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Facility Location Problem

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Evren E

The document discusses selecting an optimal hub location for flights between South America and Europe by analyzing cities' distances and passenger numbers. A linear optimization model is formulated and solved in Lingo, determining that Madrid is the optimal hub location with an objective value of 0.2743843E+11. Distances and passenger numbers between cities like Buenos Aires, Sao Paulo, Santiago, Paris, Madrid and Lisbon are provided as input data.

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ENDÜSTRİDE BİLGİSAYAR UYGULAMALARI - 2 New potential hubs in the South-Atlantic market. A problem of location Journal of Transport Geography Volume 11, Issue 2 , June 2003, Pages 139-149 FERİDE SEVLİ – 2000503052 AYTÜL ŞENER – 200503056 İLKEM YALINER – 2000503067
İçerik: Tesis Planlama Kavramı Tesis Planlamanın Amaçları Tesis Planlama Tipleri Tesisin Kurulacağı Bölgenin Seçimi Alternatiflerin Değerlendirilme Yöntemleri Havacılık Sektöründe Tesis Planlama Uygulamaları
TESİS PLANLAMA KAVRAMI: TESİS : Mal ve hizmetlerin fiilen üretildiği fiziksel birimlerdir. TESİS PLANLAMA : Tesislerin düzenleme çalışmalarının yapılması, tesisin kurulması ve faaliyete geçirilmesi aşamalarında gerçekleştirilecek olan faaliyetlerin bugünden eşgüdümlerinin yapılmasıdır.
TESİS PLANLAMA AMAÇLARI: Kaynakların kullanımı Darboğazların belirlenmesi Tesis faaliyetleri Optimum yerleşim düzeni Maliyet Mesafe Zaman
TESİS PLANLAMA TİPLERİ: Basit Yerleşim Çoklu Yerleşim Tesis Oluşturulması ve Kapasite Atama Tesis Seçimi ve Kapasite Atama Tesis İçi Yerleşim(QAP)
TESİSİN KURULACAĞI BÖLGENİN SEÇİLMESİ: Yerleşim Kararlarını Etkileyen Faktörler : Ülke seçimi (kanunlar, pazar, işgücü, tedarik, alım gücü vb.) Bölge seçimi (maliyet, çevre kuralları, müşteri&hammadde erişimi vb.) Yer seçimi (yerin alan&maliyeti, ulaşım kolaylığı vb.)
ALTERNATİFLERİ DEĞERLENDİRME YÖNTEMLERİ: Faktör Oranlama Metodu Başa-baş Analizi Ağırlık Merkezi Metodu Ulaştırma Metodu
Problem: Güney Amerika–Avrupa arasında düzenlenen uçuş seferlerinde optimum aktarma noktası seçimi
 
Veriler: 1) Şehirler arası mesafeler 0 4000 4100 24400 23100 21100 BUONES AIRES 4000 0 7000 21100 19500 17600 SAO PAULO 4100 7000 0 26200 24000 24000 SANTIAGO 24400 21100 26200 0 2500 3200 PARIS 23100 19500 24000 2500 0 1000 MADRID 21100 17600 24000 3200 1000 0 LISBON BUONES AIRES SAO PAULO SANTIAGO PARIS MADRID LİSBON TO FROM
Veriler: 2) Şehirler arası yolcu sayısı 0 0 0 83000 270051 110000 BUONES AIRES 0 0 0 143977 126851 73847 SAO PAULO 0 0 0 82000 80054 95000 SANTIAGO 85000 155036 74000 0 0 0 PARIS 240213 126137 81220 0 0 0 MADRID 121000 69238 62000 0 0 0 LISBON BUONES AIRES SAO PAULO SANTIAGO PARIS MADRID LİSBON TO FROM
min Σ (d E i H x + d H x S j ) T E i S j H x  E u S E i  E T *  S j  S
Problemin Lingo Modeli: SETS: AIRPORT/LISBON MADRID PARIS SANTIAGO SAOPAULO BUONESAIRES/:HUB,K; A(AIRPORT,AIRPORT):DIST,PASS; ENDSETS DATA: DIST=0 1000 3200 24000 17600 21100 1000 0 2500 24000 19500 23100 3200 2500 0 26200 21100 24400 24000 24000 26200 0 7000 4100 17600 19500 21100 7000 0 4000 21100 23100 24400 4100 4000 0 ; PASS=0 0 0 62000 69238 121000 0 0 0 81220 126137 240213 0 0 0 74000 155036 85000 95000 80054 82000 0 0 0 73847 126851 143977 0 0 0 110000 270051 83000 0 0 0 ; ENDDATA MIN=@SUM(AIRPORT(I):HUB(I)*K(I)); @FOR(AIRPORT(I):@SUM(A(J,K)|J#NE#I #AND# #NE#K:(DIST(J,I)+DIST(I,K))*PASS(J,K))=HUB(I)); IFA=@MIN(AIRPORT(I):HUB(I)); @FOR(AIRPORT(I)|HUB(I)#LE#IFA:K(I)=1); @FOR(AIRPORT(I):@BIN(K(I)));
Lingo Çözümü: Rows= 1 Vars= 5 No. integer vars= 5 ( all are linear) Nonzeros= 6 Constraint nonz= 0( 0 are +- 1) Density=1.000 Smallest and largest elements in absolute value= 0.317284E+11 0.481254E No. < : 0 No. =: 0 No. > : 0, Obj=MIN, GUBs <= 0 Single cols= 5 Optimal solution found at step: 0 Objective value: 0.2743843E+11 Branch count: 0 Variable Value Reduced Cost IFA 0.2743843E+11 0.0000000E+00 HUB( LISBON) 0.3455933E+11 0.0000000E+00 HUB( MADRID) 0.2743843E+11 0.0000000E+00 HUB( PARIS) 0.3879355E+11 0.0000000E+00 HUB( SANTIAGO) 0.4812541E+11 0.0000000E+00 HUB( SAOPAULO) 0.3371716E+11 0.0000000E+00 HUB( BUONESAIRES) 0.3172843E+11 0.0000000E+00 K( LISBON) 0.0000000E+00 0.3455933E+11 K( MADRID) 1.000000 0.0000000E+00 K( PARIS) 0.0000000E+00 0.3879355E+11 K( SANTIAGO) 0.0000000E+00 0.4812541E+11 K( SAOPAULO) 0.0000000E+00 0.3371716E+11 K( BUONESAIRES) 0.0000000E+00 0.3172843E+11
Lingo Çözümü: DIST( LISBON, LISBON) 0.0000000E+00 0.0000000E+00 DIST( LISBON, MADRID) 1000.000 0.0000000E+00 DIST( LISBON, PARIS) 3200.000 0.0000000E+00 DIST( LISBON, SANTIAGO) 24000.00 0.0000000E+00 DIST( LISBON, SAOPAULO) 17600.00 0.0000000E+00 DIST( LISBON, BUONESAIRES) 21100.00 0.0000000E+00 DIST( MADRID, LISBON) 1000.000 0.0000000E+00 DIST( MADRID, MADRID) 0.0000000E+00 0.0000000E+00 DIST( MADRID, PARIS) 2500.000 0.0000000E+00 DIST( MADRID, SANTIAGO) 24000.00 0.0000000E+00 DIST( MADRID, SAOPAULO) 19500.00 0.0000000E+00 DIST( MADRID, BUONESAIRES) 23100.00 0.0000000E+00 DIST( PARIS, LISBON) 3200.000 0.0000000E+00 DIST( PARIS, MADRID) 2500.000 0.0000000E+00 DIST( PARIS, PARIS) 0.0000000E+00 0.0000000E+00 DIST( PARIS, SANTIAGO) 26200.00 0.0000000E+00 DIST( PARIS, SAOPAULO) 21100.00 0.0000000E+00 DIST( PARIS, BUONESAIRES) 24400.00 0.0000000E+00 DIST( SANTIAGO, LISBON) 24000.00 0.0000000E+00 DIST( SANTIAGO, MADRID) 24000.00 0.0000000E+00 DIST( SANTIAGO, PARIS) 26200.00 0.0000000E+00 DIST( SANTIAGO, SANTIAGO) 0.0000000E+00 0.0000000E+00 DIST( SANTIAGO, SAOPAULO) 7000.000 0.0000000E+00 DIST( SANTIAGO, BUONESAIRES) 4100.000 0.0000000E+00
Lingo Çözümü: DIST( SAOPAULO, LISBON) 17600.00 0.0000000E+00 DIST( SAOPAULO, MADRID) 19500.00 0.0000000E+00 DIST( SAOPAULO, PARIS) 21100.00 0.0000000E+00 DIST( SAOPAULO, SANTIAGO) 7000.000 0.0000000E+00 DIST( SAOPAULO, SAOPAULO) 0.0000000E+00 0.0000000E+00 DIST( SAOPAULO, BUONESAIRES) 4000.000 0.0000000E+00 DIST( BUONESAIRES, LISBON) 21100.00 0.0000000E+00 DIST( BUONESAIRES, MADRID) 23100.00 0.0000000E+00 DIST( BUONESAIRES, PARIS) 24400.00 0.0000000E+00 DIST( BUONESAIRES, SANTIAGO) 4100.000 0.0000000E+00 DIST( BUONESAIRES, SAOPAULO) 4000.000 0.0000000E+00 DIST( BUONESAIRES, BUONESAIRES 0.0000000E+00 0.0000000E+00 PASS( LISBON, LISBON) 0.0000000E+00 0.0000000E+00 PASS( LISBON, MADRID) 0.0000000E+00 0.0000000E+00 PASS( LISBON, PARIS) 0.0000000E+00 0.0000000E+00 PASS( LISBON, SANTIAGO) 62000.00 0.0000000E+00 PASS( LISBON, SAOPAULO) 69238.00 0.0000000E+00 PASS( LISBON, BUONESAIRES) 121000.0 0.0000000E+00 PASS( MADRID, LISBON) 0.0000000E+00 0.0000000E+00 PASS( MADRID, MADRID) 0.0000000E+00 0.0000000E+00 PASS( MADRID, PARIS) 0.0000000E+00 0.0000000E+00 PASS( MADRID, SANTIAGO) 81220.00 0.0000000E+00 PASS( MADRID, SAOPAULO) 126137.0 0.0000000E+00 PASS( MADRID, BUONESAIRES) 240213.0 0.0000000E+00
Lingo Çözümü: PASS( PARIS, LISBON) 0.0000000E+00 0.0000000E+00 PASS( PARIS, MADRID) 0.0000000E+00 0.0000000E+00 PASS( PARIS, PARIS) 0.0000000E+00 0.0000000E+00 PASS( PARIS, SANTIAGO) 74000.00 0.0000000E+00 PASS( PARIS, SAOPAULO) 155036.0 0.0000000E+00 PASS( PARIS, BUONESAIRES) 85000.00 0.0000000E+00 PASS( SANTIAGO, LISBON) 95000.00 0.0000000E+00 PASS( SANTIAGO, MADRID) 80054.00 0.0000000E+00 PASS( SANTIAGO, PARIS) 82000.00 0.0000000E+00 PASS( SANTIAGO, SANTIAGO) 0.0000000E+00 0.0000000E+00 PASS( SANTIAGO, SAOPAULO) 0.0000000E+00 0.0000000E+00 PASS( SANTIAGO, BUONESAIRES) 0.0000000E+00 0.0000000E+00 PASS( SAOPAULO, LISBON) 73847.00 0.0000000E+00 PASS( SAOPAULO, MADRID) 126851.0 0.0000000E+00 PASS( SAOPAULO, PARIS) 143977.0 0.0000000E+00 PASS( SAOPAULO, SANTIAGO) 0.0000000E+00 0.0000000E+00 PASS( SAOPAULO, SAOPAULO) 0.0000000E+00 0.0000000E+00 PASS( SAOPAULO, BUONESAIRES) 0.0000000E+00 0.0000000E+00 PASS( BUONESAIRES, LISBON) 110000.0 0.0000000E+00 PASS( BUONESAIRES, MADRID) 270051.0 0.0000000E+00 PASS( BUONESAIRES, PARIS) 83000.00 0.0000000E+00 PASS( BUONESAIRES, SANTIAGO) 0.0000000E+00 0.0000000E+00 PASS( BUONESAIRES, SAOPAULO) 0.0000000E+00 0.0000000E+00 PASS( BUONESAIRES, BUONESAIRES 0.0000000E+00 0.0000000E+00
Lingo Çözümü: Row Slack or Surplus Dual Price 1 0.2743843E+11 -1.000000 2 0.0000000E+00 0.0000000E+00 3 0.0000000E+00 1.000000 4 0.0000000E+00 0.0000000E+00 5 0.0000000E+00 0.0000000E+00 6 0.0000000E+00 0.0000000E+00 7 0.0000000E+00 0.0000000E+00 8 0.0000000E+00 0.0000000E+00 9 0.0000000E+00 -0.2743843E+11
Problem: Avrupa – Güney Amerika arasında düzenlenen uçuş seferlerinde; biri Amerika diğeri Avrupa’da olmak üzere iki tane aktarma noktası seçimi
 
min Σ (d E j E x + d E x S y + d S y S j ) T E i S j E x ,S y  E x S E i  E T *  S j  S
Problemin Lingo Modeli: SETS: S1/LISBON MADRID PARIS/; S2/SANTIAGO SAOPAULO BUONESAIRES/; B1(S1,S1):DIST1; B2(S2,S2):DIST2; A(S1,S2):DIST,PASS,HUB,AFI; ENDSETS DATA: DIST1= 0 1000 3200 1000 0 2500 3200 2500 0 ; DIST2= 0 7000 4100 7000 0 4000 4100 4000 0 ; DIST= 24000 17600 21100 24000 19500 23100 26200 21100 24400; PASS= 157000 143085 231000 161274 252988 425087 156000 384190 168000; ENDDATA MIN=@SUM(A(I,J):HUB(I,J)*AFI(I,J)); @FOR(A(I,J):@SUM(A(K,L):(DIST1(K,I)+DIST(I,J)+DIST2(J,L))*PASS(K,L))=HUB(I,J)); IFA=@MIN(A(I,L):HUB(I,L)); @FOR(A(M,N)|HUB(M,N)#LE#IFA:AFI(M,N)=1); @FOR(A(I,J):@BIN(AFI(I,J)));
Lingo Çözümü: Rows= 1 Vars= 8 No. integer vars= 8 ( all are linear) Nonzeros= 9 Constraint nonz= 0( 0 are +- 1) Density=1.000 Smallest and largest elements in absolute value= 0.494510E+11 0.670984E No. < : 0 No. =: 0 No. > : 0, Obj=MIN, GUBs <= 0 Single cols= 8 Optimal solution found at step: 0 Objective value: 0.4630561E+11 Branch count: 0 Variable Value Reduced Cost IFA 0.4630561E+11 0.0000000E+00 DIST1( LISBON, LISBON) 0.0000000E+00 0.0000000E+00 DIST1( LISBON, MADRID) 1000.000 0.0000000E+00 DIST1( LISBON, PARIS) 3200.000 0.0000000E+00 DIST1( MADRID, LISBON) 1000.000 0.0000000E+00 DIST1( MADRID, MADRID) 0.0000000E+00 0.0000000E+00 DIST1( MADRID, PARIS) 2500.000 0.0000000E+00 DIST1( PARIS, LISBON) 3200.000 0.0000000E+00 DIST1( PARIS, MADRID) 2500.000 0.0000000E+00 DIST1( PARIS, PARIS) 0.0000000E+00 0.0000000E+00
Lingo Çözümü: DIST2( SANTIAGO, SANTIAGO) 0.0000000E+00 0.0000000E+00 DIST2( SANTIAGO, SAOPAULO) 7000.000 0.0000000E+00 DIST2( SANTIAGO, BUONESAIRES) 4100.000 0.0000000E+00 DIST2( SAOPAULO, SANTIAGO) 7000.000 0.0000000E+00 DIST2( SAOPAULO, SAOPAULO) 0.0000000E+00 0.0000000E+00 DIST2( SAOPAULO, BUONESAIRES) 4000.000 0.0000000E+00 DIST2( BUONESAIRES, SANTIAGO) 4100.000 0.0000000E+00 DIST2( BUONESAIRES, SAOPAULO) 4000.000 0.0000000E+00 DIST2( BUONESAIRES, BUONESAIRE 0.0000000E+00 0.0000000E+00 DIST( LISBON, SANTIAGO) 24000.00 0.0000000E+00 DIST( LISBON, SAOPAULO) 17600.00 0.0000000E+00 DIST( LISBON, BUONESAIRES) 21100.00 0.0000000E+00 DIST( MADRID, SANTIAGO) 24000.00 0.0000000E+00 DIST( MADRID, SAOPAULO) 19500.00 0.0000000E+00 DIST( MADRID, BUONESAIRES) 23100.00 0.0000000E+00 DIST( PARIS, SANTIAGO) 26200.00 0.0000000E+00 DIST( PARIS, SAOPAULO) 21100.00 0.0000000E+00 DIST( PARIS, BUONESAIRES) 24400.00 0.0000000E+00 PASS( LISBON, SANTIAGO) 157000.0 0.0000000E+00 PASS( LISBON, SAOPAULO) 143085.0 0.0000000E+00 PASS( LISBON, BUONESAIRES) 231000.0 0.0000000E+00 PASS( MADRID, SANTIAGO) 161274.0 0.0000000E+00 PASS( MADRID, SAOPAULO) 252988.0 0.0000000E+00 PASS( MADRID, BUONESAIRES) 425087.0 0.0000000E+00
Lingo Çözümü: PASS( PARIS, SANTIAGO) 156000.0 0.0000000E+00 PASS( PARIS, SAOPAULO) 384190.0 0.0000000E+00 PASS( PARIS, BUONESAIRES) 168000.0 0.0000000E+00 HUB( LISBON, SANTIAGO) 0.6183313E+11 0.0000000E+00 HUB( LISBON, SAOPAULO) 0.4630561E+11 0.0000000E+00 HUB( LISBON, BUONESAIRES) 0.5203010E+11 0.0000000E+00 HUB( MADRID, SANTIAGO) 0.6102913E+11 0.0000000E+00 HUB( MADRID, SAOPAULO) 0.4945099E+11 0.0000000E+00 HUB( MADRID, BUONESAIRES) 0.5538335E+11 0.0000000E+00 HUB( PARIS, SANTIAGO) 0.6709839E+11 0.0000000E+00 HUB( PARIS, SAOPAULO) 0.5427308E+11 0.0000000E+00 HUB( PARIS, BUONESAIRES) 0.5958185E+11 0.0000000E+00 AFI( LISBON, SANTIAGO) 0.0000000E+00 0.6183313E+11 AFI( LISBON, SAOPAULO) 1.000000 0.0000000E+00 AFI( LISBON, BUONESAIRES) 0.0000000E+00 0.5203010E+11 AFI( MADRID, SANTIAGO) 0.0000000E+00 0.6102913E+11 AFI( MADRID, SAOPAULO) 0.0000000E+00 0.4945099E+11 AFI( MADRID, BUONESAIRES) 0.0000000E+00 0.5538335E+11 AFI( PARIS, SANTIAGO) 0.0000000E+00 0.6709839E+11 AFI( PARIS, SAOPAULO) 0.0000000E+00 0.5427308E+11 AFI( PARIS, BUONESAIRES) 0.0000000E+00 0.5958185E+11
Lingo Çözümü: Row Slack or Surplus Dual Price 1 0.4630561E+11 -1.000000 2 0.0000000E+00 0.0000000E+00 3 0.0000000E+00 1.000000 4 0.0000000E+00 0.0000000E+00 5 0.0000000E+00 0.0000000E+00 6 0.0000000E+00 0.0000000E+00 7 0.0000000E+00 0.0000000E+00 8 0.0000000E+00 0.0000000E+00 9 0.0000000E+00 0.0000000E+00 10 0.0000000E+00 0.0000000E+00 11 0.0000000E+00 0.0000000E+00 12 0.0000000E+00 -0.4630561E+11
TEŞEKKÜRLER :))

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Facility Location Problem

  • 1. ENDÜSTRİDE BİLGİSAYAR UYGULAMALARI - 2 New potential hubs in the South-Atlantic market. A problem of location Journal of Transport Geography Volume 11, Issue 2 , June 2003, Pages 139-149 FERİDE SEVLİ – 2000503052 AYTÜL ŞENER – 200503056 İLKEM YALINER – 2000503067
  • 2. İçerik: Tesis Planlama Kavramı Tesis Planlamanın Amaçları Tesis Planlama Tipleri Tesisin Kurulacağı Bölgenin Seçimi Alternatiflerin Değerlendirilme Yöntemleri Havacılık Sektöründe Tesis Planlama Uygulamaları
  • 3. TESİS PLANLAMA KAVRAMI: TESİS : Mal ve hizmetlerin fiilen üretildiği fiziksel birimlerdir. TESİS PLANLAMA : Tesislerin düzenleme çalışmalarının yapılması, tesisin kurulması ve faaliyete geçirilmesi aşamalarında gerçekleştirilecek olan faaliyetlerin bugünden eşgüdümlerinin yapılmasıdır.
  • 4. TESİS PLANLAMA AMAÇLARI: Kaynakların kullanımı Darboğazların belirlenmesi Tesis faaliyetleri Optimum yerleşim düzeni Maliyet Mesafe Zaman
  • 5. TESİS PLANLAMA TİPLERİ: Basit Yerleşim Çoklu Yerleşim Tesis Oluşturulması ve Kapasite Atama Tesis Seçimi ve Kapasite Atama Tesis İçi Yerleşim(QAP)
  • 6. TESİSİN KURULACAĞI BÖLGENİN SEÇİLMESİ: Yerleşim Kararlarını Etkileyen Faktörler : Ülke seçimi (kanunlar, pazar, işgücü, tedarik, alım gücü vb.) Bölge seçimi (maliyet, çevre kuralları, müşteri&hammadde erişimi vb.) Yer seçimi (yerin alan&maliyeti, ulaşım kolaylığı vb.)
  • 7. ALTERNATİFLERİ DEĞERLENDİRME YÖNTEMLERİ: Faktör Oranlama Metodu Başa-baş Analizi Ağırlık Merkezi Metodu Ulaştırma Metodu
  • 8. Problem: Güney Amerika–Avrupa arasında düzenlenen uçuş seferlerinde optimum aktarma noktası seçimi
  • 9.  
  • 10. Veriler: 1) Şehirler arası mesafeler 0 4000 4100 24400 23100 21100 BUONES AIRES 4000 0 7000 21100 19500 17600 SAO PAULO 4100 7000 0 26200 24000 24000 SANTIAGO 24400 21100 26200 0 2500 3200 PARIS 23100 19500 24000 2500 0 1000 MADRID 21100 17600 24000 3200 1000 0 LISBON BUONES AIRES SAO PAULO SANTIAGO PARIS MADRID LİSBON TO FROM
  • 11. Veriler: 2) Şehirler arası yolcu sayısı 0 0 0 83000 270051 110000 BUONES AIRES 0 0 0 143977 126851 73847 SAO PAULO 0 0 0 82000 80054 95000 SANTIAGO 85000 155036 74000 0 0 0 PARIS 240213 126137 81220 0 0 0 MADRID 121000 69238 62000 0 0 0 LISBON BUONES AIRES SAO PAULO SANTIAGO PARIS MADRID LİSBON TO FROM
  • 12. min Σ (d E i H x + d H x S j ) T E i S j H x  E u S E i  E T *  S j  S
  • 13. Problemin Lingo Modeli: SETS: AIRPORT/LISBON MADRID PARIS SANTIAGO SAOPAULO BUONESAIRES/:HUB,K; A(AIRPORT,AIRPORT):DIST,PASS; ENDSETS DATA: DIST=0 1000 3200 24000 17600 21100 1000 0 2500 24000 19500 23100 3200 2500 0 26200 21100 24400 24000 24000 26200 0 7000 4100 17600 19500 21100 7000 0 4000 21100 23100 24400 4100 4000 0 ; PASS=0 0 0 62000 69238 121000 0 0 0 81220 126137 240213 0 0 0 74000 155036 85000 95000 80054 82000 0 0 0 73847 126851 143977 0 0 0 110000 270051 83000 0 0 0 ; ENDDATA MIN=@SUM(AIRPORT(I):HUB(I)*K(I)); @FOR(AIRPORT(I):@SUM(A(J,K)|J#NE#I #AND# #NE#K:(DIST(J,I)+DIST(I,K))*PASS(J,K))=HUB(I)); IFA=@MIN(AIRPORT(I):HUB(I)); @FOR(AIRPORT(I)|HUB(I)#LE#IFA:K(I)=1); @FOR(AIRPORT(I):@BIN(K(I)));
  • 14. Lingo Çözümü: Rows= 1 Vars= 5 No. integer vars= 5 ( all are linear) Nonzeros= 6 Constraint nonz= 0( 0 are +- 1) Density=1.000 Smallest and largest elements in absolute value= 0.317284E+11 0.481254E No. < : 0 No. =: 0 No. > : 0, Obj=MIN, GUBs <= 0 Single cols= 5 Optimal solution found at step: 0 Objective value: 0.2743843E+11 Branch count: 0 Variable Value Reduced Cost IFA 0.2743843E+11 0.0000000E+00 HUB( LISBON) 0.3455933E+11 0.0000000E+00 HUB( MADRID) 0.2743843E+11 0.0000000E+00 HUB( PARIS) 0.3879355E+11 0.0000000E+00 HUB( SANTIAGO) 0.4812541E+11 0.0000000E+00 HUB( SAOPAULO) 0.3371716E+11 0.0000000E+00 HUB( BUONESAIRES) 0.3172843E+11 0.0000000E+00 K( LISBON) 0.0000000E+00 0.3455933E+11 K( MADRID) 1.000000 0.0000000E+00 K( PARIS) 0.0000000E+00 0.3879355E+11 K( SANTIAGO) 0.0000000E+00 0.4812541E+11 K( SAOPAULO) 0.0000000E+00 0.3371716E+11 K( BUONESAIRES) 0.0000000E+00 0.3172843E+11
  • 15. Lingo Çözümü: DIST( LISBON, LISBON) 0.0000000E+00 0.0000000E+00 DIST( LISBON, MADRID) 1000.000 0.0000000E+00 DIST( LISBON, PARIS) 3200.000 0.0000000E+00 DIST( LISBON, SANTIAGO) 24000.00 0.0000000E+00 DIST( LISBON, SAOPAULO) 17600.00 0.0000000E+00 DIST( LISBON, BUONESAIRES) 21100.00 0.0000000E+00 DIST( MADRID, LISBON) 1000.000 0.0000000E+00 DIST( MADRID, MADRID) 0.0000000E+00 0.0000000E+00 DIST( MADRID, PARIS) 2500.000 0.0000000E+00 DIST( MADRID, SANTIAGO) 24000.00 0.0000000E+00 DIST( MADRID, SAOPAULO) 19500.00 0.0000000E+00 DIST( MADRID, BUONESAIRES) 23100.00 0.0000000E+00 DIST( PARIS, LISBON) 3200.000 0.0000000E+00 DIST( PARIS, MADRID) 2500.000 0.0000000E+00 DIST( PARIS, PARIS) 0.0000000E+00 0.0000000E+00 DIST( PARIS, SANTIAGO) 26200.00 0.0000000E+00 DIST( PARIS, SAOPAULO) 21100.00 0.0000000E+00 DIST( PARIS, BUONESAIRES) 24400.00 0.0000000E+00 DIST( SANTIAGO, LISBON) 24000.00 0.0000000E+00 DIST( SANTIAGO, MADRID) 24000.00 0.0000000E+00 DIST( SANTIAGO, PARIS) 26200.00 0.0000000E+00 DIST( SANTIAGO, SANTIAGO) 0.0000000E+00 0.0000000E+00 DIST( SANTIAGO, SAOPAULO) 7000.000 0.0000000E+00 DIST( SANTIAGO, BUONESAIRES) 4100.000 0.0000000E+00
  • 16. Lingo Çözümü: DIST( SAOPAULO, LISBON) 17600.00 0.0000000E+00 DIST( SAOPAULO, MADRID) 19500.00 0.0000000E+00 DIST( SAOPAULO, PARIS) 21100.00 0.0000000E+00 DIST( SAOPAULO, SANTIAGO) 7000.000 0.0000000E+00 DIST( SAOPAULO, SAOPAULO) 0.0000000E+00 0.0000000E+00 DIST( SAOPAULO, BUONESAIRES) 4000.000 0.0000000E+00 DIST( BUONESAIRES, LISBON) 21100.00 0.0000000E+00 DIST( BUONESAIRES, MADRID) 23100.00 0.0000000E+00 DIST( BUONESAIRES, PARIS) 24400.00 0.0000000E+00 DIST( BUONESAIRES, SANTIAGO) 4100.000 0.0000000E+00 DIST( BUONESAIRES, SAOPAULO) 4000.000 0.0000000E+00 DIST( BUONESAIRES, BUONESAIRES 0.0000000E+00 0.0000000E+00 PASS( LISBON, LISBON) 0.0000000E+00 0.0000000E+00 PASS( LISBON, MADRID) 0.0000000E+00 0.0000000E+00 PASS( LISBON, PARIS) 0.0000000E+00 0.0000000E+00 PASS( LISBON, SANTIAGO) 62000.00 0.0000000E+00 PASS( LISBON, SAOPAULO) 69238.00 0.0000000E+00 PASS( LISBON, BUONESAIRES) 121000.0 0.0000000E+00 PASS( MADRID, LISBON) 0.0000000E+00 0.0000000E+00 PASS( MADRID, MADRID) 0.0000000E+00 0.0000000E+00 PASS( MADRID, PARIS) 0.0000000E+00 0.0000000E+00 PASS( MADRID, SANTIAGO) 81220.00 0.0000000E+00 PASS( MADRID, SAOPAULO) 126137.0 0.0000000E+00 PASS( MADRID, BUONESAIRES) 240213.0 0.0000000E+00
  • 17. Lingo Çözümü: PASS( PARIS, LISBON) 0.0000000E+00 0.0000000E+00 PASS( PARIS, MADRID) 0.0000000E+00 0.0000000E+00 PASS( PARIS, PARIS) 0.0000000E+00 0.0000000E+00 PASS( PARIS, SANTIAGO) 74000.00 0.0000000E+00 PASS( PARIS, SAOPAULO) 155036.0 0.0000000E+00 PASS( PARIS, BUONESAIRES) 85000.00 0.0000000E+00 PASS( SANTIAGO, LISBON) 95000.00 0.0000000E+00 PASS( SANTIAGO, MADRID) 80054.00 0.0000000E+00 PASS( SANTIAGO, PARIS) 82000.00 0.0000000E+00 PASS( SANTIAGO, SANTIAGO) 0.0000000E+00 0.0000000E+00 PASS( SANTIAGO, SAOPAULO) 0.0000000E+00 0.0000000E+00 PASS( SANTIAGO, BUONESAIRES) 0.0000000E+00 0.0000000E+00 PASS( SAOPAULO, LISBON) 73847.00 0.0000000E+00 PASS( SAOPAULO, MADRID) 126851.0 0.0000000E+00 PASS( SAOPAULO, PARIS) 143977.0 0.0000000E+00 PASS( SAOPAULO, SANTIAGO) 0.0000000E+00 0.0000000E+00 PASS( SAOPAULO, SAOPAULO) 0.0000000E+00 0.0000000E+00 PASS( SAOPAULO, BUONESAIRES) 0.0000000E+00 0.0000000E+00 PASS( BUONESAIRES, LISBON) 110000.0 0.0000000E+00 PASS( BUONESAIRES, MADRID) 270051.0 0.0000000E+00 PASS( BUONESAIRES, PARIS) 83000.00 0.0000000E+00 PASS( BUONESAIRES, SANTIAGO) 0.0000000E+00 0.0000000E+00 PASS( BUONESAIRES, SAOPAULO) 0.0000000E+00 0.0000000E+00 PASS( BUONESAIRES, BUONESAIRES 0.0000000E+00 0.0000000E+00
  • 18. Lingo Çözümü: Row Slack or Surplus Dual Price 1 0.2743843E+11 -1.000000 2 0.0000000E+00 0.0000000E+00 3 0.0000000E+00 1.000000 4 0.0000000E+00 0.0000000E+00 5 0.0000000E+00 0.0000000E+00 6 0.0000000E+00 0.0000000E+00 7 0.0000000E+00 0.0000000E+00 8 0.0000000E+00 0.0000000E+00 9 0.0000000E+00 -0.2743843E+11
  • 19. Problem: Avrupa – Güney Amerika arasında düzenlenen uçuş seferlerinde; biri Amerika diğeri Avrupa’da olmak üzere iki tane aktarma noktası seçimi
  • 20.  
  • 21. min Σ (d E j E x + d E x S y + d S y S j ) T E i S j E x ,S y  E x S E i  E T *  S j  S
  • 22. Problemin Lingo Modeli: SETS: S1/LISBON MADRID PARIS/; S2/SANTIAGO SAOPAULO BUONESAIRES/; B1(S1,S1):DIST1; B2(S2,S2):DIST2; A(S1,S2):DIST,PASS,HUB,AFI; ENDSETS DATA: DIST1= 0 1000 3200 1000 0 2500 3200 2500 0 ; DIST2= 0 7000 4100 7000 0 4000 4100 4000 0 ; DIST= 24000 17600 21100 24000 19500 23100 26200 21100 24400; PASS= 157000 143085 231000 161274 252988 425087 156000 384190 168000; ENDDATA MIN=@SUM(A(I,J):HUB(I,J)*AFI(I,J)); @FOR(A(I,J):@SUM(A(K,L):(DIST1(K,I)+DIST(I,J)+DIST2(J,L))*PASS(K,L))=HUB(I,J)); IFA=@MIN(A(I,L):HUB(I,L)); @FOR(A(M,N)|HUB(M,N)#LE#IFA:AFI(M,N)=1); @FOR(A(I,J):@BIN(AFI(I,J)));
  • 23. Lingo Çözümü: Rows= 1 Vars= 8 No. integer vars= 8 ( all are linear) Nonzeros= 9 Constraint nonz= 0( 0 are +- 1) Density=1.000 Smallest and largest elements in absolute value= 0.494510E+11 0.670984E No. < : 0 No. =: 0 No. > : 0, Obj=MIN, GUBs <= 0 Single cols= 8 Optimal solution found at step: 0 Objective value: 0.4630561E+11 Branch count: 0 Variable Value Reduced Cost IFA 0.4630561E+11 0.0000000E+00 DIST1( LISBON, LISBON) 0.0000000E+00 0.0000000E+00 DIST1( LISBON, MADRID) 1000.000 0.0000000E+00 DIST1( LISBON, PARIS) 3200.000 0.0000000E+00 DIST1( MADRID, LISBON) 1000.000 0.0000000E+00 DIST1( MADRID, MADRID) 0.0000000E+00 0.0000000E+00 DIST1( MADRID, PARIS) 2500.000 0.0000000E+00 DIST1( PARIS, LISBON) 3200.000 0.0000000E+00 DIST1( PARIS, MADRID) 2500.000 0.0000000E+00 DIST1( PARIS, PARIS) 0.0000000E+00 0.0000000E+00
  • 24. Lingo Çözümü: DIST2( SANTIAGO, SANTIAGO) 0.0000000E+00 0.0000000E+00 DIST2( SANTIAGO, SAOPAULO) 7000.000 0.0000000E+00 DIST2( SANTIAGO, BUONESAIRES) 4100.000 0.0000000E+00 DIST2( SAOPAULO, SANTIAGO) 7000.000 0.0000000E+00 DIST2( SAOPAULO, SAOPAULO) 0.0000000E+00 0.0000000E+00 DIST2( SAOPAULO, BUONESAIRES) 4000.000 0.0000000E+00 DIST2( BUONESAIRES, SANTIAGO) 4100.000 0.0000000E+00 DIST2( BUONESAIRES, SAOPAULO) 4000.000 0.0000000E+00 DIST2( BUONESAIRES, BUONESAIRE 0.0000000E+00 0.0000000E+00 DIST( LISBON, SANTIAGO) 24000.00 0.0000000E+00 DIST( LISBON, SAOPAULO) 17600.00 0.0000000E+00 DIST( LISBON, BUONESAIRES) 21100.00 0.0000000E+00 DIST( MADRID, SANTIAGO) 24000.00 0.0000000E+00 DIST( MADRID, SAOPAULO) 19500.00 0.0000000E+00 DIST( MADRID, BUONESAIRES) 23100.00 0.0000000E+00 DIST( PARIS, SANTIAGO) 26200.00 0.0000000E+00 DIST( PARIS, SAOPAULO) 21100.00 0.0000000E+00 DIST( PARIS, BUONESAIRES) 24400.00 0.0000000E+00 PASS( LISBON, SANTIAGO) 157000.0 0.0000000E+00 PASS( LISBON, SAOPAULO) 143085.0 0.0000000E+00 PASS( LISBON, BUONESAIRES) 231000.0 0.0000000E+00 PASS( MADRID, SANTIAGO) 161274.0 0.0000000E+00 PASS( MADRID, SAOPAULO) 252988.0 0.0000000E+00 PASS( MADRID, BUONESAIRES) 425087.0 0.0000000E+00
  • 25. Lingo Çözümü: PASS( PARIS, SANTIAGO) 156000.0 0.0000000E+00 PASS( PARIS, SAOPAULO) 384190.0 0.0000000E+00 PASS( PARIS, BUONESAIRES) 168000.0 0.0000000E+00 HUB( LISBON, SANTIAGO) 0.6183313E+11 0.0000000E+00 HUB( LISBON, SAOPAULO) 0.4630561E+11 0.0000000E+00 HUB( LISBON, BUONESAIRES) 0.5203010E+11 0.0000000E+00 HUB( MADRID, SANTIAGO) 0.6102913E+11 0.0000000E+00 HUB( MADRID, SAOPAULO) 0.4945099E+11 0.0000000E+00 HUB( MADRID, BUONESAIRES) 0.5538335E+11 0.0000000E+00 HUB( PARIS, SANTIAGO) 0.6709839E+11 0.0000000E+00 HUB( PARIS, SAOPAULO) 0.5427308E+11 0.0000000E+00 HUB( PARIS, BUONESAIRES) 0.5958185E+11 0.0000000E+00 AFI( LISBON, SANTIAGO) 0.0000000E+00 0.6183313E+11 AFI( LISBON, SAOPAULO) 1.000000 0.0000000E+00 AFI( LISBON, BUONESAIRES) 0.0000000E+00 0.5203010E+11 AFI( MADRID, SANTIAGO) 0.0000000E+00 0.6102913E+11 AFI( MADRID, SAOPAULO) 0.0000000E+00 0.4945099E+11 AFI( MADRID, BUONESAIRES) 0.0000000E+00 0.5538335E+11 AFI( PARIS, SANTIAGO) 0.0000000E+00 0.6709839E+11 AFI( PARIS, SAOPAULO) 0.0000000E+00 0.5427308E+11 AFI( PARIS, BUONESAIRES) 0.0000000E+00 0.5958185E+11
  • 26. Lingo Çözümü: Row Slack or Surplus Dual Price 1 0.4630561E+11 -1.000000 2 0.0000000E+00 0.0000000E+00 3 0.0000000E+00 1.000000 4 0.0000000E+00 0.0000000E+00 5 0.0000000E+00 0.0000000E+00 6 0.0000000E+00 0.0000000E+00 7 0.0000000E+00 0.0000000E+00 8 0.0000000E+00 0.0000000E+00 9 0.0000000E+00 0.0000000E+00 10 0.0000000E+00 0.0000000E+00 11 0.0000000E+00 0.0000000E+00 12 0.0000000E+00 -0.4630561E+11