PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-1 MATRIKULASI PMM UNLAM METODE KUANTITATIF Linear Programming Formulasi dan Solusi Metode Grafis Kuliah 03

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-2 Outline  Persyaratan Masalah Linear Programming  Formulasi Masalah Liniear Programming  Contoh Shader Electronics  Solusi Grafis untuk suatu Masalah Linear programming  Representasi grafis pembatas-pembatas  Metode Solusi Garis Iso-Laba  Metode Solusi Titik-Sudut

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-3 Outline - lanjutan Solusi Masalah Minimasi  Aplikasi-aplikasi LP  Contoh Bauran Produk  Contoh Masalah Diet  Contoh Penjadwalan Produksi  Contoh Penjadwalan Pekerja

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-4 Setelah anda belajar materi ini, anda harus mampu :  Mengidentifikasi atau Mendefinisikan :  Fungsi tujuan  Batasan-batasan  Daerah layak  Metode Iso-laba/ Iso-biaya  Metode Solusi Titik-sudut Tujuan Belajar

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-5 Setelah anda belajar materi ini, anda harus mampu :  Mendeskripsikan atau Menjelaskan:  Bagaimana memformulasikan model linear  Metode grafis dari linear programming Tujuan Belajar - lanjutan

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-6  Teknik matematis  Bukan pemrograman komputer  Alokasi sumberdaya-sumberdaya langka untuk mencapai tujuan  Dipelopori oleh George Dantzig dalam Perang Dunia II  Dibangun dapat-bekerja memberi solusi dalam 1947  Disebut metode sompleks Apa yang dimaksud Linear Programming?

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-7  Penjadwalan bus sekolah untuk memanimasi total jarak pengantaran ketika antar-jemput pelajar  Pengalokasian satuan polisi patroli untuk menanggapi panggilan 911  Penjadwalan kasir bank untuk menyesuaikan kebutuhan setiap jam dari hari-kehari yang minimalisasi total biaya pekerja. Contoh-contoh keberhasilan Aplikasi LP

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-8 Contoh-contoh keberhasilan Aplikasi LP - lanjutan  Memilih bahan baku dalam adonan bahan makanan untuk kombinasi pakan pada biaya minimum  Memilih bauran produk dalam pabrik untuk membuat mesin terbaik dan jam-buruh yang ada yang memaksimumkan keuntungan perusahaan  Mengalokasikan ruang bagi suatu bauran penyewaan dalam suatu mall pembelanjaan baru sedemikian rupa untuk memaksimumkan pendapatan perusahaan penyewaan

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-9 Persyaratan Masalah Linear Programming 1 Harus cari maksimasi atau minimasi berapa kuantitas (fungsi tujuan) 2 Menghadirkan batasan-batasan/ constraints – batas yg mungkin utk mencapai tujuan 3 Harus ada alternatif pilihan dari tindakan dari sesuatu yang dipilih 4 Tujuan dan batasan hrs dpt dinyatakan sebagai persamaan linear atau pertidaksamaan

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-10 Komponen Formulasi Masalah Program Linier 1. Elemen Fungsi Tujuan LP (Objective Function) disingkat OF 2. Elemen Fungsi Pembatas LP (Constraints Function) disingkat CF atau ST (Subject To) 3. Elemen Fungsi Pembatas Bukan- Negatif LP (Non-Negative Constraints Function) disingkat NNCF

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-11 Formulasi Masalah Linear Programming  Umpama:  Anda memproduksi 2 produk (1) Walkman AM/FM/Cassette (x1) dan (2) Watch-TV (x2)  Walkman perlu 4 jam kerja elektronik dan 2 jam perakitan  Watch-TV perlu 3 jam kerja elektronik dan 1 jam perakitan  Tersedia 240 jam waktu kerja elektroni dan 100 jam waktu perakitan  Laba tiap Walkman $7; laba tiap Watch-TV $5

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-12 Formulasi Masalah Linear Programming - lanjutan  Misal:  X 1 = kuantitas Walkman  X 2 = kuantitas Watch-TV  Lalu:  4X 1 + 3X 2  240electronics constraint  2X 1 + 1X 2  100assembly constraint  7X 1 + 5X 2 = profitmaximize profit

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-13  Gambar absis dan ordinat (hanya kuadran pertama)  Plot garis pembatas, gambar garisnya  Gunakan koordinat garis ( X 1,0), (0, X 2 )  Dapatkan area layak/ feasible  Dapatkan solusi optimal  Metode titik-sudut  Metode Iso-laba Metode Solusi Grafis

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-14 Shader Electronic Company Problem Hours Required to Produce 1 Unit DepartmentX1X1 Walkmans X2X2 Watch-TV’s Available Hours This Week Electronic43240 Assembly21100 Profit/unit$7$5 Constraints: 4x 1 + 3x 2  240 (Hours of Electronic Time) 2x 1 + 1x 2  100 (Hours of Assembly Time) Objective:Maximize: 7x 1 + 5x 2

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-15 Shader Electronic Company Constraints Number of Walkmans (X 1 ) Number of Watch-TVs (X 2 ) Electronics (Constraint A) Assembly (Constraint B)

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-16 Shader Electronic Company Feasible Region Number of Walkmans (X 1 ) Number of Watch-TVs (X 2 ) Electronics (Constraint A) Assembly (Constraint B) Feasible Region

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-17 Shader Electronic Company Iso-Profit Lines Number of Walkmans (X 1 ) Number of Watch-TVs (X 2 ) Electronics (Constraint A) Assembly (Constraint B) 7*X 1 + 5*X 2 = 210 7*X 1 + 5*X 2 = 420

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-18 Shader Electronic Company Solution Number of Walkmans (X 1 ) Number of Watch-TVs (X 2 ) Electronics (Constraint A) Assembly (Constraint B) ISO-Profit Line Solution Point (X 1 =30, X 2 =40)

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-19 Shader Electronic Company Solution Corner Point Solution Number of Walkmans (X 1 ) Number of Watch-TVs (X 2 ) Electronics (Constraint A) Assembly (Constraint B) Possible Corner Point Solution Optimal solution

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-20  Decision variables  X 1 = tons of BW chemical produced  X 2 = tons of color chemical produced  Objective  Minimize Z = 2500 X X 2  Constraints  X 1  30 (BW); X 2  20 (Color)  X 1 + X 2  60 (Total tonnage)  X 1  0; X 2  0 (Non-negativity) Formulation of Solution

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-21 Sensitivity Analysis  Projects how much a solution might change if there were changes in variables or input data.  Shadow price (dual) - value of one additional unit of a resource

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-22 You’re an analyst for a division of Kodak, which makes BW & color chemicals. At least 30 tons of BW and at least 20 tons of color must be made each month. The total chemicals made must be at least 60 tons. How many tons of each chemical should be made to minimize costs? BW: $2,500 manufacturing cost per month Color: $ 3,000 manufacturing cost per month © 1995 Corel Corp. Minimization Example

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-23 Graphical Solution FeasibleRegion Tons, Color Chemical (X 2 ) Tons, BW Chemical (X 1 ) BW Color Total Find values for X 1 + X 2  60. X 1  30, X 2  20.

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-24 FeasibleRegion Tons, Color Chemical Tons, BW Chemical BW Color Total Find corner points. A B Optimal Solution: Corner Point Method

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J B-25