Formulation and Solving Integer Programming Models by Using Microsoft Excel. Instructor.Mushtaq.T. Hussain

Transcription

1 Formulation and Solving Integer Programming Models by Using Microsoft Excel. Instructor.Mushtaq.T. Hussain / /... solver. Abstract Integer programming can be applied in numerous business situations. however, integer programming problems are generally more complex and time consuming to solve than linear programming problems. The aim of this research focuses upon the importance using spreadsheet software package in solving the different models of integer linear programming, this spreadsheet is simplifying in defeating the difficulty opposite to researchers and graduate students who have competence and interest in operations research scopes. In this research we identification four basis of integer programming :pure integer programming model, mixed integer programming model, binary integer programming model and mixed binary integer programming model. After that we take case study for each model and formulate the mathematical model for each case study and so find the optimal integer solution for each case study by using (solver tool) in excel program by presenting different solver windows. Keywords: LinearProgramming, Integer Programming, Microsoft Excel. 347

2 Introduction - Program Linearity.. ) (.... : ).(.... (Maximize).(Minimize). Homogenous...( ) (divisibility) (Andrew J.Mason2007). The research problem

3 . The importance of research 3- Solver.. The aim of research 4- Integer Linear Programming (ILP) - 2. ).(... (John A. Lawrence & others 998:63) Types of Integer Programming Models -.2 : total integer programming models. : mixed integer programming models. : pure binary (zero- one) integer programming models.(0-) 349

4 : mixed binary integer programming models (binary) (Joern 2009) Integer Linear Programming With -3 Microsoft Excel.. (Solver). () ( lathe press ). $50 $ 00 (). :. (ft 2 200) ($40000) () () (Taylor 2006). (Bernard W. Taylor Introduction to management science, Ninth Edition Virginia Polytechnic Institute and State University.publisher:Prentice Hall 2006). : Maximize or Minimize Z= Subject to: machine Press Lathe n j = n j= c jx a x, =, b i=, 2,.., m ij Required space ft j x j 0 & integer i j Purchase price j=,2,..,n : =X 350

5 MaximizeZ= 00X s.t : 5X 8000X X, X 4000X X X 0&integer =X 2 : -4 Solution of integer programming models by using excel program. (a. (b (c. (d. (e. (f ( Andrew J.Mason 2007 ). (g (a) (C5,B5) () ) (C4,B4) target ) ( ) D5 ( changing cells. ( cell ( input cells ) (C9:B9,C8:B8) (D9:D8). (F9:F8).(output cells) () / 35

6 D5 X ( ) X ).. X2 ( ) X2 D5=sumproduct(B4:C4,B5:C5) (b sumproduct (2) =Sumproduct(B4:C4,B5: =Sumproduct(B8:C8,$B$4:$C$4) = Sumproduct(B9:C9,$B$4:$C$4) Solver (optimization) (2) Solver parameter. Tools solver (c : :.( 3) (set target cell) * sumproduct sumproduct(b4:c4,b5:c5). 2 C4*C5 B4*B5 (LHS ). ( ) 2 ( RHS ) $ $C$4,$B$4 $ $ D9 D8 D9 D8. D9 =sumproduct(b5:c5,b9:c9) Andrew ). D9=sumproduct($B$4:$C$4,B9:C9): D9 $ of J.Mason Solving linear programs using Microsoft excel solver university (Auckland

7 set target D5 ( 3 ) cell Min Max Enter.. Solver (3) / Changing Cell ( ) (d X2 X (). ( ) (C4:B4) C4:B4 Solver parameter (By changing cell).( 4) By changing cell ) القرار (4) (e : (C4:B4) 353

8 . Add constraint Add Solver parameter (Cell Reference ) Add constraint ( $C$4:$B$4) integer ( int). Add.( 5 ) constraint Cell ) ($D$9:$D$8 ) Add constraint ( Reference $F$9:$F$8 constraint (<=) ( 6 ) (5) Cell ) ($D$9:$D$8 ) Add constraint ( Reference $F$9:$F$8 constraint (<=) ( 6 ) (6) 354

9 Add ok ( 7 ) Solver parameter constraint solver parameter (7) / f : Assume linear model Solver option Solver parameter Option (8) Assume Non-. Assume linear model Solver Option. ok Negative (8) 355

10 (g. Solve parameter ok Assume linear model. (9) Solver Results. Solve (9) Keep Solve Solution ( 9) Restore Original Values.(0).. X= ) (0) (000$) (X2=6 (0) / 356

11 (Mixed Integer Programming) (2) * () : $ (3) (2) $2000 acre $9000 $50000 $8000 $ $000 (MIP) (Bernard W. Taylor. Introduction to management science, Ninth Edition Virginia Polytechnic Institute and State University.publisher:Prentice Hall 2006) (4). Solver Maximize or Minimize Z= Subject to: n j= n j= c jx a x, =, b i=, 2,.., m ij j x j 0 & some integer i j j=,2,..,n Max Z=9000X +500X X 3 S.T: 5000X +2000X X X 4 X 2 5 X 3 20 X, X3 0and integer X = (acre) 357

12 = : X =X 2 =X 3 E5=sumproduct(B4 :D4,B5:D5) آمية المورد المستهلك للقيد( E7=sumproduct(B7:D7,$B$4:$D$4)( آمية المورد المستهلك للقيد( 2 ) E8=sumproduct(B8:D8,$B$4:$D$4) آمية المورد المستهلك للقيد( 3 ) E9=sumproduct(B9:D9,$B$4:$D$4) آمية المورد المستهلك للقيد( 4 ) E0=sumproduct(B0:D0,$B$4:$D$4) () (2 ) (2) 358

13 (2 ) Add قيمة X يجب ان تكون عددية قيمة X2 يجب ان تكون مستمرة (3) Option (4 ) عدم السالبية للمتغيرات (4) / 359

14 Solver Parameter Solver Options Ok.(5 ) Solve (5) / نحدد الخيار الاول لاظهار النتاي ج في ورقة اآسل (6) 4 (7).$ (7) 360

15 Binary Integer Programming BIP (3) *.( ) ( ) ( ) (3) (2) / (Bernard W. Taylor Introduction to management science, Ninth Edition Virginia Polytechnic Institute and State University.publisher:Prentice Hall 2006) (2) $ (BIP) :. (Bernard W. Taylor Introduction to management science, Ninth :. Edition Virginia Polytechnic Institute and State University.publisher:Prentice Hall 2006) Maximize or Minimize Z= Subject to: n j= n j= c jx j a x, =, b i=, 2,.., m ij X j =0 or Max Z=300X +90X X 3 +50X 4 S.T: 35000X +0000X X X X + 2X 2 + 7X 3 + 3X 4 2 j i. j=,2,..,n : 36

16 = X + X 2 X, X 2, X 3, X 4 =0 or X 4 = X 3 = X 2 = X. Sumproduct(B5:E5,B6: E6) () sumproduct(b8:e8,$b$5:$e$5) (2) sumproduct(b9:e9,$b$5:$e$5) (3) sumproduct(b0:e0,$b$5:$e$5) (8) $F$6 $B$5:$E$5 ( 0) (9) / 362

17 (20) (2) (22) / 363

18 X=. 700 X3= Fixed Cost -5 (... ). (John A. Lawrence & others,998:82).. (4) * ( John A. Lawrence : : & others Applied management science john wiley & sons,inc.998 ) (3) ($000) ( 000) ( John A. Lawrence & others Applied management science john wiley & sons,inc.998 ) (4) B 0 A 2 ( 000) ( John A. Lawrence & others Applied management science john wiley & sons,inc.998 ) (5) B A / 2 3 ( John A. Lawrence & others Applied management science john wiley & sons,inc.998 ) 364

19 Min ST. : m m i= n n X X ij C ij X X ij ij i= j= i= = d 0 Yi( ij j= j= Yi= 0or j + m n i=,2,.,m j=,2,,n FiYi d j ) 0 : i 0=Yi i j i i=,2,...,m j=,2,...,n =X ij : j i =X ij i =fi j i =C ij j =dj i=,2,3 j=a,b, i i 0=Yi j i =X ij Minimize Z = 8x A + 5x B +3x 2A + 0x 2B + 6x 3A + 4x 3B Y Y Y3 Subject to: x A + x B -.2 Y 0 x 2A + x 2B -0.5 Y2 0 x 3A + x 3B Y

20 x A + x 2A + x 3A =2 x B + x 2B + x 3B = 0 xij 0 yi = 0 or Solver (23) =SUMPRODUCT(B 2:J2,B3:J3) آمية المورد المستهلك للقيد ( ( =SUMPRODUCT(B5:J5,$B$2:$J$2) آمية المورد المستهلك للقيد (2 ( =SUMPRODUCT(B6:J6,$B$2:$J$2) آمية المورد المستهلك للقيد (3 ( =SUMPRODUCT(B7:J7,$B$2:$J$2) آمية المورد المستهلك للقيد (4 ( =SUMPRODUCT(B8:J8,$B$2:$J$2) آمية المورد المستهلك للقيد (5 ( =SUMPRODUCT(B9:J9,$B$2:$J$2) (24) 366

21 و 4 202/ تحدي د خلي ة الهدف L$2$ تحدي د خلاي ا متغي رات الق رار(الخلاي ا المتغي رة) $B$2:$J$2 خلاي ا الق رار الت ي يج ب ان تك ون ذات قيمة اآبر او تساوي الصفر خلاي ا الق رار الت ي يج ب ان تك ون ذات قيمة( 0 او ) (25) الط رف الاي سر للقي ود( 2 3 ) يج ب ان يك ون اق ل او م ساوي للم ورد المتي سر(الط رف الايم ن) لكل قيد. الطرف الايسر للقيود( 5) يج ب ان يساوي المورد المتسير لكل منها. (26) (26) / Solver Parameters (Solver Options) ok (27) Solve 367

22 (27) (28) :.A 500. B 0000.A 500.Y=0.$ 30.5= 368

23 Conclusions and Recommendations -6 Conclusions -6 Solver Recommendations 2-6 solver. 369

24 Arabic References - " " " " " " " " English References -Andrew J. Mason "Solving linear programs using Microsoft Excel solver" University of Auckland Bernard W. Taylor Introduction to management science, Virginia Polytechnic Institute and State University.publisher:Prentice Hall. Ninth Edition Bronson, Richard "schaum's OutLine Series Theory and Applications of Operations Research" McCraw-Hill Book Company John A. Lawrence & others Applied management science john wiley & sons,inc Meissner,Joern" An introduction to spreadsheet optimization using excel solver"lancaster University Taha,Hamdy A. Operations Research An Introduction Prentice-Hall, Inc eighth edition