MATRIKULASI PMM UNLAM METODE KUANTITATIF Linear Programming Formulasi

MATRIKULASI PMM UNLAM METODE KUANTITATIF Linear Programming Formulasi dan Solusi Metode Grafis Kuliah 03 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 1 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

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 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 2 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Outline - lanjutan Solusi Masalah Minimasi ¨ Aplikasi-aplikasi LP ¨ Contoh Bauran Produk ¨ Contoh Masalah Diet ¨ Contoh Penjadwalan Produksi ¨ Contoh Penjadwalan Pekerja Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 3 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Tujuan Belajar 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 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 4 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Tujuan Belajar - lanjutan Setelah anda belajar materi ini, anda harus mampu : ¨ Mendeskripsikan atau Menjelaskan: Bagaimana memformulasikan model linear ¨ Metode grafis dari linear programming ¨ Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 5 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Apa yang dimaksud Linear Programming? ¨ 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 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 6 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Contoh-contoh keberhasilan Aplikasi LP ¨ 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. Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 7 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

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 8 penyewaan Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

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 9 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

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 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 10 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Formulasi Masalah Linear Programming ¨ Umpama: ¨ Anda memproduksi 2 produk (1) Walkman AM/FM/Cassette (x 1) dan (2) Watch-TV (x 2) ¨ 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 11 $5 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Formulasi Masalah Linear Programming lanjutan ¨ Misal: ¨ X 1 = kuantitas Walkman ¨ X 2 = kuantitas Watch-TV ¨ Lalu: + 3 X 2 240 electronics constraint ¨ 2 X 1 + 1 X 2 100 assembly constraint ¨ 7 X 1 + 5 X 2 = profit maximize profit ¨ 4 X 1 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 12 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Metode Solusi Grafis ¨ 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 ¨ Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 13 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Shader Electronic Company Problem Hours Required to Produce 1 Unit Department X 1 X 2 Available Hours Walkmans Watch-TV’s This Week Electronic 4 3 240 Assembly 2 1 Profit/unit $7 $5 100 Constraints: 4 x 1 + 3 x 2 240 (Hours of Electronic Time) 2 x 1 + 1 x 2 100 (Hours of Assembly Time) Objective: Maximize: 7 x 1 + 5 x 2 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 14 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Number of Watch-TVs (X 2) Shader Electronic Company Constraints Electronics 120 (Constraint A) 100 Assembly (Constraint B) 80 60 40 20 0 0 10 20 30 40 50 60 70 80 Number of Walkmans (X 1) Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 15 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Number of Watch-TVs (X 2) Shader Electronic Company Feasible Region Electronics (Constraint A) 120 100 Assembly (Constraint B) 80 60 40 Feasible Region 20 0 0 10 20 30 40 50 60 70 80 Number of Walkmans (X 1) Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 16 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Number of Watch-TVs (X 2) Shader Electronic Company Iso-Profit Lines Electronics (Constraint A) Assembly (Constraint B) 120 100 80 7*X 1 + 5*X 2 = 420 60 7*X 40 1 +5 *X 2 20 =2 10 0 0 10 20 30 40 50 60 70 80 Number of Walkmans (X 1) Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 17 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Number of Watch-TVs (X 2) Shader Electronic Company Solution Electronics (Constraint A) 120 ISO-Profit Line 100 80 Assembly (Constraint B) Solution Point (X 1=30, X 2=40) 60 40 20 0 0 10 20 30 40 50 60 70 80 Number of Walkmans (X 1) Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 18 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Number of Watch-TVs (X 2) Shader Electronic Company Solution Corner Point Solution Electronics (Constraint A) Assembly (Constraint B) 120 100 Possible Corner Point Solution Optimal solution 80 60 40 20 0 0 10 20 30 40 50 60 70 80 Number of Walkmans (X 1) Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 19 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Formulation of Solution ¨ Decision variables X 1 = tons of BW chemical produced ¨ X 2 = tons of color chemical produced ¨ ¨ Objective ¨ Minimize Z = 2500 X 1 + 3000 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) ¨ Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 20 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

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 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 21 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Minimization Example BW: $2, 500 You’re an analyst for a manufacturing division of Kodak, which makes BW & color chemicals. cost per month 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 Color: $ 3, 000 be made to minimize costs? manufacturing cost per © 1995 Corel Corp. month Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 22 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Graphical Solution 80 Find values for X 1 + X 2 � 60. BW 60 X 1 30, X 2 20. Total Feasible Region Tons, 40 Color Chemical 20 (X 2) Color 0 0 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 20 40 60 80 Tons, BW Chemical (X 1) 23 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Optimal Solution: Corner Point Method 80 Find corner points. BW 60 Total Tons, 40 Color Chemical 20 Feasible Region B Color A 0 0 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 20 40 60 Tons, BW Chemical 24 80 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 25 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458