R O S L E N A B I N T I A B D U L R A H I M R A J A N A D I A B I N T I R A J A A H M A D E R N A R U H A I L Y B I N T I R U S L I ENGINEERING MATHEMATICS THREE LLIINNEEAARR PPRROOGGRRAAMMMMIINNGG WWOORRKKBBOOOOKK
ENGINEERING MATHEMATICS THREE Linear Programming Workbook
FIRST ISSUE 2023 Publication@Polytechnic Nilai AUTHOR(S) ROSLENA BINTI ABDUL RAHIM RAJA NADIA BINTI RAJA AHMAD ERNA RUHAILY BINTI RUSLI EDITOR SUHANA BINTI BAHARUDDIN
A PUBLICATION BY: e ISBN 978-967-2742-24-1 ENGINEERING MATHEMATICS 3: Linear Programming Workbook A student workbook that contains a step-by-step answer guide as well as enrichment exercises related to specific topics and subtopics. Department of Mathematics, Science and Computer Polytechnic Nilai Negeri Sembilan Kompleks Pendidikan Nilai 71760, Bandar Enstek Negeri Sembilan Darul Khusus COPYRIGHT DECLARATION Al l r i g h t s r e s e r v e d . N o p a r t o f t h i s p u b l i c a ti on m ay b e r e p r o d u c e d , s t o r e d f o r p r o d u c ti on, o r c onv e r t e d int o any f o r m o r by m e ans o f d i s s e m ina ti on, wh e t h e r by e l e c t r oni c m e ans , p h o t o g r a p h s and r e c o r d ing s , and t h e l i k e wit h o u t p r i o r wr itt en p e r m i s s i on o f t h e c o pyr i g h t h o l d e r fi r s t. PUBL IS H ED BY Po lyt e c hni c N i l a i N e g e r i Se m b i l an Mini s t ry O f H i g h e r Ed u c a ti on Ma l ays i a
A C K N O W L E D G E M E N T A l l p r a i s e s t o A l m i g h t y A l l a h , T h e M o s t M e r c i f u l a n d T h e M o s t C o m p a s s i o n a t e . N o n e o f t h i s w o u l d h a v e b e e n p o s s i b l e w i t h o u t H i s b l e s s i n g s . A l l o f t h i s s t a r t e d w i t h a d r e a m . A d r e a m t o c r e a t e l o t s o f w o n d e r f u l t h i n g s a n d s u c c e s s i n m a n y t h i n g s . W e a r e s o g r a t e f u l t o b e p a r t n e r t o g e t h e r w i t h d e a r e s t f r i e n d s .
P R E F A C E T h i s L i n e a r P r o g r a m m i n g E n g i n e e r i n g M a t h e m a t i c s 3 W o r k b o o k p r o v i d e s s t u d e n t s w i t h g u i d e l i n e s a n d v a r i o u s q u e s t i o n s o n s o l v i n g l i n e a r p r o g r a m m i n g T h e e x e r c i s e s p r o v i d e d i n t h i s w o r k b o o k a r e b a s e d o n t h e f o l l o w i n g t o p i c s : L i n e a r I n e q u a l i t i e s , F o r m u l a t i n g M a t h e m a t i c a l M o d e l s , S h a d e d R e g i o n s , G r a p h i c a l M e t h o d a n d S i m p l e x M e t h o d . A s h o r t a n d b r i e f n o t e a s g u i d e l i n e s i s i n c l u d e d i n t h i s w o r k b o o k . W i t h t h e s e s t e p - b y - s t e p g u i d e s , i t w o u l d b e a g r e a t h e l p f o r t h e s t u d e n t s t o e n h a n c e t h e i r k n o w l e d g e a n d m a s t e r t h i s t o p i c . C o n g r a t u l a t i o n s t o t h e a u t h o r s f o r c o n t r i b u t i n g t h e i d e a s i n t h e m a k i n g o f t h i s w o r k b o o k
TABLE OF CONTENT Topic/Subtopic Page(s) PREFACE IDENTIFYING SIGNS IN LINEAR PROGRAMMING STEP BY STEP GUIDED EXERCISE FORMULATING MATHEMATICAL MODEL STEP BY STEP GUIDED EXERCISE TUTORIAL SHADED REGION / GRAPHICAL METHOD STEP BY STEP GUIDED EXERCISE TUTORIAL SIMPLEX METHOD STEP BY STEP GUIDED EXERCISE TUTORIAL FINAL SESSION II 2021/2022 FINAL SESSION I 2022/2023 FINAL SESSION I1 2022/2023 REFERENCE I 1 1 - 2 3 3 - 8 9 10 - 11 11 - 15 16 17 18 - 24 25 26 - 29 30 - 32 33 - 36 37
IDENTIFYING SIGNS IN LINEAR PROGRAMMING SIGN MEANING EXAMPLE LESS THAN LESS THAN or EQUAL TO NOT GREATER THAN AT MOST GREATER THAN GREATER THAN or EQUAL TO NOT LESS THAN AT LEAST LINEAR INEQUALITIES NOTE: a > b and a < b is called “STRICT INEQUALITIES” HINT: The “open” (larger) part of the inequality symbol always faces the larger quantity. WORKBOOK LINEAR PROGRAMMING 1
LINEAR INEQUALITIES 2 WORKBOOK LINEAR PROGRAMMING
FORMULATING MATHEMATICAL MODEL 3 WORKBOOK LINEAR PROGRAMMING
FORMULATING MATHEMATICAL MODEL 4 WORKBOOK LINEAR PROGRAMMING
FORMULATING MATHEMATICAL MODEL 5 WORKBOOK LINEAR PROGRAMMING
FORMULATING MATHEMATICAL MODEL 6 WORKBOOK LINEAR PROGRAMMING
FORMULATING MATHEMATICAL MODEL 7 WORKBOOK LINEAR PROGRAMMING
FORMULATING MATHEMATICAL MODEL 8 WORKBOOK LINEAR PROGRAMMING
TUTORIAL 4A 9 WORKBOOK LINEAR PROGRAMMING Answer: Answer:
SHADED REGION and and 10 WORKBOOK LINEAR PROGRAMMING
SHADED REGION 11 WORKBOOK LINEAR PROGRAMMING
GRAPHICAL METHOD NOTE: Only Linear Programming with TWO VARIABLES can be analyzed graphically. 12 WORKBOOK LINEAR PROGRAMMING
GRAPHICAL METHOD 13 WORKBOOK LINEAR PROGRAMMING
GRAPHICAL METHOD NOTE: Only Linear Programming with TWO VARIABLES can be analyzed graphically. 14 WORKBOOK LINEAR PROGRAMMING
GRAPHICAL METHOD 15 WORKBOOK LINEAR PROGRAMMING
TUTORIAL 4B 16 WORKBOOK LINEAR PROGRAMMING
THE SIMPLEX METHOD 17 WORKBOOK LINEAR PROGRAMMING
THE SIMPLEX METHOD 18 WORKBOOK LINEAR PROGRAMMING
THE SIMPLEX METHOD 19 WORKBOOK LINEAR PROGRAMMING
THE SIMPLEX METHOD 20 WORKBOOK LINEAR PROGRAMMING
THE SIMPLEX METHOD 21 WORKBOOK LINEAR PROGRAMMING
THE SIMPLEX METHOD 22 WORKBOOK LINEAR PROGRAMMING
THE SIMPLEX METHOD 23 WORKBOOK LINEAR PROGRAMMING
THE SIMPLEX METHOD 24 WORKBOOK LINEAR PROGRAMMING
TUTORIAL 4C 25 WORKBOOK LINEAR PROGRAMMING
FINAL SESSION II 2021/2022 QUESTION 4 [A] 26 WORKBOOK LINEAR PROGRAMMING
FINAL SESSION II 2021/2022 QUESTION 4 [B] 27 WORKBOOK LINEAR PROGRAMMING
FINAL SESSION II 2021/2022 QUESTION 4 [C] 28 WORKBOOK LINEAR PROGRAMMING
FINAL SESSION I 2022/2023 QUESTION 4 [A] 29 WORKBOOK LINEAR PROGRAMMING
FINAL SESSION I 2022/2023 QUESTION 4 [B] 30 WORKBOOK LINEAR PROGRAMMING
FINAL SESSION I 2022/2023 QUESTION 4 [B] 31 WORKBOOK LINEAR PROGRAMMING
FINAL SESSION I 2022/2023 QUESTION 4 [C] 32 WORKBOOK LINEAR PROGRAMMING
FINAL SESSION II 2022/2023 QUESTION 4 [A] 33 WORKBOOK LINEAR PROGRAMMING
FINAL SESSION II 2022/2023 QUESTION 4 [B] 34 WORKBOOK LINEAR PROGRAMMING
FINAL SESSION II 2022/2023 QUESTION 4 [B] 35 WORKBOOK LINEAR PROGRAMMING
FINAL SESSION II 2022/2023 QUESTION 4 [C] 36 WORKBOOK LINEAR PROGRAMMING
REFERENCE https://www.toppr.com/guides/maths/linear-programming/linearprogramming-problem/ https://www.gurobi.com/resources/linear-programming-tutorial/ INTRODUCTION TO OPERATIONS RESEARCH, FREDERICK S. HILLIER, Stanford University Hillier, F.S. , and G.J. Lieberman. Introduction to Operations Research, sixth edition (McGraw-Hill, Inc. , New York, 1995). 998 pp. Ravindran, A. , D.T. Phillips, and J.J. Solberg. Operations Research: Principles and Practice, second edition (John Wiley & Sons, New York, 1987). 637 pp. 37
"The Only Way to Learn Mathematics is to Do Mathematics" ~ P a u l Ha lmo s ~ Hu n g a r i a n -Ame r ic a n M a t h ema t ici a n D e p a r t m e n t o f M a t h e m a t i c s , S c i e n c e a n d C o m p u t e r P o l y t e c h n i c N i l a i N e g e r i S e m b i l a n K o m p l e k s P e n d i d i k a n N i l a i 7 1 7 6 0 , B a n d a r E n s t e k N e g e r i S e m b i l a n D a r u l K h u s u s