Modified Ant Colony Algorithm for the Vehicle Routing Problem with Time Windows

M Taghavifard, K. Sheikh & A. Shahsavari,

M Taghavifard, Assistant Prof., Allame Tabatabaei Univ., Tehran-Iran, [email protected]
K. Sheikh, Industrial Engineering Dept., Faculty of Graduate Studies, Azad Univ., Tehran South Branch,[email protected]
A. Shahsavari, Industrial Engineering Dept., Faculty of Graduate Studies, Azad Univ., Tehran South Branch, [email protected]

Keywords ABSTRACT

Keywords ABSTRACT

Vehicle Routing
Problem with
Time Windows,
Ant Colony
Algorithm, Solomon testproblems Vehicle Routing Problem with Time Windows (VRPTW) is an NP-Complete Optimization Problem. Even finding an optimal solution for small size problems is too hard and time-consuming. The objective of VRPTW is to use a fleet of vehicles with specific capacity to serve a number of customers with dissimilar demands and time window constraints at minimum cost, without violating the capacity and time window constraints. This problem has been solved with a number of heuristic and meta-heuristic solution algorithms and optimal or near optimal solutions gained. In this paper, a modified Ant Colony algorithm is proposed. In this algorithm we tried to simplify the solution procedure and computational complexities of ant colony metaheuristic. To gain this capability, we sacrificed some computational accuracy. Testing the solution procedure on the Solomon test-problems showed that this algorithm is capable of generating relatively good solutions.
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ƾƳŚƯŻƵźŬƴěƽŚƷŢƿŵƹŶŰƯíƾƫřëšLJŵŚƘƯŶƃŚŝƾƯƶǀƬƤƳƶƬǀſƹ
ŶƃŚŝƾƯƽżĩźƯƺěŵƹƱŚƿźŤƄƯƽřźŝ
ƱŚĮģŹƺƯƾƳƺƬĩƮŤƿŹƺĮƫřè
ƽřźŝŶƿŶūƽŹŚĪŤŝřřźƟƁƹŹĨƿ(ACO)ƱŚĮģŹƺƯƾƳƺƬĩƮŤƿŹƺĮƫř ƺĩŹŚƯ Ǝſƺţ ŹŚŝ Ʋǀƫƹř įřźŝ ƹ ŶƃŚŝƾƯ ƽŻŚſ ƶƴǀƸŝ ƪŗŚƀƯ ƪů ƽŚƷŹŚŤƟŹźŝƾƴŤŞƯįŹŚĪŤŝřřźƟƮŤƿŹƺĮƫřƲƿř>ê@ŶƃŭźƐƯƺĮƿŹƹŵ ŚƴǀŝŚƳ ƾƘƣřƹ ƽŚƷƶģŹƺƯ Ɩƣřƹ Źŵ Ţſř ƾƘƣřƹ ƽŚƷƶģŹƺƯ ƾƳƺƬĩ řŹřŵřŹƾƿřŸƛƖŞƴƯŢưƀŝźǀƀƯƲƿźŤƷŚţƺĩƲŤƟŚƿƾƿŚƳřƺţŚƯřŶƴŤƀƷ
ƾƯƭŚŬƳřƱƺƯźƟźŧřƽŹřŸĭƽŚūźŝĨưĩƶŝřŹŹŚĪƴƿřƹŶƴƃŚŝƾƯ æå ƹ î šLJŵŚƘƯ ƩŚƤŤƳř ƲǀƳřƺƣ Żř ƾƗƺƴƈƯ ƽŚƷƶģŹƺƯ ŶƴƷŵ ŶƈƤƯƹŶǀƬƤţřŹƾƘǀŞƏƽŚƷƶģŹƺƯŹŚŤƟŹƲƿřƹƵŵƺưƳƵŵŚƠŤſř ƲǀƳřƺƣ Żř ƾƗƺƴƈƯ ƽŚƷƶģŹƺƯ ƲǀƴĤưƷ ŶƴŝŚƿƾƯ řŹ ƽŶƘŝ
ƱōŻřƹŶƴƿŚưƳƾƯƵŵŚƠŤſřæçƹææšLJŵŚƘƯƱƺƯźƟźŧřƾƳŚſŹŻƹźŝ ŶƴƴĩƾƯƵŵŚƠŤſřŚƷźǀƀƯƽƹŹƱƺƯźƟźŧřƾƳŚſŹŻƹźŝƽřźŝ
j  arg rmaxTK i i,rvi,r if q  q0 î
Jo.w.
ĨƿŵŚŬƿřƱōƶƠǀƓƹƶĩŶƃŚŝƾƗƺƴƈƯƶģŹƺƯĨƿ k ŶǀƴĩƉźƟ ƶŝ ƹ ŶƿŚưƳƾƯ šŚƣLjƯ řŹ ƱŚƿźŤƄƯ ƾƯŚưţ ƶģŹƺƯ Ʋƿř Ţſř źǀƀƯ
ŢŰţ ƾŤƀǀƫ ƭř k ƶģŹƺƯ Śŝ ƵřźưƷ ŵŵźĭƾƯ ŻŚŝ ƶǀƫƹř ƶƐƤƳ ŶƳřƵŶƄƳ šŚƣLjƯ ŻƺƴƷ ƶĩ ƾƳŚƿźŤƄƯ ƾƯŚưţ ƪƯŚƃTkiƱřƺƴƗ ŽŚſřźŝŵŹřŵŹřźƣƭř i ƽźŤƄƯƪŰƯŹŵƶĩƭř k ƶģŹƺƯŵŹřŵŵƺūƹ ƵŶƴƷŵ ƱŚƄƳ i,r ŵƹŹƾƯ ƭř j ƽźŤƄƯ ƪŰƯ ƶŝ èæ ƶƫŵŚƘƯ
ŶƃŚŝƾƯ r ƹ i ƽźŤƄƯ ƹŵ Ʋǀŝ ƱŚưĩ ƽƹŹ źŝ ƱƺƯźƟ źŧř ŹřŶƤƯ
ƹŵ Ʋǀŝ ƶƬƇŚƟ ŽƺĪƘƯ Żř ƹ Ţſř ƽŹŚĪŤŝř ŹřŶƤƯ Ĩƿ vi,r
ŢǀưƷřƶĩŶƃŚŝƾƯźŤƯřŹŚěĨƿ ƹŵƺƃƾƯƪƇŚů r ƹ i ƽźŤƄƯ ƾƟŵŚƈţŹƺƐŝƶĩŢſřƽŹřŶƤƯ q ŶƷŵƾƯƱŚƄƳřŹ vi,rƾŞƀƳ
źŝŹŚĩƎſƺţƶĩŢſřƽźŤƯřŹŚěżǀƳ q0 ƹŵƺƃƾƯŶǀƫƺţæƹåƲǀŝ ƾƟŵŚƈţźǀƜŤƯĨƿżǀƳ J ŶƃŚŝƾƯæƹåƲǀŝƹŵƺƃƾƯƲǀǀƘţ
ŶǀƫƺţæåƶƫŵŚƘƯŹŵƵŶƃƶŗřŹřƩŚưŤůřƖƿŻƺţƖŝŚţŽŚſřźŝƹŢſř
ƾƯƭř j ƽźŤƄƯśŚŴŤƳřƩŚưŤůř Pk i, jæåƶƫŵŚƘƯŹŵŵƺƃƾƯ
ŶƃŚŝ
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Pk  i,ji,rviv,ij,r if jTk  i æå i, j rTK i
0o.w.
Ʋƿř ŵƺƃƾƯ ƵŵŚƠŤſřƱƺƯźƟ źŧř ƾƬŰƯ ƾƳŚſŹŻƹźŝ ƽřźŝ ææ ƶƫŵŚƘƯ ƪŰƯ Żř ƭř k ƶģŹƺƯ ƶĩ ŶŤƟřƾƯ ơŚƠţř ƾƳŚƯŻ ƾƬŰƯ ƾƳŚſŹŻƹźŝ ƺŰƯźŤƯřŹŚě ƶƫŵŚƘƯƲƿřŹŵŵƹŹƾƯ j ƽźŤƄƯƪŰƯƶŝ i ƽźŤƄƯ

KN
 X ijk 1for
k1 i0 j1,2,…Nè
KN
 X ijk 1for
k1 j0
i1,2,…Né
N
N mi X ijk  qk
i0 j0
KN fork 1,2,…,Kê
k1 j0 X ijk t i ti, j f  i  w i  tj ë
forj1,2,…N
ƱōŹŵƶĩ t0 f 0 w0 0
N N X ijk ti, j f  i  w i  l0 ìi0 j0
fork 1,2,…K
eiti wi li fori1,2,…,Ní
ƶĩƽŹƺƐŝ
k 1,2,…,Kƹƭř k ƶǀƬƤƳƶƬǀſƹ k
i1,2,…Nƹƭř i ƽźŤƄƯ Ci
ƽżĩźƯƺěŵ C0
ƶĩ Cj ƹ Ci Ʋǀŝ ƾſŶǀƬƣř ƎŴƫř ƮǀƤŤƀƯ ƶƬƇŚƟ di, j
i  j ƹ i, j1,2,…N
i, j1,2,…Nƶĩ Cj ƶŝ Ci ŻřƾƿŚŬŝŚūƶƴƿżƷ ci, j
i  j ƹ
ƹ i, j1,2,…N ƶĩ Cj ƹ Ci Ʋǀŝ źƠſ ƱŚƯŻ ti, j
i  j
ƭř i ƽźŤƄƯƽŚƋŚƤţ mi
ƭř k ƶǀƬƤƳƶƬǀſƹŢǀƟźƓ qk ƭř i ƽźŤƄƯƪŰƯƶŝŵƹŹƹƱŚƯŻƲƿźţŵƹŻ ei
ƭř i ƽźŤƄƯƪŰƯƶŝŵƹŹƹƱŚƯŻƲƿźţźƿŵ li
ƭř i ƽźŤƄƯƪŰƯŹŵžƿƹźſƱŚƯŻ f i
ƭř i ƽźŤƄƯƪŰƯƶŝŵƹŹƹƱŚƯŻ ti
ƭř i ƽźŤƄƯƪŰƯŹŵŹŚƔŤƳřƱŚƯŻ wi
ƶŝ i ƵźĭŻřƶǀƬƤƳƶƬǀſƹĨƿƶĩƾƳŚƯŻƽřźŝĨƿ X ijk 1or0
šŹƺƈƴƿřźǀƛŹŵźƠƇƹŶƃŚŝƵŵźĩŢĩźů j Ƶźĭ
ƽŹřƺƀƸƃƲƿŹōƹŲǀƃƱřƺǀĩŵźƟƽƺƤţƾƤţŶưŰƯƪƿŚſƹƾŝŚƿźǀƀƯƶƬŘƀƯƪůŢƸūƱŚĮģŹƺƯƾƳƺƬĩƵŶƃŭLjƇřƁƹŹƶŗřŹřçë
ç ƹ æ šLJŵŚƘƯ ŶƃŚŝƾƯ ƪĩ ƶƴƿżƷ ƱŵƺưƳ ƶƴǀưĩ ƝŶƷ k ŢƫŚůƲƿźŤƄǀŝŹŵŶƴƷŵƾƯƵŻŚūřƶĩŶƴƃŚŝƾƯƾƿŚƷŢƿŵƹŶŰƯ ƱŚưƷƶŝľřŵŶŬƯŢƿŚƸƳŹŵƹƵŵƺưƳħźţřŹƽżĩźƯƺěŵƶǀƬƤƳƶƬǀſƹ ƎſƺţƱŶƃšŚƣLjƯƶŝřŹƽźŤƄƯźƷéƹèšLJŵŚƘƯŶƳŵźĭŻŚŝƺěŵ ŢƃřŵŵŚƿ ƾƳƺŤſ šŹƺƈŝ ŚƸƳō x, yšŚƈŤŴƯ ƹŚƷƵźĭ ƱŚƿźŤƄƯ Ƶźĭ Ʋƿř Żř řŹ ŚƷƵźĭ ƾƯŚưţ ƶƬƇŚƟ Ŷǀƴĩ ŻŚƛō ƺěŵ Ƶźĭ Żř ŶǀƴƧ ĨƿŹŵřŹƵźĭƭŚƳƹŶƿźǀĮŝźƔƳŹŵřŹƶƬƇŚƟƲƿźŤưĩŶǀŗŚưƳƶŞſŚŰƯ
ŶǀƴĩřŶǀěƪŞƣƶƬůźƯƵźĭŻřřŹŚƷƵźĭƾƯŚưţƶƬƇŚƟŶǀƀƿƺƴŝŢƀǀƫ ƶƯřŵřřŹŹŚĪƴƿřŶǀƴĩƶƟŚƋřŢƀǀƫƶŝƹƶŤƟŚƿřŹƶƬƇŚƟƲƿźŤưĩľřŵŶŬƯ ƽřƵźĭźĮƿŵƹŶƳƺƃŵŹřƹŢƀǀƫŹŵŵƺūƺƯƽŚƷƵźĭƾƯŚưţŚţƵŵřŵ ƲƿźųōŚţŶƿřƵŵƺưƳƕƹźƃƱōŻřƶĩƽřƵźĭŻřŜǀţźţƶŝŶƳŚưƳƾƣŚŝ ƱŚưƷ ŹřŶƤƯ ƲƿřŶǀŗŚưƳ Ɩưū ƮƷ Śŝ řŹ ŚƷƶƬƇŚƟ ƲƿźŤưĩ Ƶźĭ
ƲŤƟŚƿŻřžěŶƃŚŝƾƯƶǀƫƹřƶūƺƯźǀƛŚƿƶūƺƯźǀƀƯƲƿźŤƷŚţƺĩ
ƶǀƫƹřƱƺƯźƟŹřŶƤƯƹƶŤƟźĭźƔƳŹŵřźƳōƩƺƏƶǀƫƹřźǀƀƯƲƿźŤƷŚţƺĩ
ŹŵƶĩŵźǀĭƾƯŹřźƣźƿŻƶƐŝřŹŚŝźŝřźŝƶƬŘƀƯƽŚƷźǀƀƯƾƯŚưţƽřźŝ ƽŚƷƵźĭ ƱŚƿźŤƄƯ ŵřŶƘţ n ƹ źǀƀƯ ƲƿźŤƷŚţƺĩ ƩƺƏ L Ʊō ŢſřźǀƀƯƽƹŹƶǀƫƹřƱƺƯźƟŹřŶƤƯżǀƳ0 ŶƃŚŝƾƯƪůŹŵŵƺūƺƯ
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ŻřžěƾĪƿřŹŚƷƵźĭŜǀţźţƲƿŶŝƹƶŤƟŚƿƽŶƘŝŢĩźůƽřźŝřŹƵźĭ
ƹŶƳŚưƳƾƣŚŝƽřƵźĭźĮƿŵƶĩƾƳŚƯŻŚţŵŹřŸĭƾƯźſŢƄěƽźĮƿŵ ƱŶƃ źě źƏŚŴŝ ƶģŹƺƯ Ţſř ƲĪưƯ ƶŤŞƫř ŵŵźĭ ŻŚŝ ƺěŵ ƶŝ ƶģŹƺƯ ŵŵźĭŻŚŝƺěŵƵźĭƶŝŚƷŹŚŝƹŚƷŹŚŝƾƳŚƯŻƽŚƷƵźŬƴěƊƤƳŚƿŢǀƟźƓ ƪǀĪƄţřŹŵƺųƽŚƷźǀƀƯƶƬǀſƹƶģŹƺƯƕƺƳ k ƾƯŚưţƶĪƳōŻřžě ƹŶƳźǀĭƾƯŹřźƣƾŝŚƿŻŹřŵŹƺƯƾƿŚƷŹŚǀƘƯŽŚſřźŝŚƷźǀƀƯƲƿřŶƳŵřŵ ŵŹƺƯƶƬŘƀƯƽŚƷźǀƀƯƱƺƯźƟŹřŶƤƯƾƳŚſŹŻƹźŝƽřźŝźǀƀƯƲƿźŤƸŝ ŹŚŞĪƿ ŚƸƴţ ƮŤƿŹƺĮƫř Ʋƿř Źŵ Ɩƣřƹ Źŵ ƾƴƘƿ ŵźǀĭƾƯ Źřźƣ ƵŵŚƠŤſř
ƮŤƿŹƺĮƫřŻřŹřźĪţźƷƱŚƿŚěŹŵƮƷƱōƹŵźǀĭƾƯƭŚŬƳřƾƳŚſŹŻƹźŝ
Ţſř
ŵƺưƳ ƞƿźƘţ Ʊō ƞƣƺţ ƽřźŝ ƽŹŚǀƘƯ ŶƿŚŝ ƮŤƿŹƺĮƫř ƭŚưţř ŹƺƔƴưŝ
ŶƳřƵŶƃƶŗřŹřźƿŻŹŵƞƿźƘţƪŝŚƣƽŚƷŹŚǀƘƯŻřƾƿŚƷƶƳƺưƳ
ŹřźĪţ æååå ƩŚŨƯ ŹƺƐŝ ŵƺƃ ƽźĜſ ƲǀƘƯ ƽŚƷŹřźĪţ ŵřŶƘţ æ
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ƪů Źŵ ƽŵƺŞƸŝ ĢǀƷ ŚƷŹřźĪţ Żř ƾƴǀƘƯ ŵřŶƘţ ŢƃŸĭ Żř žě è
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ƵŶƃ ƶŗřŹř ľLjƿŷ ƮŤƿŹƺĮƫř Ʋƿř ƪůřźƯ ŶƴƴĩƾƯ ƵŵŚƠŤſř ĬƴƿźŤƬǀƟ
Ţſř ƽŶƘŝƽźŤƄƯśŚŴŤƳřŢƸūĬƴƿźŤƬǀƟƮŤƿŹƺĮƫřƽŚƷƭŚĭ æƹåƲǀŝƱōŹřŶƤƯƹŶƃŚŝƾƯźǀƀƯƽƹŹƾƬŰƯƱƺƯźƟźŧřƱŶƃ
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ƾƳŚſŹŻƹźŝ ŶƳŵƺưƳ Ŷǀƫƺţ řŹ ŵƺų ƽŚƷźǀƀƯ ŚƷƶģŹƺƯ ƶĪƳō Żř žě
šŹƺƇ æç ƶƫŵŚƘƯ ŽŚſřźŝ ŹŚĪƴƿř ŵźǀĭƾƯ ƭŚŬƳř ÎƱƺƯźƟ źŧř ƾƬĩ źǀƀƯƽƹŹƾƬĩƱƺƯźƟźŧřƱŶƃƺŰƯźŤƯřŹŚěƱōŹŵƶĩŵźƿŸěƾƯ
ƩƺƏ Lgb ƲǀƴĤưƷŵƺƃƾƯƞƿźƘţæƹåƲǀŝƱōŹřŶƤƯƹŶƃŚŝƾƯ ŢſřƪůƽřŶŤŝřŻřźǀƀƯƲƿźŤƸŝ
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ƵŶƃŭLjƇřƱŚĮģŹƺƯƾƳƺƬĩƮŤƿŹƺĮƫřé
ƱŚĮģŹƺƯƾƳƺƬĩƽŻŚſƶƴǀƸŝŻřƶŤƟźĭźŝƮŤƿŹƺĮƫřƲƿřŹŵƾţŚŞſŚŰƯ
ƾƳƺƬĩ ƽŻŚſ ƶƴǀƸŝ ƮŤƿŹƺĮƫř ƵŶƃ ŭLjƇř ƕƺƳ ƖƣřƹŹŵ ŶƃŚŝƾƯ
šŹƺƈŝ ƱŚĭƹŚƳ ƕƺƳ ƮŤƿŹƺĮƫř Ʋƿř Źŵ ŵƺƃƾƯ ŭźƐƯ ƱŚĮģŹƺƯ
Ʊƺģ ŵźĪƿƹŹ Ʋƿř Źŵ >ìë@Ţſř ƵŶƃ ƶŤƟźĭ źƔƳŹŵ ÏƲĮưƷŚƳ ŶƳřƵŶƃ Ƶŵřŵ Źřźƣ ƶǀƬƤƳ ƪŗŚſƹ ƽřźŝ ƾƳŚĭŶƴƿŚưƳ ƱřƺƴƘŝ ŚƷƶģŹƺƯ ŢƃřŵƮǀƷřƺųƶģŹƺƯƕƺƳ k ƶǀƬƤƳƶƬǀſƹƕƺƳ k ƲŤƃřŵŚŝžě źƷ Ţƃřŵ ƮǀƷřƺų ƕƺƳ k ŻřƶģŹƺƯ n ƶƬǀſƹ n ƪĩ Źŵ ƾƴƘƿ
ŶƴƃŚŝƾƯƲǀƘƯŢǀƟźƓĨƿƽřŹřŵƶƬǀſƹƶģŹƺƯƕƺƳ k ƲƿřŻřĨƿ
šŹŚģƺƬƟƶƫŚƤƯŢſƺǀěæƪĪƃŹŵŵƺưƳŶƴƷřƺųƪůřŹƶƬŘƀƯƹ
ŢſřƵŶƃƶŗřŹřśřƺūŶǀƫƺţƮŤƿŹƺĮƫř
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قیمت: تومان


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