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Unformatted text preview: OPERATIONS RESEARCHI _ W
' ILE 321 HM 1’ / I ,/‘ ' During next three monthsAircoﬂgo. Must meet/(on time) the following demands for
air conditioners: month 1, 3007month 2, 400; month ~3, 500. Air conditioners can be produced in either New York or Los Angles. IT takes 1.5 hours of skilled labor to
produce an air conditioner in Los Angles and 2 hours in New York. It costs $400 to
produce an air conditioner in Los Angles and $350 in New York. During each month W each city has 420 hours of skilled labor available. It cost‘$100 to hold an air
conditioner in inventory for a month. At the beginning of month 1 Airco has 200 air
conditioners in stock. Formulate an LP whose solution will tell Airco how to minimize ' the cost of meeting air conditioner demands for the next three months. I _ b’ L}. Cit.” (mi/f kill/7W3 m [new L09 far men 319:) 3:1; 2)}: 1 . l ‘ ‘ I ' ‘ I ’ r ' . r 5/1” * "Ti 1 v r210":
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.3 c3 a CYPRUS INTERNATIONAL UNIVERSITY c I u ' ENGINEERING FACULTY C Y P R U S Department of Industrial Engineering
Course Code: ILE 321 Academic Year: 2007—2008 Fall
Course Title: Operations Research I Exam: Midterm II
Instructor: Nidai KORDAL Duration: 120 minutes
Student Name & Surname : Part I :
Student Number ' Part II :
Signature Grade : of 100 Part 1: Brieﬂy explain below questions. Use clear hand writing ! (5 points each)
1. Why do we need to use Methods to solve LP/IPs, instead of Graphical Solution Method? 2. What creates the difference between Simplex and Big M Method and leads to selection of one of them to solve an LP? 3. What is the purpose of Ratio Test in the Simplex Method? 4. For a given LP, if we cannot make a selection at the end of Ratio Test because of
negative results, what can be saying about the Model? 5. If any Artiﬁcial variable (a) is positive in the optimal Big M tableau, what can be saying
about the original LP? Why? Part II: Please solve the Model with suitable method. State the optimal solution or any
special cases obtained. Case of alternative optima, ﬁnd the line segment. (25 Points each) 1. The following LP model is used to determine the optimum product mix of interior (X1) and
exterior (X2) paints (tons) that maximizessdaily profit of Reddy Mikks Company. / Max z= 5X1+4X2
s. t. 6X1+4X2_<_24 LIM‘$~%/M'§I_ \ X1+2X236 K ﬂ , X1+X251 ‘ 1}
5m X232 ,/
3 XlURS,X2_>_O r a, z
2. Solve; MinZ=4X1+X2 " x ‘
$.11. 3X1+ X2: 3 I Lug: 2.» 4m3,(1/\/4W’) 4X1+3X336 5’ Ct "
‘1 X1+2X2 . g Xlaxzzo '
3. Solve; :
MaxZ=2X1+4X2 _ '
s.t. X‘ + 2x2 35 x1+ x254 £3 Xc‘qu’L'! Li X1 ’ X2 2 0 ‘I
,_ » M— ‘4!
l“ '“ 33 a ' " I g . GOOD LUCK © ‘_ 047. @7500? ILE 321 a) What is the difference between the Feasible Solution and Optimal Solution?
b) A Decision Variable can be Real or Integer type. Does this cause a difference in the
solution? How?
0) What is the difference between Multioptimality and Multiobjective? I
_ If a problem is classiﬁed as infeasible or Unbounded, what should be done? 6) State two assumptions of Linear Mathematical Pri~ and explain them. Wu, 1. The following LP model is used to determine the optimum product mix of interior (X1) and
“exterior (X2) paints that maximizes daily proﬁt of Reddy Mikks Company. V X1 = tons produced daily of exterior paint X2 = tons produced daily of interior paint Maximize Z= 5X1 + 4X2
Subject to: 6X1 + 4X2 5 24
X1 + 2X2 5 6
—X1 + X; S 1
X2 5 2
thz 2 0 By using graphical method ﬁnd all extreme points (state feasible or infeasible), draw
isoproﬁt line and ﬁnd the optimum product mix. 1. The Reggio Advertisement Company wishes to plan an advertising campaign in three
different media; television, radio and magazines. The purpose of the advertising program is to
reach as many potential customers as possible. Results of a market study are given below: If ‘ Prime 4/ ' Radio CostofAdvertisin unit $40,000 v $75,000 $30,000 $15,000 Number of Potential '  customers reached er unit 400,000 900,000 500,000. 200,000
Number of Women ff” _ i ' ‘
customers reached 300,000 400,000 200,000 100,000 The company does not want to spend more than $ 800,000 on advertising. It further requires that:
(a) at least Lmillion,,_exposures take place among women, (b)advertising on television be limited
to?500,000,1c) atj’least 3 advertising units be bought on daytime television, and two units during
prime time: (d)theﬂiﬁumb’er of advertising units on radio and magazine should be each be between
5 and 10. Formulate a LP model to ﬁnd the optimum advertising campaign that maximizes the
number of potential customers reached. 2. A Tomato Cannery has 5,000 kilos of grade A tomatoes and 10,000 kilos of grade B  Itomatoes, from which they will make whole canned tomatoes and tomato paste. Whole tomatoes 
must be composed of at least 80 percent grade A tomatoes, whereas tomato paste must be made , grade A tomatoes. Because of Governments Health production should be twice of tomato paste..If the company needs for grade A & B
tomatoes, they can buy at most 1,000 kilos of grade A for $0.06 per kilo and atmost 2,000 kilos
of grade B tomatoes for $0.04 per kilo. Whole tomatoes sell for $0.08 per kilo and paste sells for
$0.05 per kilo. Formulate a LP model to solve for how much of each product to make, if the
company wants to maximize proﬁt. . Each day Eastinghouse produces capacitors during three shifts: 8 A.M.4 P.M.; 4 RM.
Midnight; Midnight8 AM. The hourly salary paid to the employees on each shift, the price
charged for each capacitor made during each shift and the number of defects in each capacitor
produced during a shift are shown in the below table. Each of the company’s 25 workers can be
assigned to one of the three shifts. A worker produces 10 capacitors during a shift, but due to
machinery limitations, no more than 10 workers can be assigned to any shift. Each day at most
250 capacitors can be sold, and the average number of defects per capacitor for the day’s
production cannot exceed 3. Formulate an LP to maximize Eastinghouse’s daily profit. \ Defects
Shift Hourly Salary (per capacitor) Price
‘8 A.M.—4 P.M. $12 4 $18
4 P.M.Midnight $16 3 $22 4
Midnight8AM. $20 2 $24 4. BNU Co. manufactures sedans and wagons. The number of Vehicles that can besgldsageh of
the next three months are listed below. Each sedan sells for $8000 and each wagon sells for $9000. It costs $6000 to produce a sedan and $7500 to produce a wagon. To holda vehicle in inventory for one month costs $150 per sedan and $200 per wagon. During each month, at most 1500 vehicles can be produced. Production line restrictions dictate that during monthl: at least two thirds of all cars produced must be sedans. At the beginning/ofrnwgnthml, 200 “7:3 sedans Wagons are available. Formulate an, LPTliaf’caﬁmbewused to maximize , Co.’s proﬁt during next three moths. (25 Points) $005 AM” 2"“ “it”? " ‘395 i“ “1' WWW? ____,,_ . x
 N15C) {flit film—F1131 _, 7,3, {H i, of?» X“ '9' “J 4 W. 0) x ; ~ ~i~ a . Month 1 Sedans Wa ans \
»;;::1ioo.~ ~ 600  7 twin 3 “"1500 _ 700
1200 500 @EA Pastryibakes Kazandibi and Ekmek Kadayiﬁ. During any month, he can bake at most 65 {tipsserts The costs per. d demands foredesserts, which must be met on timeﬁare _
7 listed below. It (Est; 0 cen I hold a Kazandibi arid 40 cents to hold an Ekmek Kadyiti in
2 (7;; "x * * .r I i I :
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It “ inventory for a month. Formulate an LP to minimize the total cost of meeting the next three
months’ demands. (25 Points) ‘ ' ‘ 1 ' ii ‘ [ﬁx Month 2 I Kazandibi
Ekmek
ViKadayifi l
f 3. [5A cargo plane has three compartments for storing cargo: front, center, and back. These compartments have capacity limits on both weight and space, as summarized below: ' The four cargoes have been offered for shipment on an upcoming ﬂight as space is available. (Tons) (ma/Ton) ($IT on)
.
' m
— 400 Any proportion (tons) of these cargoes can be accepted. The objective is to .Hdgemtermine how much (if any) of each cargo shouldb'é‘a'c’cwepted aﬁa'ﬁdw to disuibute’Each amongh'the compartments to._ maximize the total roﬁt for the ﬂight. Formulate the linear programming
model for this problem. (25 Points) ‘  CENTER BONUS: (5 points) To maintain the balance of the airplane, weight capacity utilization (loaded
weight/weight capac'gr) should equally be distributed amongtheﬁ compartments. ‘ e.g. front utilization = center utilization = back utilization "' " Construct additional set of constraints fpr question 4 in order to maintain the balance of the
plane. '  _CYPRUS INTERNATIONAL UNIVERSITY
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 Spring '12
 NidaiKordal
 Operations Research, Optimization, Kilogram, Kilo

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