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ORC711S - Operations Research 313 - 1st Opp - June 2022 |
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nAmlBIA unlVERSITY
OF SCIEnCE Ano TECHnOLOGY
FACULTYOF ENGINEERINGAND SPATIALSCIENCE
DEPARTMENTOF Mining and Process Engineering
QUALIFICATION : Bachelors of Engineering in Mining Engineering
QUALIFICATION CODE: BEMIN
LEVEL: 8
COURSE CODE: OPC 711S
COURSENAME: OPERATIONSRESEARCH
SESSION: JUNE 2022
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 100
EXAMINER(S)
MODERATOR:
FIRSTOPPORTUNITYQUESTION PAPER
Lawrence Madziwa
Dr Mallikarjun Rao Pillalamarry
INSTRUCTIONS
1. Answer all questions.
2. Read all the questions carefully before answering.
3. Marks for each questions are indicated at the end of each question.
4. Please ensure that your writing is legible, neat and presentable.
PERMISSIBLEMATERIALS
1. Examination paper.
THIS QUESTION PAPER CONSISTS OF 5 PAGES (Including this front page)
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1. All loaded mine trucks traveling from the mine pit to the plant are required to stop at a
weigh and scan station. Trucks arrive at the station at a rate of200 per 8-hour shift, and the
station can weigh, on the average, 220 trucks per shift.
a. Determine the following
I. the average number of trucks waiting
[3]
II. the average time spent waiting and being weighed at the station by each
truck, and
[5]
III. the average waiting time before being weighed for each truck. [5]
b. Truck drivers find out they are queueing at the station longer than 15 minutes
as planned, on average, they will start driving slower thus reducing mine
productivity. The mine estimates that it loses $10,000 in downtime per year for
each extra minute the trucks must wait at the station. A new set of scales would
have the same service capacity as the present set of scales, and it is assumed that
arriving trucks would line up equally behind the two sets of scales. It would cost
$50,000 per year to operate the new scales. Should the mine install the new set
of scales?
[8]
2. You are responsible for transporting five items on a limited space of 13 tons from
Windhoek to Swakopmund. There are five different items that your company can
transport between Windhoek and Swakopmund. Each item has a weight in tons, a net
profit in thousands of dollars, and an unlimited number of items that are available for
shipping as shown in Table below. Use dynamic programming to determine how
many of each item should be shipped to maximize profits.
[20]
Item
Weight
1
7
2
5
3
4
4
3
5
1
Profit I Unit ($)
9
4
3
2
1/2
Number Available
unlimited
unlimited
unlimited
unlimited
unlimited
3. A new Uranium mine, in Namibia has decided, as a matter of urgency, to build a new
road as an upgrade to the existing one to facilitate efficient transportation of materials.
The road works has been divided into five stages which are built simultaneously. Within
Namibia, there are six large companies to undertake the construction of any of the five
stage and each company has been invited to submit a tender for each stage of the project.
The tenders (in millions of Namibian dollars) are as follows:
Company Stage
1
2
3
4
5
A
49
84
63
82
68
B
53
92
62
No bid
67
C
54
86
67
78
68
D
46
86
62
76
No bid
E
57
94
66
83
70
F
50
82
65
80
72
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a. Assuming that none of the companies is large enough to undertake the work of
more than one stage, advise the mine how the five contracts should be allocated.
What is the minimum total cost for the project?
[10]
b. The mine is installing power lines to different areas on the mine. The mine wants
to minimize the total length of the power line. The network of all positions
requiring electricity is shown below with the associated distances between all
points. Recommend how to best connect all the positions in the network. [10]
c. Discuss three network models that can be used to resolve some of the mining
challenges and give examples of how you can apply them in mining.
[6]
4. Show that the average inventory of an inventory model with backorder is:
Average inventory= (Q-S)2/2Q
[10]
b. Patterson Electronics supplies your mine with micro processors for your processing
plant. Currently, Patterson orders components from various suppliers. One of the
components is ordered in batches of 150 units. It has been estimated that annual
demand for these components is 250. Furthermore, carrying cost is estimated to be $1
per unit per year. For the order policy to be optimal, determine what the ordering cost
would have to be.
[5]
c. Flemming Accessories produces paper slicers used in offices and art stores. The
mini slicer has been one of the most popular items: Annual demand is 6750 units. The
company produces the mini slicers in batches. On average the company can
manufacture 125 units per day. Demand for these slicers during the production
process is 30 per day. The set up cost for the equipment necessary to produce the mini
slicers is $150. Carrying costs are $1 per mini slicer per year. How many mini slicers
should the company manufacture in each batch?
[4]
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5. The following details pertain to a job, and indirect cost for the project is N$ 1100 per day.
Scheduled the project to optimal cost.
Job Predecessor
Normal
Crash
Time in hrs Cost in N$ Time in hrs Cost in N$
A
-
B
C
C
-
D
A,B
E
C
F
D
G
E,F
H
D
3
1400
2
2100
6
2150
5
2750
2
1600
1
2400
4
1300
3
1800
2
1700
1
2500
7
1650
4
2850
4
2100
3
2900
3
1100
2
1800
a. Draw the network diagram and determine the project completion time? [7]
b. What is the total cost of the project?
[1]
c. Given that the project has been given a deadline by management, to what extent
can the project be crashed and what is the corresponding cost? [7]
End of Exam
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TABLE1
Stan<l:irdnonn:ilcun-c arc:i.~
~kd
=,a P,fl., cl
'
0
'
0.00
0.01
0.0:? 0.03
o.~ 0.05
0.06
0.07
o.os 0.09
0.0 0.5000 0.50-IO 0.5080 0.5120 0.5160 0.51'19 0.5231) U.5279 0.5319 0.5359
0.1 ll.5398 ll.5-138 U.5478 0.5517 0.5557 0.55% 0.5636 ll.56 5 0.5714 0.575'
0.2 0.571)
0.5:m 0.5!i71 0.5•JI0 0.5948 U.5987 0.6026 0.60M 0.6103 0.6141
0.3 0.617') 0.6217 0.6255 0.6293 0.6331 0.6'68 0.6-106 0.6+13 0.6-180 0.6517
0.655-1 0.6591 0.6628 0.6664 0.6700 0.67 6 0.6772 0.6S0S ll.6'+l 0.6S7')
0.5 0.61)1.5 0.61)50 0.6%5 ll.7011J 0.7054 0.70SS U.712' 0.7157 0.7l9o
0.6 0.7257 0.7291 0.7 24 0.7357 0.7381) U.7422 0.7454 0.7-l.6 0.7517 0.7549
0.7 0.7580 0.761l U.76-12 0.7673 ()_77(}.l 0.773-l 0.776-1 0.7794 0.7823 0.7S52
0.X ll.7881
0.9 0.8159
0.7910
0.SI '6
0.7<J13J
0.8212
0.7'J67
0.82'8
0.7995 0.8023
O.!L6-I 0.;'281)
0.8051
U.S 15
0.S07S 0.SI~
0. 3-10 0. 365
0.81'3
0..'J '<J
1.0 lUi413 0.S.1'S U.S46I 0.!i4'5 0.8508 0.8531 0.8554 ll.!i577 0. '599 U.,'6_1
I.I 0.XMJ lUl665 0.S6S6 0.S70S U.8729 0..'74') o.:mo 0.S790 U.S,'10 O.,'I"')
1.2 0.!i849 0.S!i61J o.sss. 0.89()7 ll.SIJ25 llS944 lUl%2 0.S9S0 0.,'997 0.9015
1.3 0.9032 0.
o.•J(Ki6 0.90S2 0.'J()<)<J U.91!5 0.9131 0.')147 0.')16_ 0.91?7
IA 0.9192 0.9207 ll.9222 0.92'6 ll.9251 ll.9265 ll.9279 O.'J2lJ2 0.9306 U.9.il'J
1.5 0.'JJ:2 o.•J~4.5 0.1J357 0.9'70 0.9 .'2 0.9'94 0.9406 ll.94IS 0.9-112) 0.'1441
1.6 0.9452 0.946 0.9474 o.94S4 U.9495 0.9505 ll.9515 U.952 0.9535 0.9545
1.7 0.9554 0.956-1 U.9573 U.9'S2 0.')591 0.959') 0.960,' U.9616 ()_96_5 U.963'
l.!i ll.1)6-1l 0.9M9 0.%56 0.9664 0.%71 U.967.' 0.%,'6 U.969' 0.%r.,<J 0.9 06
1.9 o.•m3 0.IJ711J 0.9726 0.9732 0.'>73S 0.'J7+l 0.9750 0.97. 6 0.97(11 0.976
2.0 lJ.9772 0.9778 lJ.97S' 0.97SS 0.'J7'J3 ll.979S 0.9S0J 0.9 '0' U.'J,12 0.9.'l7
2.1 0.'Jli21 o.•J,_6 ll.9S'lJ 0.9S34 0.'JSJS 0.'J!i-12 ll.9S46 0.9S5lJ U.'J'>1 0.98.57
2.2 0.9li6I 0.9S6-I U.9X6S 0.9S71 0.9875 0.9'7.' 0.9.'Sl ll.9S:4 0.9 '8
lJ.9 'I}()
2.J ll.9S9J 0.'J:i'.16 U.9S1JS 0.9901 0.')9()4 0.')')()6 0_<)')()9 U.9911 0.')913 ll.'JJl6
2.4 0.'J9IS ()_l)'}_l) 0.9922 0.')')25 U.'J927 0.')<2J9 0.9931 lJ.'J<J3_ 0.91J34 0.')<)36
2.5 0.'NJ,
o.•./')41 0.')')43 0.')9-15 0.1JH6 0.'JHS 0.')1.>49 0.')')51 0.IJ<J-
2.6 0.995 ll.9'J55 0.9956 ll.'J'J57 ll.9959 U.'N60 ll.'JJ61 0!J'J62 U.9963 0.9%-l
_.7 0.9%5 0.'JJM U1.JJ67 0.'J'J6!i 0.1J969 (}_l)'}i() 0.'./'J7l 0.')972 0.99 J lJ.')<J4
2.!i 0.'JIJ74 0.9'J75 lJ.9'J76 0.9977 0.')977 0.9978 0.')'J79 lJ.99 9 lJ.99.'0 0.99 'I
2.9 0.99Sl 0.9'nL 0.'JJS2 ll.1JIJS' lJ.'J9S-l 0.'.l'JS-l ll.9'JS5 U.91J,'5 0.99S6 lJ.')'J'6
J.0 U.99S7 0.9'J.'7 0.'.1')87 ll.9'J.'S 0.')9SS lJ.')IJ,'1) ll.'.l'JS1J 0.1)9.'9 U.9990 lJ.9')1)()
.I
0.91)1)()
0.')')91 ll.'.l'J9I 0.')')'JI U.99'J2 (J_')l)'J2 0.9')92
0.')')')2 U.91)')3 U.'.1993
J.2 0.99'J.I 0.9')9.1 0.'JJ1H 0.'11)1).j 0.')9114 0.9'J'J4 0.9')94 ll.99'J5 0.'JW5 0.'J<J95
3.' 0.1J91J. 0.1J<1J5 0.'JJ95 ll.')')'J6 0.91)96 U.99% 0.'.1')% ll.9'JJ6 0.9996 ()_<)<J97
3.4 l).')')')7
LJ.'JJ97 ll.9'J'J7 ().')')')7
0.'J91J7 (J_\\)(J<J7 ll.'.l'J•)7 0.')')')7 0.'J9'J7 0.999.'
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Queuing Model, Single Server Formulas
J p0
=
Prob
[system
empty
i~.
(idle)
=I- A
µ
L = average number
q in the queue
µ(µ - A)
L = average number = /4
in the system
µ- A
W = average time=
A
4 in the queue µ(µ - A)
W = average time = I
in the system µ - }"
Note:
)1, is the arrival rate.
µ is the service rate.