Second Semester Examination 

2016/2017 Academic Session 

June 2017

Duration : 3 hour

PPSM, Pusat Pengajian Sains Matematik

Repository@USM

UNIVERSITI SAINS MALAYSIASecond Semester Examination2016/2017 Academic SessionJune 2017MAT251 - Introduction to Operations Research[Pengantar Penyelidikan Operasi]Duration : 3 hours[Masa : 3 jam]Please check that this examination paper consists of EIGHT pages of printed materialbefore you begin the examination.[Sila pastikan bahawa kertas peperiksaan ini mengandungi LAPAN muka surat yangbercetak sebelum anda memulakan peperiksaan ini.]Instructions: Answer all four [4] questions.[Arahan: Jawab semua empat [4] soalan.]In the event of any discrepancies, the English version shall be used.[Sekiranya terdapat sebarang percanggahan pada soalan peperiksaan, versi BahasaInggeris hendaklah diguna pakai].. . .2/−- 2 -[MAT251]Question 1(a) Table Tsar Sdn. Bhd. manufactures 2 models of tables with 4 legs. Both models usethe same tabletop, but model A has shorter legs (1.5 feet) while model B has longerlegs (2.5 feet). It takes 0.10 labour hours to cut and shape a short leg from stock,0.15 labour hours to do the same for a long leg, and 0.50 labour hours to produce atabletop. An additional 0.30 labour hours is needed to attach a set of legs to a tabletopfor either model. The estimated profit is RM30 for each model A sold and RM45for each model B sold. Table Tsar has access to unlimited material for tabletops butonly have 500 feet of leg stock and 80 labour hours at their disposal. Help Table Tsarformulate a Linear Program that maximises profit, assuming that they sell all tablesmade and there are zero leftover table legs or tabletops.[13 marks](b) Solve the following Linear Program:maximise z = 2x1 + x2 + x3subject tox1 + x2 + 2x3 = 8−3x1 + 2x2 + 5x3 ≥ 16x1, x2, x3 ≥ 0.[12 marks]Soalan 1(a) Table Tsar Sdn. Bhd. membuat 2 jenis meja berkaki 4. Kedua-dua jenis meja meng-gunakan ‘tabletop’ yang sama, tetapi model A berkaki pendek (1.5 kaki) manakalamodel B berkaki panjang (2.5 kaki). Kerja memotong dan mengukir satu kaki pen-dek mengambil masa sebanyak 0.10 jam, kerja yang sama untuk satu kaki panjangmengambil masa 0.15 jam, dan kerja penghasilan satu ‘tabletop’ mengambil masasebanyak 0.50 jam. Kerja penyambungan kaki-kaki meja pada satu ‘tabletop’ pulamengambil masa sebanyak 0.30 jam. Anggaran keuntungan bagi setiap model A di-jual adalah RM 30, dan RM 45 bagi setiap model B dijual. Table Tsar boleh menda-patkan bahan yang tidak terhad untuk membuat ‘tabletop’, tetapi hanya boleh mem-peroleh stok bahan kaki meja sebanyak 500 kaki, dan 80 jam bekerja. Bantu TableTsar merangka suatu model Pengaturcaraan Linear yang memaksimumkan keuntun-gan, dengan anggapan bahawa mereka menjual semua meja yang dibuat dan tiadalebihan kaki meja dan ‘tabletop’ dibuat.[13 markah]. . .3/-- 3 -[MAT251](b) Selesaikan model Pengaturcaraan Linear berikut:memaksimumkan z = 2x1 + x2 + x3terhadapx1 + x2 + 2x3 = 8−3x1 + 2x2 + 5x3 ≥ 16x1, x2, x3 ≥ 0.[12 marks]Question 2Chick Chick Boom Sdn. Bhd. manufactures organic chicken feed and duck feed from corn,wheat and oats. Currently, 40kg of corn, 30kg of wheat and 40kg of oats are available. Asack of chicken feed sells for RM40 and requires 1kg of corn, 1kg of wheat and 2kg of oats.A sack of duck feed sells for RM50 and requires 2kg of corn, 1kg of wheat and 1kg of oats.Chick Chick Boom knows they can sell all feed produced. Chick Chick Boom’s Linear Pro-gram that maximises revenue is as follows:Let:x1 = Sacks of chicken feed produced.x2 = Sacks of duck feed produced.The Linear Program:maximise z = 40x1 +50x2 (Revenue)subject tox1 + 2x2 ≤ 40 (Corn)x1 + x2 ≤ 30 (Wheat)2x1 + x2 ≤ 40 (Oats)x1, x2 ≥ 0.The optimal Simplex tableau for this Linear Program is as follows:Basic x1 x2 s1 s2 s3 Solutionz 0 0 20 0 10 1200x2 0 1 2/3 0 −1/3 40/3s2 0 0 −1/3 1 −1/3 10/3x1 1 0 −1/3 0 2/3 40/3. . .4/-- 4 -[MAT251](a) Based on the optimal tableau given, how much chicken and duck feed should ChickChick Boom produce? What is the revenue if they sell those quantities of feed?[3 marks](b) Half of Chick Chick Boom’s corn supply was stolen. Would the current basis still befeasible? If yes, what is the new production plan?[11 marks](c) Chick Chick Boom would like to increase the price of duck feed to RM60. Would thecurrent basis still be optimal? If yes, what is the new revenue?[11 marks]Soalan 2Chick Chick Boom Sdn. Bhd. membuat makanan organik untuk ayam dan itik menggunakanjagung, gandum dan oat. Mereka mempunyai stok jagung sebanyak 40kg, gandum sebanyak30kg dan oat sebanyak 40kg. Seguni makanan ayam dijual dengan harga RM40, dan memer-lukan 1kg jagung, 1kg gandum dan 2kg oat. Seguni makanan itik dijual dengan harga RM50,dan memerlukan 2kg jagung, 1kg gandum dan 1kg oat. Chick Chick Boom boleh menjualsegala makanan yang mereka keluarkan. Model Pengaturcaraan Linear Chick Chick Boomyang memaksimumkan pulangan adalah seperti berikut:Biarkan:x1 = Bilangan guni makanan ayam dikeluarkan.x2 = Bilangan guni makanan itik dikeluarkan.Model Pengaturcaraan Linear:memaksimumkan z = 40x1 +50x2 (Pulangan)terhadapx1 + 2x2 ≤ 40 (Jagung)x1 + x2 ≤ 30 (Gandum)2x1 + x2 ≤ 40 (Oat)x1, x2 ≥ 0.Tablo optimum Simplex bagi model Pengaturcaraan Linear di atas adalah seperti berikut:Asas x1 x2 s1 s2 s3 Penyelesaianz 0 0 20 0 10 1200x2 0 1 2/3 0 −1/3 40/3s2 0 0 −1/3 1 −1/3 10/3x1 1 0 −1/3 0 2/3 40/3. . .5/-- 5 -[MAT251](a) Berdasarkan tablo optimum yang diberi, berapa banyakkah makanan ayam dan itikpatut Chick Chick Boom hasilkan? Berapakah pulangan yang didapati jika kesemuamakanan ayam dan itik dapat dijual?[3 markah](b) Separuh daripada stok jagung Chick Chick Boom telah dicuri. Adakah asas semasamasih tersaur? Jika ya, apakah pelan produksi baru?[11 markah](c) Chick Chick Boom mahu menaikkan harga makanan itik kepada RM60. Adakah asassemasa masih optimum? Jika ya, berapakah nilai pulangan baru?[11 markah]Question 3Mooptech Sdn. Bhd. manufactures satellite phones for consumer use. Mooptech has plants inPulau Pinang, Kulim and Johor Bahru. Currently, there are 30 phones in Pulau Pinang, 40 inKulim and 30 in Johor Bahru. Mooptech’s customers Andy, Sandy, Mandy and Randy haveput in their orders of 20, 20, 25 and 35 phones respectively. The delivery costs per phone (inRM) are as follows:From \To Andy Sandy Mandy RandyPulau Pinang 7 10 14 8Kulim 7 11 12 6Johor Bahru 5 8 15 9(a) Formulate Mooptech’s problem as a balanced transportation model.[3 marks](b) Use the North West Corner Rule on the transportation model in (a).[3 marks](c) Solve Mooptech’s problem using the Transportation Simplex Method. Use the answerin (b) as the initial Basic Feasible Solution. Report the optimal delivery plan and cost.[11 marks](d) Mooptech has to change couriers for the Pulau Pinang - Mandy delivery, and theydo not want to recalculate their delivery plan from scratch as this would incur morecosts. Which of the following couriers should Mooptech choose and why?• FedEz (RM20/phone).• UPZ (RM15/phone).• SkyZet (RM12/phone).• GDez (RM8/phone). [8 marks]. . .6/-- 6 -[MAT251]Soalan 3Mooptech Sdn. Bhd. membuat telefon satelit untuk pengguna akhir. Mooptech mempunyaikilang di Pulau Pinang, Kulim dan Johor Bahru. Kini, terdapat 30 telefon di Pulau Pinang,40 di Kulim dan 30 di Johor Bahru. Pelanggan-pelanggan Mooptech, Andy, Sandy, Mandydan Randy telah membuat tempahan sebanyak 20, 20, 25 dan 35 telefon masing-masing. Kospenghantaran bagi setiap telefon (dalam RM) adalah seperti berikut:Dari \Ke Andy Sandy Mandy RandyPulau Pinang 7 10 14 8Kulim 7 11 12 6Johor Bahru 5 8 15 9(a) Rangkakan suatu model pengangkutan seimbang bagi masalah Mooptech.[3 markah](b) Gunakan Petua Sudut Barat Laut pada model pengangkutan dalam (a).[3 markah](c) Selesaikan masalah Mooptech menggunakan Kaedah Simplex Pengangkutan. Gu-nakan penyelesaian dalam (b) sebagai penyelesaian asas permulaan tersaur. La-porkan pelan penghantaran optimum dan kos.[11 markah](d) Mooptech perlu menukar syarikat penghantaran bagi penghantaran Pulau Pinang- Mandy, dan mereka tidak mahu membuat semula pelan penghantaran kerana iaakan menyebabkan penambahan kos. Syarikat yang manakah patut Mooptech pilihdan kenapa?• FedEz (RM20/telefon).• UPZ (RM15/telefon).• SkyZet (RM12/telefon).• GDez (RM8/telefon). [8 markah]. . .7/-- 7 -[MAT251]Question 4Action Comics Studios is about to begin production on its most important movie this year,Condiment Man. Production begins this week, and the producer Kenneth Feike has decidedto use CPM/PERT to plan his project. Kenneth has identified the activities necessary to occurbefore the movie can be released. The activities, their precedence relations, durations andcosts are as follows:Activity A B C D E F G HPreceding Activities - - A A B B C, E D, FDuration (weeks) 5 3 4 6 5 7 9 8Cost ($million) 20 10 16 25 22 30 25 30(a) Construct an arrow diagram for this project. Show the earliest and latest event timeson the diagram.[5 marks](b) What is the cost (in $million) of producing Condiment Man? How long (in weeks)before Condiment Man can be released when production starts? Which activities arein the critical path?[3 marks](c) Kenneth recently learned that Action Comics Studios’ rival BS Comics Studios arealso releasing a movie when Condiment Man is scheduled to be released. After a longdiscussion with Action Comics boss Stan Zee, the project duration has been changedto 15 weeks. Given the following information, help Kenneth determine the least costlyway of crashing his project.Activity A B C D E F G HCrash Time (week) 3 2 2 3 4 4 5 6Crash Cost ($million) 30 20 24 43 30 48 45 44Report the list of activities crashed and their reduction in number of weeks, alongwith the new cost to produce Condiment Man.[17 marks]. . .8/-- 8 -[MAT251]Soalan 4Dalam masa terdekat, Action Comics Studios akan memulakan proses penghasilan filem ter-penting tahun ini buat mereka, Condiment Man. Proses tersebut bermula minggu ini danpenerbit filem Kenneth Feike telah membuat keputusan untuk menggunakan CPM/PERT un-tuk merancang projeknya. Kenneth telahpun mengenal pasti aktiviti-aktiviti yang perlu di-jalankan sebelum filemnya boleh ditayang di pawagam. Aktiviti-aktiviti tersebut, aktiviti-aktiviti pendahulu, tempoh dan kos adalah seperti berikut:Aktiviti A B C D E F G HAktiviti Dahulu - - A A B B C, E D, FTempoh (minggu) 5 3 4 6 5 7 9 8Kos ($juta) 20 10 16 25 22 30 25 30(a) Lakarkan gambarajah aliran projek ini. Tunjukkan masa terawal and terlewat buatsetiap acara di atas gambarajah.[5 markah](b) Berapakah kos (dalam $juta) untuk menghasilkan Condiment Man? Berapa lamakah(dalam minggu) sebelum Condiment Man boleh ditayangkan di pawagam? Apakahaktiviti-aktiviti yang genting?[3 markah](c) Kenneth mendapat tahu bahawa pesaing Action Comics Studios, BS Comics Studiosjuga akan mengeluarkan suatu filem pada masa yang sama dengan Condiment Man.Setelah berbincang panjang dengan bos Action Comics, Stan Zee, tempoh projek di-ubah kepada 15 minggu. Diberi maklumat di bawah, bantu Kenneth mengenalpasticara memampatkan projeknya dengan kos terendah.Aktiviti A B C D E F G HTempoh Mampat (minggu) 3 2 2 3 4 4 5 6Kos Mampat ($juta) 30 20 24 43 30 48 45 44Laporkan senarai aktiviti yang dimampat berserta tempoh pengurangan dalam minggu,dan kos penghasilan baru buat Condiment Man.[17 markah]- ooo O ooo -

MAT251 - Introduction To Operations Research

[Pengantar Penyelidikan Operasi] Jun 2017

Second Semester Examination

2015/2016 Academic Session

June 2016

Duration : 3 hour

       UNIVERSITI SAINS MALAYSIA  Second Semester Examination 2015/2016 Academic Session   June 2016  MAT 251 – Introduction to Operations Research [Pengantar Penyelidikan Operasi] Duration : 3 hours [Masa : 3 jam]     Please check that this examination paper consists of EIGHT pages of printed material before you begin the examination.  [Sila pastikan bahawa kertas peperiksaan ini mengandungi LAPAN muka surat yang bercetak sebelum anda memulakan peperiksaan ini.]   Instructions:  Answer EIGHT (8) questions.  [Arahan: Jawab LAPAN (8) soalan].     In the event of any discrepancies, the English version shall be used.  [Sekiranya terdapat sebarang percanggahan pada soalan peperiksaan, versi Bahasa Inggeris hendaklah diguna pakai.]            …2/- [MAT251] - 2 -    1.  Standardize the following Linear Programming formulation:   Maximize Z  =  5x1  +  20x2   +  15x3   Subject to 2x1  + 5x2   +  x3  +  2x4        20  3x1  + 2x2   +  4x3  +   x4       -10 x1  + 3x2   +  5x3  +   x4    =   15   x1   0,   x2   0,  -5   x3   5,  x4 is unresctricted     [ 7 marks ]    1.  Piawaikan rumus Pengaturcaraan Linear yang berikut:   Maksimumkan  Z  =  5x1  +  20x2   +  15x3   Terhadap  2x1  + 5x2   +  x3  +  2x4        20  3x1  + 2x2   +  4x3  +   x4       -10 x1  + 3x2   +  5x3  +   x4    =   15    x1   0,   x2   0,  -5   x3   5,   x4  tak tersekat     [ 7 markah ]    2.  Consider the following Linear Programming formulation:  Maximize  z  =    8x1  +   6x2     Subject to  2x1     +     4x2     20   (Resource 1) 4x1     +     2 x2    40    (Resource 2) x1      -     2x2      8    (Resource 3) -x1     +       x2      3   (Resource 4)           x1 ,  x2    0   (i) Solve the problem using the graphical method.  (ii) Get the range of Resource 2 so that the optimal solution remains.  (iii) Get the range for the coefficient of the objective function for variable x2 so that current solution remains basic.   [ 15 markah ]      ...3/- [MAT251] - 3 -    2.  Pertimbangkan rumus Pengaturcaraan Linear yang berikut:  Maksimumkan  z  =    8x1  +   6x2     Terhadap  2x1     +     4x2     20   (Sumber 1) 4x1     +     2x2     40    (Sumber 2) x1      -     2x2      8    (Sumber 3) -x1     +       x2      3   (Sumber 4)           x1 ,  x2    0   (i) Dapatkan penyelesaian dengan menggunakan kaedah bergraf.  (ii) Berikan julat sumber 2 supaya penyelesaian asas kekal.  (iii) Berikan julat pekali fungsi matlamat pembolehubah x2 supaya penyelesaian asas yang sedia ada kekal.   [ 15 markah ]    3.   Consider the problem,  Minimize Z  =  5x1  +  20x2   +  15x3       Subject to 2x1  + 5x2   +   x3       10  x1               +   x3   =   4            x1   0,  x2   0,   x3   0               Using the two-phase method, do two iterations towards solving it. [ 8 marks ]    3.  Pertimbangkan masalah berikut,    Minimumkan  Z  =  5x1  +  20x2   +  15x3       Terhadap  2x1  + 5x2   +   x3        10  x1                +   x3    =   4          x1   0,  x2   0,  x3   0            Dengan menggunakan kaedah dua-fasa, lakukan dua lelaran ke arah menyelesaikan masalah ini. [ 8 markah ]    ...4/- [MAT251] - 4 -    4.    A factory produces three products X, Y and Z using raw materials A and B. The  following table gives the necessary data for the problem:                          The labour time per unit of product X is three times that of product Y and twice that of product Z. The entire labour force of the factory can produce an equivalent 200 units of product X. Market requirements specify the ratios 4:3:5 for the production of the three respective products. Formulate the problem as a linear program. [ 10 marks ]    4.    Sebuah kilang menghasilkan tiga produk X, Y dan Z menggunakan bahan mentah A and B. Jadual berikut memberikan data berkenaan masalah tersebut:                          Keperluan masa buruh seunit produk X adalah tiga kali produk Y dan dua kali  produk Z. Keupayaan keseluruhan masa buruh kilang ini adalah bersamaan dengan menghasilkan 200 unit produk X.  Keperluan pemasaran menetukan nisbah  pengeluaran 4:3:5 untuk ketiga-tiga produk tersebut. Rumuskan masalah ini sebagai suatu pengaturcaraan linear. [ 10 markah ]             …5/- Raw material Requirements per unit Availability         X           Y            Z  A B          3           2            6          5           3            8    5000        8000 Minimum demand Profit per unit (RM)        300       350        250         40         30          50  Bahan mentah Keperluan per unit Kedapatan         X           Y            Z  A B          3           2            6          5           3            8    5000        8000 Permintaan Minimum  Keuntungan seunit (RM)        300       350        250         40         30          50  [MAT251] - 5 -    5. A problem to determine the level of production of 3 products (x1, x2 and x3 )  using 4 resources (the constrainsts are in the form of   and the slack variables are  respectively s1, s2, s3, and s4) results in the following tableau. Z represents profit  (in RM).                    Basic x1 x2 x3 s1 s2 s3 s4 Solution Z 0 0 3 0 1 0 0 32 x1 1 0 -2 2 -1 0 0 2 x2 0 1 1 0 -2 0 0 5 s3 0 0 0 -1 0 1 0 4 s4 0 0 1 3 -1 0 1 3  (i) Describe the status of each resource (binding, non-binding, scarce or surplus)?  (ii) What is the remainder of each of the resources above?  (iii) Find the range of  resource 1 that would not affect the solution above.  (iv) How much will  the profit reduce if 1 unit of product 3 is produced?  (v) How much will change in  per unit profit of product 2 before production of product 3 starts?  (vi) How much change in resource 2 that would make the above solution infeasible?  (vii) Get an alternative basic optimal solution. Give the general expression for all of the alternative optimal solutions.  (viii) If an additional constraint, 3x1 + 2x2 + 8x3   20 is imposed, will the original solution remain optimal? [15 marks]    5. Suatu masalah untuk menentukan aras pengeluaran 3 jenis produk (x1, x2 dan x3 )  yang menggunakan 4 sumber (dengan kekangan asal semuanya berbentuk   dan  pembolehubah lalainya ialah s1, s2, s3, dan s4) menghasilkan tablo maksimum berikut. Z ialah keuntungan (dalam RM).                    Asas x1 x2 x3 s1 s2 s3 s4 Penyelesaian Z 0 0 3 0 1 0 0 32 x1 1 0 -2 2 -1 0 0 2 x2 0 1 1 0 -2 0 0 5 s3 0 0 0 -1 0 1 0 4 s4 0 0 1 3 -1 0 1 3    ...6/- [MAT251] - 6 -    (i) Nyatakan status setiap sumber (terikat, tak terikat, berkurangan atau berlebihan)?  (ii) Berapakah baki setiap  sumber bagi penyelesaian di atas?  (iii) Dapatkan julat amaun sumber 1 supaya tidak menjejaskan penyelesaian di atas.  (iv) Berapakah jumlah keuntungan akan menyusut jika sebanyak 1 unit produk 3 dihasilkan?  (v) Berapakah keuntungan seunit produk 2 perlu berubah untuk mula menghasilkan produk 3?  (vi) Berapakah amaun sumber 2 perlu berubah untuk menjadikan penyelesaian di atas tak tersaur?  (vii) Dapatkan satu penyelesaian alternatif yang asas. Berikan ungkapan umum untuk semua penyelesaian optimum alternatif.  (viii) Sekiranya suatu kekangan tambahan  3x1  + 2x2 + 8x3    20 dikenakan, adakah penyelesaian asal masih optimum? [15 markah]    6.  A machine operator processes 5 types of items on his machine each week, and must choose a sequence for them. The set-up cost per change depends on the item presently on the machine according to the following table:  Items  E D C B A 8 6 14 8  A 8 6 12  8 B 10 14  12 14 C 14  14 6 6 D  14 10 8 8 E  How should the machine operator sequence the items on his/her machine in order to minimize the total set-up cost?  [ 15 marks ]       ...7/- [MAT251] - 7 -    6.  Seorang pengendali jentera memproses 5 jenis barangan menggunakan jenteranya setiap minggu dan mesti memilih suatu jujukan baginya. Kos penyediaan bagi setiap tukaran bergantung pada barangan yang berada di atas jentera seperti jadual berikut:   Jenis Barangan  E D C B A 8 6 14 8  A 8 6 12  8 B 10 14  12 14 C 14  14 6 6 D  14 10 8 8 E  Bagaimanakah pengedali jentera ini menyusun barangan bagi jentera supaya dapat meminimumkan jumlah kos penyediaan?   [ 15 markah ]    7.  Seven types of packages are to be delivered by five trucks. There are three packages of each type, and the capacities of the five trucks are 6, 4, 5, 4, and 3 packages respectively. Set up a maximum-flow problem that can be used to determine whether the packages can be loaded so that no truck carries two packages of the same type.   [ 15 marks ]    7.  Terdapat tujuh jenis pakej perlu dihantar dengan menggunakan lima buah trak. Terdapat tiga pakej bagi setiap jenis pakej, dan kapasiti kelima-lima trak adalah 6, 4, 5, 4, dan 3 pakej masing-masing. Sediakan suatu masalah aliran maksimum yang dapat digunakan untuk menentukan sama ada pakej-pakej tersebut dapat dimuatkan supaya tiada trak yang membawa dua pakej dari jenis pakej yang sama.  [ 15 markah ]           ...8/- [MAT251] - 8 -    8.  Consider the list of write-up activities that are involved in your final year project.  Activity Description Immediate Predecessors Duration (weeks) A Identifying Research Context - 5 B Defining Problem Statement A 8 C Literature Review B 10 D Data Collection B 5 E Analysis B 4 F Interpreting Results E 6 G Final Write-up C,F 3  Draw the project network, and find the following information: critical path, total float and free float of each activity.  [ 15 marks ]    8.  Pertimbangkan senarai aktiviti yang terlibat dalam penyediaan projek tahun akhir anda.   Aktiviti Keterangan Aktiviti Sebelumnya Tempoh (minggu) A Mengenalpasti Konteks Penyelidikan - 5 B Penentuan Kenyataan Masalah A 8 C Kajian Literatur B 10 D Pengumpulan Data B 5 E Analisis B 4 F Mentafsirkan Keputusan E 6 G Penulisan Projek C,F 3  Lukiskan rangkaian projek, dan berikan maklumat-maklumat berikut: laluan kritikal, jumlah apungan, dan apungan bebas untuk setiap aktiviti di atas.   [ 15 markah ]      - ooo O ooo - 

MAT 251 - Introduction To Operations Research [Pengantar Penyelidikan Operasi] Jun 2016

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MAT251 - Introduction To Operations Research [Pengantar Penyelidikan Operasi] Jun 2017

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