I
DECLARATION
I dec lare t hat t he d iss ert at io n here by su bmit t ed t o t he U niver s it y o f Limpo po fo r t he degre e o f Mast ers S c ie nce has no t pre vio u s ly be e n su bmit t ed by me fo r a degree at t his o r any o t her unive rs it y, t hat it is my o w n wo rk in de s ig n a nd e xecut io n, a nd t hat a ll mat er ia l co nt a ine d t here in ha s bee n du ly ack no w led ged.
S ig ned:___ ______ ______
Dat e:________ ______ ___
II
I wo uld lik e t o ackno w ledg e a ll t he peo p le w ho co nt ribut ed in ad vis ing a nd as s ist ing me w hile wo rk ing o n t his pro je c t . I t hank my sup er viso rs Pro f. M A Le sao a na a nd Pro f. V S S Y ada va lli fo r t he ir gu id a nc e, va lu a ble co mme nt s, pat ie nc e a nd t he ir a ll t ime acc es s ib ilit y in super vis ing t his d is s ert at io n t hro ug h it s var io u s p ha se s.
T he fina nc ia l as s ist a nce o f t he Nat io na l R es earc h Fo und at io n (NR F) i n co nju nct io n w it h t he U n iver s it y o f Limp o po , to wards t his res earc h is here b y ack no w ledg ed.
Spec ia l t ha nk s go to my mo t her, T ho ro met ja ne a nd my s ist e r Ngao anap ho kwa ne fo r t he ir fina nc ia l a ss ist ance t hro ugho ut my sc ho o l life. T o my lo ve ly w ife, Prec io u s a nd my lit t le g ir l, T u me lo , I a ffa bly t ha nk yo u fo r t he e nco urage me nt , pat ie nc e a nd a ll t ime supp o rt .
Abo ve a ll, I g ive g lo r y t o Almig ht y Go d who is a ble.
III
ABSTRACT
T his st ud y is fo c us ed o n t he sc hedu ling o f min ibu s (t ax i) t ra nspo rt bet wee n t he C it y o f Po lo kwa ne a nd M a nkw e ng T o wns hip in t he Limpo po pro vinc e o f So ut h Afr ic a. New fo r mu lat io ns o f int eger pro gramming fo r t axi t imet a blin g pro ble ms w ere pre se nt ed. T he pro ble m is mo d e led a s a s ing le link t rans it . Har d a nd so ft co nst ra int s ha ve bee n id e nt ified.
T he ma in o bje ct ive o f t he st ud y is t o desig n t he be st sc hed u le t hat min imize s t he w a it ing t ime o f pa ss e nger s a nd pro vid es bet t er ser vice t o t he pu blic fo r t axi t ra nspo rt o n t he a fo re me nt io ned link.
In t his st ud y t he pro po sed so lut io n met ho ds ma nage d t o pro duce a t imet ab le w it h a no n- co nflict ing set o f t axis a nd no co nse cut ive as s ig nme nt o f o ne t a x i to t he t rip s w it hin a dur at io n t ime (t ot a l t ime t rave led o n a ret urn t r ip bet wee n t wo lo cat io ns). Lo ca l s earc h a lgo r it hms suc h as t he G e net ic Algo r it hm ar e used in t he r es earc h, ma in ly du e t o it s fle x ib ilit y a nd po wer t o pro duce t he best so lut io n t o t he t imet a bling pro ble ms. T he a lgo r it hm st art s w it h a po pu lat io n o f a fea s ib le s earc h sp ac e. T wo o perato rs: mut at io ns a nd cro sso ver s, were de s ig ned in su c h a wa y t hat t he y do no t pro duce infe as ib le o ffs pr ing.
IV
Table of contents
DECLAR AT ION ……… ……… …… ……… ……… I AC KN O WLEDGEMENT S ……… … ……… …………. . II ABST R ACT ……… ……… ……… ………. III ACR ONY MNS ……… ……… ……… ……… IV T AB LE O F CO NT ENT S ……… ……… ……… V LIST OF FIGURES……….. ……... VII APPENDI CES……… ……… ……… ………. . VII
CHAPTER 1 ……… ……… ……… ……… 1
INT RODU CT ION……… ……… ……… …………. . 1
1. 1 Int ro duct io n . . ……… …… ……… …… 1
1. 2 Geo graphy ……… ……… ………… ……… ……. 3
1. 3 Pro ble m st at eme nt ……… ……… ……. . 5
1. 4 Obje ct ive Fu nct io n ……… ……… ……… ……. 10
1. 5 Pre liminar y d at a ……… ……… ……. 10
1. 6 Organiz at io n o f t he repo rt ……… ………. 13
CHAPTER 2 ……… ……… ……… ……… 14
LIT ER AT URE REVIE W……… ……… ……… ………. 14
2. 1 Int ro duct io n…… ……… ……… ……… ……. . . 15
2. 2 T imet a bling pro ble m……… ………… ……… ……… 15
2. 2. 1 Hard a nd So ft co nst ra int s……… …… ………. 15
2. 2. 2 Fe as ib le so lut io n o r t imet a ble ………… … ………. 16
V
2. 3 Univers it y t imet ab ling pro ble ms ……… … ……… ……… 20
2. 3. 1 Univers it y co urse t imet a bling pro ble ms … ………. . . . . 21
2. 3. 2 Univers it y e xa minat io n t imet a bling pro b le ms ……… ……. 25
2. 4 Public T ra nspo rt Schedu ling pro ble ms … ……… ………. 28
2. 5 So lut io n t echn ique s o f u niver s it y t imet a bling ……… ……… 34
CHAPTER 3……… ……… ……… ……… 45
SCHEDULIN G MO DEL: AN APP LIC AT I ON……… 45
3. 1 Int ro duct io n…… ……… ……… ……… ……. . . 46
3. 2 Ap p licat io n o f U n iver s it y T imet a bling pro ble m……… ………… 46
3. 2. 1 Pro ble m s it uat io n……… ……… … ……… 46
3. 2. 2 T axi t imet a bling ……… ……… ………. 48
3. 2. 3 T he hard co nst ra int s ……… ……… ………. 50
3. 2. 4 T he so ft co nst ra int s……… ……… ………. . 51
3. 2. 5 T he o bje ct ive fu nct io n… ……… ………. . 52
3. 2. 6 Eva lu at io n o f t he fea s ib le sc hedu le ……… ……… 53
3. 3 Mat he mat ic a l pro gra mming mo d e l……… ……… ………. 54
3. 3. 1 As su mpt io ns……… ……… ………… ………. . 54
3. 3. 2 De fin it io ns………… ……… ……… ………. 55
3. 3. 3 Not at io ns……… ……… ………… ……… 56
CHAPTER 4……… ……… ……… ………….. 60
AN ALYSIS OF T HE MO DEL……… …… ……… ………. 60
4. 1 Co mput at io na l e xper ime nt s a nd re su lt s… ……… ……. 61
CHAPTER 5……… ……… ……… ……… 64
CON CLUSION…. . ……… ……… ……… ………. . 64
REFER ENC ES………… ……… … ……… ……… 67
VI 1. CBD: Ce nt ra l B us ine s s D ist r ict . 2. UL: Univers it y o f Limpo po .
3. GMT A: Great er Ma nkw e ng T a xi As so c iat io n.
4. UT P: Univers it y T imet a bling Pro ble m.
5. CT P: Co urse T imet a bling Pro ble m.
6. ET P: Exa minat io n T imet a bling Pro ble m.
7. G A: Ge net ic Algo r it hm 8. SA: S imu lat ed Annea ling 9. T S: T abu Se arc h.
10. T T P: T ra in T imet ab ling Pro ble m.
11. VT P: Ve hic le T imet a bling Pro ble m.
12. VSP: Ve hic le S c hedu ling Pro ble m.
13. AT P: Air line T imet a bling Pro ble m.
14. BSP: Bus Sc hedu ling Pro ble m.
15. BCSP: Bus Cr ew Sc he du ling Pro ble m.
16. RCP: Ro bu st Co lo ur ing Pro ble m.
17. IP: Int eger Pro gra mming 18. SPP: Set Part it io ning Pro ble m
VII
LIST OF FIGURES
1. Figur e 1. 1: T he So ut h Afr ic a map……… ……… ………. 4 2. Figur e 1. 2: T he Limpo po pro vinc ia l ma p……… … ………. 6 3. Figur e 1. 3: Est imat ed nu mber o f p as se ngers t ranspo rted
da ily bet wee n Ma nkwe ng a nd Po lo kwa ne ………. . 12 4. Figur e 1. 4: Nu mber o f a cc id e nt s acco rd ing t o t he fact o rs…… 13 5. Figur e 2. 1: T he c la s se s in t he ge nera l t imet a bling mo de l……. 19 6. Figur e 3. 1: A ve h ic le blo ck de ve lo p me nt ……… ………. . 48 7. Figur e 3. 2: Est imat ed nu mber o f p as se ngers t ranspo rted
da ily ……… ……… ………… ………. 49 8. Figur e 3. 3: Schedu ling a nd res c hed u ling o f t imet a ble ………. . 50 9. Figur e 4. 1: T he o bje ct ive fu nct io n fo r inst a nce o f 30 t r ip s…. 62 10. Figur e 4. 2: T he o bje ct ive fu nct io n fo r inst a nce o f 50 trips…… 63
APPENDICES
1. Ap pe nd ix 1: A st ruct ured quest io nna ir e fo r pre limina r y dat a…73
