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Unformatted text preview: ENMA 421 Midterm Exam Spring 2011 Name: Agibmﬁgz A}; by}. [a 5 Instructions 1. You have until 5:00 pm. on Friday Feb. 18 to return this exam. Please emaii it to me (cgarciaQoduedu). Time iimit: 3.5 hours to be taken in one sitting. (Builtin 3&minute break ©). Only time
used actually taking the test counts toward this limit — use as much time as you need
afterward for scanning, typing, emailing, etc. Please state your start and end time in the designated space below. Show ALL worir. Explicitly state your assumptions. Make SURE to define your constants
and decision variables. Explain your reasoning where needed. This will enable maximum
partial credit to be given. The exam is open book and open notes. Calculators are permitted but Excel Solver {or other solver tools) is not permitted.
Please attach extra sheets if needed. MAKE SURE to put YOUR NAME on all extra
attached sheets. By turning in your exam you pledge that you have neither given nor received any
unauthorized help (i.e. any heip other than your book and notes). Additionally, you
pledge that you have accurately recorded the start time and end time. Good Luck! TimeStarted: Em Time Finished: I l :00 P W. QM” y/XDU 9roblem 1: A furniture manufacturer produces two types of tables (country and contemporary) using three types
of machines. The time required to produce the tables on each mathine is given in the following table: Available Hours Polisher Country tables sell for $350 and contemporary tables sell for 5450. Management has determined that at least 20%
of the tables made should be country and at least 30% should be contemporary. Formulate a linear programming
model to determine the numberof each table to make in order to maximize revenue. A“‘“ "$ ljrfd ‘iuoh Jcixt {1’15
, , ‘1 Md 7 my)“
anyn. [75 {(“pvu'( (fulﬁlyl ‘lr‘ik'; '— X
((:r\+(‘\.v)c~l»'! 4(1“ L l
1» Va
{’lf‘iL—i" (orjlvlaw'iﬁ  Sub)f(‘l v“, 'i’f'.) c L JO 7 _
Z J.( ‘11 1 1': \i {,1 ﬂIAIO éjjcoo
30): + H y .L. (moo Awfy + JFQYi [jam
O::l\(¥'l\/]_/X '.
O‘3(¥l\i) 5—7 gay .2 0 ﬂL‘xf; \I‘n{(.x(' ‘éjxijlouupﬁq “Ac‘(l\e( .J g r or W V a
i m ' . r r I”; r( \ (r; ‘, ‘\'l\\ I‘ 7‘ a is“ U 'v"\ l‘ t L
l 4 _ . WWW,“ mm"... mm .ﬁ. ._......m._
y "" Problem 2: Holiday Fruit Company buys oranges and processes them into gift fruit baskets and fresh juice. The
gradES offruit it has on are a state from 1 (lowest quality) to 5 (highest). The following table summarizes Holiday’s
current fruit inventory: m__ 3 5
s... at...” em_ Each pound of fruit turned into fruit baskets gives a $2.50 profit, and each pound turned into juice gives a 5175
proﬁt. The fruit in baskets requires an average quality (by total fruit weight) of at least 3.75, and the juice requires
an average quality of at least 2.50. Formuiate an LP model to determine how to maximize proﬁt. {If\Di‘jAfﬂj u a \‘a a 53/1“ 3
Wm (mi buska L‘ X
‘ JL; Hf. : n 5‘ l (7i....§v {1 j
ML "_;,,;;___. “ my” 40 vi _: avg?) g 4.. I.7§\/ _ )L) 1/ 7/ ((73
iww ;L 5. 9,000 (+7 4 3mm
(“:er L” I «a 5. $616 ﬁr.»
C, Jam—“H 1" diploma .
6 WM ‘ 2]: (‘0 X 3 § .7": Problem 3: There are three factories on the Elizabeth River: 1, 2. and 3. Each emits 2 types of pollutants! and 2. If
the waste from each factory is processed, the pollution in the river can be reduced. It costs $15 to process a ton of
factory 1 waste, and each ton processed reduces the amount of pollutant 1 by0.10 ton and pollutant 2 by 0.45
ton. It costs $1010 process a ton of factory 2 waste, and each ton processed reduces the amount of pollutant 1 by
0.20 ton and pollutant 2 by 0.25 ton. It costs $20 to process a ton offacmry3 waste, and each ton pracessed
reduces the amount of pollutant 1 by 0.40 ton and pollutant 2 by 0.30m. The state wants to reduce the amount
of pollutant 1 in the river by at least 30 tons, and pollutant 2 by at least 40 tons. Formulate an LP model that will
minimize the cost of reducing the pollutants by the desired amounts. Amﬁéi") Minamza Hm. C0
ollgluwls by 4+ 0‘? Vt’éutmx‘c) H’V’v [7
in Aw? [Mlou‘ J. g 4' v ,4 7O k'i‘iﬁ
liqlo“: a My 4* I
I E Problem 4: A company makes products A, B, and C and can sell these prodmts in unlimited quantitiaillalo
following prices per unit: A, $10; B, 556; and C, 5100. Producing a unit of A requires 1 hour of labor. Prbil'ming one
unit of 8 requires 2 labor hours plus 2 units of A. Ptodmim one unit 0“: requires 3 labor hours and 1 unit of B. Any
Athat is usedto produce Brannot besold; llkewlseany B madmprodmeCcannot besold. Atotaloﬂo labor hours are available. Formulate an LP model to maximize the W: resultus. AWOKKll") l5 Muyllmiig, CWmels Nile ._‘ l. ,
isn't: was R)“; (506 ‘ £Lr l hour
3k hen—9*” *’ A‘ A
g \noulﬁ 4 jig 1 \‘Tlorl" X5, *&¥3 “a My Mao 5,; non WiﬁOA‘“ 1 VIP.) Yup, 5kg; ¥Bpicpi20 Y
‘ L v” “()Q‘ Pmblem 5: Solve the folan Iinr program using thew solution method. Maximize z = 5A + 53
SubjectTo: As 100 BS 80 2A+4B 5 409 A. B: 0 ZISA‘VgB 4A4. 148;“ #00 M Ag (MoALB
'wkin A:O Bzmo
3 glad—JA' WNW B: o 4:;00 Von/v1 A : 6,000) 4 §(@) : {00
Few; g = {(0)4{(80)
: 1100
[ﬁnd (.3 D
3):. (TC “hf. : — A
w (5108) I re
:nh‘cpf+ : $00
;icr314aon 2—D : A = —3.P+ 3.00
CDiBAiatoo “loo 71°F” Ego RiAzmo
”  _: It‘eC A
3:40, A: {60 ﬂ“ 3° ° "
OpJnml ' sow“ a s Azjoa 8 '1'. Pawng '72 :— §(goo)+ {(50)
— Z: 7§O Problem 6: Gepbab Production Company uses labor and raw material to produce 3 products. The resounte i
requirements are as follows: 00.00"...) — lﬂﬂiﬂﬂﬂﬂﬂﬂllllllllIIIIIIIIEIIIIIIIIIIIIIIIIJIIIIIIIIIIIIIIIIIEIIIIIII Currently, 60 units of raw material are available. Up to 90 hoursof labor can be purchased at $1 per hour. To
maximize proﬁts, Gepbab solved the following LP (where x, denotes the number product 1' produced and L dens
the number of labor hours purchased} and obtained the following sensitivity analysis report: MAX 6X1 + 8X2 + l3X3  L
SUBJECT TO
2) L  3X1 ~ 4X2 — 6X3 >= 0
!(ENSURE L = SUM OF LABOR HOURS FOR EACH PRODUCT) 3) 2X1 + 2X2 + 5X3 <= 60
{(DON'T EXCEED RAW MATERIAL LIMIT) 4) L <= 90 !(DON'T EXCEED LABOR LIMIT) END
LP OPTIMUM FOUND AT STEP 0
OBJECTIVE FUNCTION VALUE
1) 97.50000 i
VARIABLE VALUE REDUCED COST
x1 0.000000 0.250000
x2 11.250000 0.000000
x3 7.500000 0.000000
L 90.000000 0.000000
Row SLACK 0R SURPLUS DUAL PRICES
2) 0.000000 —1.750000
3) 0.000000 0.500000
4) 0.000000 0.750000
N0. ITERATIONS: 0 RANGES IN WHICH THE BASIS IS UNCHBNGED: OBJ COEFFICIENT RANGES  VARIABLE CURRENT AIJDIEBLE ALLOWABLE “’,
COEF INCREASE DECREASE
X1 6.000000 0.250000 INFINITY
X2 8.000000 0.666667 0.666667
X3 13.000000 3.000000 1.000000
L 1.000000 INFINITY 0.750000 RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE 2 0.000000 18.000000 30.000000 3 60.000000 15.000000 15.000000 4 90.000000 30.000000 18.000000 a) What is the most the company should pay for another unit of raw material? cubs»
awn, 4 M + Muﬂ: W. (:é'uw c. La?“ b) What isthemostthecompanvshouldpavfotanotherhouroflabor?
: L  31K; '14“ —6x3‘2pr »
L ;__ 3(3) 4 2103.4”: "‘07" 3  5’ Clef/0T5 —‘_,,,,_, c) what would product 1 need to sell for to make it dairable for the company to produce it? 7 clot/65 d) If 100 hours of labor could be purchased, what would the Ms proﬁt be?
63:, +3)“ 4(?¥ _L_
6 3 3
Y, + gym/3 — m3 : 16? Cid/ms e) Find the new optimal solution if product 3 sold for $15. [email protected]£+ : xxf‘céﬁas ...
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 Spring '11
 Khajah

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