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
Introduction - Program Linearity.. ) (.... : ).(.... (Maximize).(Minimize). Homogenous...( ) (divisibility) (Andrew J.Mason2007). The research problem 2-. 348
. 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
: mixed binary integer programming models (binary) (Joern 2009) Integer Linear Programming With -3 Microsoft Excel.. (Solver). () ( lathe press ). $50 $ 00 (). :. (ft 2 200) ($40000) () () (Taylor 2006). (Bernard W. Taylor Introduction to management science, Ninth Edition Virginia Polytechnic Institute and State University.publisher:Prentice Hall 2006). : Maximize or Minimize Z= Subject to: machine Press Lathe n j = n j= c jx a x, =, b i=, 2,.., m ij Required space ft 2 5 30 j x j 0 & integer i j Purchase price 8000 4000 j=,2,..,n : =X 350
MaximizeZ= 00X s.t : 5X 8000X X, + + 30X 4000X X 2 2 2 + 50X 0&integer 2 200 40000 =X 2 : -4 Solution of integer programming models by using excel program. (a. (b (c. (d. (e. (f ( Andrew J.Mason 2007 ). (g (a) (C5,B5) () ) (C4,B4) target ) ( ) D5 ( changing cells. ( cell ( input cells ) (C9:B9,C8:B8) (D9:D8). (F9:F8).(output cells) () / 35
D5 X ( ) X ).. X2 ( ) X2 D5=sumproduct(B4:C4,B5:C5) (b sumproduct (2) =Sumproduct(B4:C4,B5: =Sumproduct(B8:C8,$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 2007 352
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
. 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$8 constraint (<=) ( 6 ) (6) 354
Add ok ( 7 ) Solver parameter constraint solver parameter (7) / f : Assume linear model Solver option Solver parameter Option (8) Assume Non-. Assume linear model Solver Option. ok Negative (8) 355
(g. Solve parameter ok Assume linear model. (9) Solver Results. Solve (9) Keep Solve Solution ( 9) Restore Original Values.(0).. X= ) (0) (000$) (X2=6 (0) / 356
(Mixed Integer Programming) (2) * () : $250000. (3) (2) $2000 acre $9000 $50000 $8000 $500 4. $000 (MIP). 20 5 (Bernard W. Taylor. Introduction to management science, Ninth Edition Virginia Polytechnic Institute and State University.publisher:Prentice Hall 2006) (4). Solver Maximize or Minimize Z= Subject to: n j= n j= c jx a x, =, b i=, 2,.., m ij j x j 0 & some integer i j j=,2,..,n Max Z=9000X +500X 2 +000X 3 S.T: 5000X +2000X 2 +8000X 3 250000 X 4 X 2 5 X 3 20 X, X3 0and integer X 2 0 2 4000= (acre) 357
= : X =X 2 =X 3 E5=sumproduct(B4 :D4,B5:D5) آمية المورد المستهلك للقيد( E7=sumproduct(B7:D7,$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
(2 ) Add قيمة X يجب ان تكون عددية قيمة X2 يجب ان تكون مستمرة (3) Option (4 ) عدم السالبية للمتغيرات (4) / 359
Solver Parameter Solver Options Ok.(5 ) Solve (5) / نحدد الخيار الاول لاظهار النتاي ج في ورقة اآسل (6) 4 (7).$64750. 7 4.5 (7) 360
Binary Integer Programming BIP (3) *.( ) 4 2 7 3 ( ) ( ) (3) (2) 35000 0000 25000 9000 / (Bernard W. Taylor Introduction to management science, Ninth Edition Virginia Polytechnic Institute and State University.publisher:Prentice Hall 2006) (2) $ 20000.. (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 2 +400X 3 +50X 4 S.T: 35000X +0000X 2 +25000X 3 +9000X 4 20000 4X + 2X 2 + 7X 3 + 3X 4 2 j 300 90 400 50 i. j=,2,..,n : 36
= 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
(20) (2) (22) / 363
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) 3 450 2 390 405 ($000) 2.8 0.5.2 ( 000) ( John A. Lawrence & others Applied management science john wiley & sons,inc.998 ) (4) B 0 A 2 ( 000) ( John A. Lawrence & others Applied management science john wiley & sons,inc.998 ) (5) B 5 0 4 A 8 3 6 / 2 3 ( John A. Lawrence & others Applied management science john wiley & sons,inc.998 ) 364
Min ST. : m m i= n n X X ij C ij X X ij ij i= j= i= = d 0 Yi( ij j= j= Yi= 0or j + m n i=,2,.,m j=,2,,n FiYi d j ) 0 : i 0=Yi i j i i=,2,...,m j=,2,...,n =X ij : j i =X ij i =fi j i =C ij j =dj i=,2,3 j=a,b, i i 0=Yi j i =X ij Minimize Z = 8x A + 5x B +3x 2A + 0x 2B + 6x 3A + 4x 3B + 405 Y + 390 Y2 + 450 Y3 Subject to: x A + x B -.2 Y 0 x 2A + x 2B -0.5 Y2 0 x 3A + x 3B - 2.8 Y3 0. 365
x A + x 2A + x 3A =2 x B + x 2B + x 3B = 0 xij 0 yi = 0 or Solver (23) =SUMPRODUCT(B 2:J2,B3:J3) آمية المورد المستهلك للقيد ( ( =SUMPRODUCT(B5:J5,$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
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$2 خلاي ا الق رار الت ي يج ب ان تك ون ذات قيمة اآبر او تساوي الصفر خلاي ا الق رار الت ي يج ب ان تك ون ذات قيمة( 0 او ) (25) الط رف الاي سر للقي ود( 2 3 ) يج ب ان يك ون اق ل او م ساوي للم ورد المتي سر(الط رف الايم ن) لكل قيد. الطرف الايسر للقيود( 5) يج ب ان يساوي المورد المتسير لكل منها. (26) (26) / Solver Parameters (Solver Options) ok (27) Solve 367
(27) (28) :.A 500. B 0000.A 500.Y=0.$ 30.5= 368
Conclusions and Recommendations -6 Conclusions -6 Solver 2 8 2 6.. Recommendations 2-6 solver. 369
Arabic References - " " -.2006 5 " ". -2.2005 986 " ". -3.200 " ". -4 - English References -Andrew J. Mason "Solving linear programs using Microsoft Excel solver" University of Auckland 2007 2-Bernard W. Taylor Introduction to management science, Virginia Polytechnic Institute and State University.publisher:Prentice Hall. Ninth Edition 2006 3-Bronson, Richard "schaum's OutLine Series Theory and Applications of Operations Research" McCraw-Hill Book Company 982 4- John A. Lawrence & others Applied management science john wiley & sons,inc.998 5-Meissner,Joern" An introduction to spreadsheet optimization using excel solver"lancaster University 2009. 6-Taha,Hamdy A. Operations Research An Introduction Prentice-Hall, Inc eighth edition. 2007. 370