-( cb '.i. ol CJ5g :( 9s-g q4'3 1 3 s- 94g CD 3 5 :11 4 <: s- p!--1--"-'---+t''-"-1 > p f-7--hf--h- PJ.:--f.b,I--H---'+-+-tHt

YORK UNIVERSITY. Faculty of Science and Engineering Faculty of Liberal Arts and Professional Studies. MATH B Test #1. September 28, 2012

YORK UNIVERSITY. Faculty of Science and Engineering Faculty of Liberal Arts and Professional Studies. MATH B Test #1. September 28, 2012 YORK UNIVERSITY Faculty of Science and Engineering Faculty of Liberal Arts and Professional Studies MATH 1013 3.00 B Test #1 September 28, 2012 Surname (print):------------ Given Name: Student No: --------------