Robust Optimization of Stochastic Revenue Management in Hotel Industry

Mohammad Modarres* & Mehdi Najafi

Mohammad Modarres Sharif University of Technology,
Mehdi Najafi, Sharif University of Technology

Keywords ABSTRACT

Robust Optimization;
Revenue Management;
Robust Solution;
Robust Model,
Stochastic Problem,
Hotel Management The successful application of “revenue management” in airline industry motivated hotel industry management as well as some researchers to apply the concept and techniques of revenue management to enhance the income of hotels. In this paper, we develop a multi objective model for hotel management in stochastic environments. Since demand for rooms, and in some cases the supply, is random we introduce a robust model, in order to manage its uncertain parameters. Therefore, the objectives of the model are robustness as well as income enhancement. Furthermore, some other new assumptions such as overbooking, early (or late) departure of a passenger and different prices are also included in the model
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1268732040128

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ƪŤƷŶƯōŹŵŹřŶƿŚěƽŻŚſƶƴǀƸŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝ
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ƵŶǀĪģƽŶǀƬĩšŚưƬĩ
30467-5391

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ŢƸūƽźŤƄǀŝšŚƤǀƤŰţƾŗŚưǀěřƺƷŢƘƴƇŚŝƱōżƿŚưţƹƽŹřŵƪŤƷ ŵƺƃƭŚŬƳřŶƿŚŝƽŹřŵƪŤƷŢƘƴƇŹŵŶƯōŹŵŢƿźƿŶƯŚƸƫŶƯƶƘſƺţ ŹƺƏƶŝƱřƺţƾƯřŹƵŻƺůƲƿřŹŵƵŶƃƭŚŬƳřšŚƤǀƤŰţŻřƾƌƘŝ>æ@ ƹ ƱřźŤǀŝ ƹ >ì@ŵŹƺƟŹŶƿƹ ŵźƧ ƱŚǀŝ źƿŻ ŭźƃ ƶŝ ƶƇLjų ƂƿŚƯŻōŢƸūƽŻŚſƶǀŞƃƽŚƷƩŶƯŻřƵŵŚƠŤſřƶŝ>í@ƲǀĤſŵƺƯ ŵŹƺƯŹŵƮǀưƈţƽřźŝƱŚƃŵƺųƎſƺţƵŶƃƶŗřŹřƽŹŚƨŤŝřƽŚƷƁƹŹ ƭŚŬƳřšŚƘƫŚƐƯƶƘſƺţƶŝ>î@źĪǀŝŶƳřƶŤųřŵźěŚƷŚƋŚƤţŵŹŚƿƁźƿŸě ŵƺųƽŹŚƨŤŝřƽŚƷƩŶƯŚŝŚƷƩŶƯƲƿřƶƀƿŚƤƯƶŝƹƶŤųřŵźěơƺƟƵŶƃ ƶƀƿŚƤƯƶŝ>æå@ƺǀĭƹŶǀŤſřŢſřƶŤųřŵźěƞƗŚƌƯƹŹŻŹŵŹƺƯŹŵ ƶĩŶƳŵřŵƱŚƄƳƹƶŤųřŵźěƪŤƷŹŵƽŹřŸĭŢưǀƣƞƬŤŴƯƽŚƷƁƹŹ ŹřŻŚŝƎƿřźƃŹŵŶƿŚŝŶƯōŹŵŢƿźƿŶƯƮǀƷŚƠƯŽŚſřźŝƽŹřŸĭŢưǀƣ ƾſŹźŝƶŝŵƺųƶƘƫŚƐƯŹŵżǀƳ>ææ@ĬƳřƺƷŵźƿŸěšŹƺƇƾŤŝŚƣŹ
ƱōƾƬƇřŌżūƂƃƹƶŤųřŵźěƪŤƷŹŵŶƯōŹŵŢƿźƿŶƯƾƬƇřƽřżūř

Overbooking
Nested
Non-Nested
æèƾƠŬƳƽŶƸƯƹŽŹŶƯŶưŰƯƪŤƷŶƯōŹŵŹřŶƿŚěƽŻŚſƶƴǀƸŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝ

ŹŵƽŵƹŹƹƱřźƟŚƀƯƶŝŢŞƀƳƽŵƺƘƇźǀƛƖŝŚţƶƏƺŝźƯƖŝŚţƶĩŢſř
ŢſřŻƹŹƱō ƶŝƞƗŚƌƯƹŹŻŹƮŤƀǀſŹŵŢǀƟźƓźŝŵřŻŚƯƽŵƹŹƹƱřźƟŚƀƯ 
ŢųřŵźěŚƷƱōƶŝƽřƶưƿźūĢǀƷƹŶƳƺƃƾƯƵŵŚŤſźƟźĮƿŵƽŚƷƪŤƷ
ŵƺƃƾưƳ

ƩŶƯƽŚƷźĭŚſŚƴƃçç
ŻřŶƴţŹŚŞƗƩŶƯƽŚƷźĭŚſŚƴƃƵŶƃŭźƐƯšŚǀƋźƟƶŝƶūƺţŚŝ
ƾſŹźŝŵŹƺƯƵŹƹŵŹŵƽŵƹŹƹƽŚƷŻƹŹƶƗƺưŬƯI
ƾſŹźŝŵŹƺƯƵŹƹŵŹŵƾūƹźųƽŚƷŻƹŹƶƗƺưŬƯJ
ƾſŹźŝŵŹƺƯƵŹƹŵŹŵŚƷŻƹŹƶƗƺưŬƯK
ƾſŹźŝŵŹƺƯƵŹƹŵƩƺƏT
ŚƋŚƤţƞƬŤŴƯƽŚƷƺƿŹŚƴſƶƗƺưŬƯS ƪŤƷŢǀƟźƓƞƬŤŴƯƽŚƷƺƿŹŚƴſƶƗƺưŬƯM ƾſŹźŝŵŹƺƯƵŹƹŵŹŵƱřźƟŚƀƯŵƹŹƹŻƹŹƶŝƍƺŝźƯźĭŚſŚƴƃi ƾſŹźŝŵŹƺƯƵŹƹŵŹŵƱřźƟŚƀƯũƹźųŻƹŹƶŝƍƺŝźƯźĭŚſŚƴƃj ƾſŹźŝŵŹƺƯŻƹŹƶŝƍƺŝźƯźĭŚſŚƴƃk
ŚƋŚƤţƽƺƿŹŚƴſƶŝƍƺŝźƯźĭŚſŚƴƃs
ŢǀƟźƓƽƺƿŹŚƴſƶŝƍƺŝźƯźĭŚſŚƴƃm
ŚƷźŤƯřŹŚěèç
ŻřŶƴţŹŚŞƗƩŶƯƽŚƷźŤƯřŹŚě
ŹŵƭřjŻƹŹŹŵũƹźųƹƭřiŻƹŹŹŵźƟŚƀƯŵƹŹƹŻřƾƃŚƳŶƯōŹŵRij
ƱŚƀĪƿŢưǀƣŢƫŚů
ƭřjŻƹŹŹŵũƹźųƹƭřiŻƹŹŹŵźƟŚƀƯŵƹŹƹŻřƾƃŚƳŶƯōŹŵRijs
sƺƿŹŚƴſŢŰţ
ƪŤƷƾƬƇřŢǀƟźƓC
ƭřmƽƺƿŹŚƴſŢŰţƪŤƷŢǀƟźƓC m
ŢƫŚůƭřjŻƹŹŹŵũƹźųƹƭřiŻƹŹŹŵŵƹŹƹƽřźŝŚƋŚƤţƱřżǀƯUij
ƶƿŚě
ŢŰţƭřjŻƹŹŹŵũƹźųƹƭřiŻƹŹŹŵŵƹŹƹƽřźŝŚƋŚƤţƱřżǀƯUijs ƭřsƺƿŹŚƴſ
ƮǀưƈţƽŚƷźǀƜŤƯéç
šƹŚƠŤƯŚƿƹƱŚƀĪƿŢưǀƣƽřŹřŵŚƷƺƿŹŚƴſƶĩƲƿřƶŝƶūƺţŚŝ
ŻřŶƴţŹŚŞƗƮǀưƈţƽŚƷźǀƜŤƯŶƴƃŚŝ
ũƹźųƹƭřiŻƹŹŹŵŵƹŹƹŢƸūƵŶƃƶŤƟźƿŸěƱřźƟŚƀƯŵřŶƘţxij
ƱŚƀĪƿŢưǀƣŢƫŚůŹŵƭřjŻƹŹŹŵ
ũƹźųƹƭřiŻƹŹŹŵŵƹŹƹŢƸūƵŶƃƶŤƟźƿŸěƱřźƟŚƀƯŵřŶƘţ xijs
šƹŚƠŤƯŢưǀƣŢƫŚůŹŵƺƿŹŚƴſŢŰţƭřjŻƹŹŹŵ
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ŵŹřƹƭřkŻƹŹŹŵƶĩƾƳřźƟŚƀƯƪĩŵřŶƘţŢƠĭƱřƺţƾƯƹźƴƿřŻř ŚŝŢſřźŝřźŝŶƳƺƃƾƯ ƍƺŝźƯƶĩŶƴŤƀƷƾţŚǀƋźƟƭƹŵƶŤſŵŶƳƺƃƾƯƪƯŚƃřŹŚƸƫŶƯƶǀƬĩ
ŶƴŤƀƷƢǀƤŰţƲƿřŹŵƵŶƃƶŗřŹřƽŚƸƫŶƯŻřƾųźŝƶŝ
ŻřŶƴţŹŚŞƗƩƹřƶŤſŵšŚǀƋźƟ
ƲǀƴĤưƷƹƶŤƃřŶƳŹƺƌůƪŤƷŹŵźƟŚƀƯĢǀƷźƠƇƱŚƯŻŻřƪŞƣ ƶŝŶƳŵźĭũŹŚųƪŤƷŻřƵŹƹŵƽŚƸŤƳřŚţŶƿŚŝƽŵƹŹƹƱřźƟŚƀƯƶǀƬĩ
ƶŤƃřŵŵƺūƹƪŤƷŹŵźƟŚƀƯĢǀƷŶƿŚŞƳƵŹƹŵƽŚƸŤƳřŹŵźĮƿŵšŹŚŞƗ
ŶƃŚŝ
ĢǀƷƹŢſřźƿŸěƱŚĪƯřƱřźƟŚƀƯŵƹŹƹŚƸƴţźƠƇƱŚƯŻŻƹŹŹŵ 
ŵŵźĭƾưƳũŹŚųƪŤƷŻřźƟŚƀƯ ƶƯŚƳźŝƵŹƹŵƽŚƷŻƹŹŹŵũƹźųƹŵƹŹƹƽřźŝŚƋŚƤţƖƿŻƺţƖŝŚţ ŹŵũƹźųƹƮŬƴěŻƹŹŹŵŵƹŹƹƽřźŝŚƋŚƤţŵřŶƘţƩŚŨƯƱřƺƴƗƶŝƽżƿŹ
ŢſřƾƴǀŝƂǀěƪŝŚƣƶŤƃŸĭƽŚƷƵŵřŵƶŝƶūƺţŚŝƆŴƄƯƮƸƳŻƹŹ
ŚƿƱŚƀĪƿƽŚƷŢưǀƣƽřŹřŵŶƴƳřƺţƾƯŚƋŚƤţƞƬŤŴƯƽŚƷƺƿŹŚƴſ 
ŶƴƃŚŝšƹŚƠŤƯ
ŻřŶƴţŹŚŞƗëŚţéƽŚƸƄŴŝƶŝƍƺŝźƯƭƹŵƶŤſŵšŚǀƋźƟ
ŻřŻƹŹźƷŹŵƶĩŵŹřŵŵƺūƹŚƋŚƤţƽřźŝƾƠƬŤŴƯƽŚƷƺƿŹŚƴſ  ƭŻLJƶŤŞƫřŵŹřŵƕƺƣƹƱŚĪƯřƾƇŚųƩŚưŤůřŚŝŚƷƱōŻřĨƿźƷƵŹƹŵ ƶŝƵŹƹŵŻřƆŴƄƯŻƹŹĨƿŹŵƺƿŹŚƴſĨƿƕƺƣƹƶĩŢſřźĩŷƶŝ
ƲĪưƯźĮƿŵšŹŚŞƗƶŝŢƀǀƳŶƘŝƽŚƷŻƹŹŹŵƺƿŹŚƴſƱōŹřźĪţƾƴƘƯ
sƽƺƿŹŚƴſtæŻƹŹŹŵƾƫƹŶƷŵűŹƭřsƺƿŹŚƴſtŻƹŹŹŵƶĩŢſř
ŵŶƳƺǀĜŝƕƺƣƹƶŝ
ƶƯŚƳźŝƱŚƯŻŹŵƶĩƵŶƄƳƾƴǀŝƂâěƵĦƿƹƽŚƷƶƯŚƳźŝƶŝƶūƺţŚŝ  ƅŚƈŤųřƪǀƫŵƶŝŚƿƹƽŹřźƐƋřźǀưƘţźǀƔƳŢƀǀƳƆŴƄƯƽżƿŹ
ƽŚƷơŚţřŵřŶƘţƾŤƫƹŵƽŚƷƱŚĭŹřƶŝƪŤƷƽŚƷơŚţřŻřƾųźŝƽŹŚŞūř ƽŚƷƺƿŹŚƴſƲƨǀƫŢƀǀƳƆŴƄƯLjƯŚĩƱřźƟŚƀƯƽřźŝƵŵŚƠŤſřƪŝŚƣ ĨƿƕƺƣƹŵƺưƳŹƺƔƴƯƱřƺţƾƯřŹƆŴƄƯšLJŚưŤůřŚŝƾƠƬŤŴƯ
ƽŚƷŻƹŹŹŵƱōŹřźĪţƾƴƘƯƶŝƵŹƹŵŻřƆŴƄƯŻƹŹĨƿŹŵƺƿŹŚƴſ
ŢƀǀƳźĮƿŵ
ƽŚƷŢưƀƣŹŵƶĩŶƴŤƀƷźƿŻšŚǀƋźƟƭƹŵƶŤſŵšŚǀƋźƟźĮƿŵŻř
ŶƳƺƃƾƯƶŤƟźĭŹŚƧƶŝƶƫŚƤƯƮƄƃƹƮŬƴě šŶƯƶŝźŨĩřŶůŢƯŚƣřšŶƯƱŵźĩƾƳLJƺƏŚƿźţƵŚţƺĩƱŚĪƯř ƹƭŚĮƴƷŵƹŻũƹźųŜǀţźţƶŝƶĩŵŹřŵŵƺūƹƲƿźƟřŚƀƯƽřźŝŻƹŹƹŵ ƽŵƹŹƹƽŚƷźƟŚƀƯƩŚŨƯƱřƺƴƗƶŝŶƳƺƃƾƯƵŶǀƯŚƳƭŚĮƴƷźƿŵũƹźų řŹƮƸƳŻƹŹŹŵũƹźųƾƴƘƿƮƸƳŻƹŹŚţŢƯŚƣřŶƈƣƶĩƮŬƴěŻƹŹŹŵ
ũƹźųŶƴƃŚŝƶŤƃřŵŹƺƌůƮŤƄƷŚƿƮŤƠƷŻƹŹŚţŢſřƲĪưƯŶƳŹřŵ
ŚƿƮƷŵŻƹŹŚţƹƵŵźĩźţƾƳLJƺƏřŹŵƺųŢƯŚƣřšŶƯŚƿƹƭŚĮƴƷŵƹŻ
ƭŚĮƴƷźƿŵũƹźųŶƳƺƃũŹŚųƮƷŵŻŚƿ
ƶŝƹåæŚŝźŝřźŝŻƹŹĨƿšŶƯƶŝŢƯŚƣřƱŚƯŻŹŵźǀǀƜţƩŚưŤůř 
ŵƺƃƾƯƉźƟååêŚŝźŝřźŝŻƹŹƹŵšŶƯ ƹŹŻŹƱŚĪƯřŢƀǀƳƆŴƄƯƪƯŚĩŹƺƐŝŚƷŚƋŚƤţƶĪǀƿŚŬƳōŻř ŻƹŹƽřźŝƱōŹřŶƤƯƶĩŵŹřŵŵƺūƹƪŤƷŢƿźƿŶƯƽřźŝżǀƳƞƗŚƌƯ
ƾƸƿŶŝŢſřŻƹŹƱōŹŵƽŵƹŹƹƱřźƟŚƀƯƽŚƋŚƤţƶŝƶŤƀŝřƹƆŴƄƯ
xij Uij i  0,…,T 1 j 1,…,T í
xij  0 i  0,…,T 1 j 1,…,T î
ƾƠŬƳƽŶƸƯƹŽŹŶƯŶưŰƯƪŤƷŶƯōŹŵŹřŶƿŚěƽŻŚſƶƴǀƸŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝæé
ƱŵźĩƶƴǀƄǀŝƩŚŞƳŵƶŝêƝŶƷƖŝŚţŵƺƃƾƯƵŶƷŚƄƯƶĩŹƺƐƳŚưƷ ë ŢƿŵƹŶŰƯ Ţſř ƱřźƟŚƀƯ ƶŝ ƾƷŵžƿƹźſ Żř ƶƬƇŚů ŵƺſ ŢǀƟźƓźŨĩřŶůŻřƪŤƷŹŵźƋŚůƱřźƟŚƀƯŵřŶƘţƶĩŶƴĩƾƯƲǀưƌţ ŻřƂǀŝƱřźƟŚƀƯŵƹŹƹƭŶƗźĮƳŚǀŝìŢƿŵƹŶŰƯŶƿŚưƴƳŻƹŚŬţƪŤƷ ƲǀưƌţíŢƿŵƹŶŰƯŚŤƿŚƸƳƹƵŵƺŝźƠƇŻƹŹŹŵƪŤƷŢǀƟźƓ ƹƭřiŻƹŹŹŵŵƹŹƹŢƸūƵŶƃƶŤƟźƿŸěƱřźƟŚƀƯŵřŶƘţƶĩŶƴĩƾƯ
ŶƃŚŞƳ źŤƄǀŝ ƵŶƃ ƾƴǀŝƂǀě ƽŚƋŚƤţ Żř ƭř j ŻƹŹ Źŵ ũƹźų
ŢſřƶƏƺŝźƯƩŶƯźǀƜŤƯźĮƳŚǀŝżǀƳîŢƿŵƹŶŰƯ
ŹřŶƿŚěƽŻŚſƶƴǀƸŝƹƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƩŶƯé
ƽżƿŹƶƯŚƳźŝêîƾƋŚƿŹƩŶƯŵƺƃƾƯƵŶƷŚƄƯƶĩŹƺƐƳŚưƷ Źŵ CźŤƯřŹŚěƶĩŢƃřŵƶūƺţŶƿŚŝŢſřƾƐųƹŮǀŰƇŵŶƗ íŢƿŵƹŶŰƯŹŵ Uij ƲǀƴĤưƷƹìƹëƽŚƷŢƿŵƹŶŰƯ
ŶƳŹřŵƞƬŤŴƯƽŚƷƺƿŹŚƴſƶŝƾĮŤƀŝƱōźƿŵŚƤƯƹŶƴŤƀƷƾƘƐƣźǀƛ ŢƟźĭŹŚĪŝƱřƺţƾƯřŹƾƠƬŤŴƯƽŚƷŵźĪƿƹŹƪĪƄƯƲƿřƪůŢƸū
ƾƴƘƿŚƷźŤƯřŹŚěƲƿřƾƋŚƿŹŶǀƯřŻřƵŵŚƠŤſřŚƷŵźĪƿƹŹƲƿřŻřƾĪƿ źƔƳƶŝ ƵŵŚſ ŹŚǀƀŝ ƶģźĭř ŵźĪƿƹŹ ƲƿřŢſř EC ƹ EUij  ƶūƺƯƽŚƷśřƺūŶǀƫƺţƲǀưƌţƭŶƗƱōƵŶưƗƞƘƋŚƯřŶſŹƾƯ żǀƫŚƳōŻřƽŵŹřƺƯŹŵƞƘƋƲƿřƱŵƺưƳƖƠţźƯŢƸūƹźƴƿřŻřŢſř ƲǀƴģŚƯřŵƺƃƾƯƵŵŚƠŤſřšŚůLjƇřƾųźŝƭŚŬƳřƽřźŝŢǀſŚƀů ŢƸūŚƷŵźĪƿƹŹźĮƿŵŻř>ç@ŢſřƾƫŚƘƠƳřŵźĪƿƹŹĨƿƽŵźĪƿƹŹ żǀƳƵŶƳźǀĭƂǀěƽŚƷŵźĪƿƹŹŌżūƶĩƱŚƴǀưƏřƭŶƗƲƿřŚŝŵŹƺųźŝ
ƩŶƯŹŵƱōƽźǀĭŹŚĪŝƹƱōħŹŵƱŚƴǀưƏřƭŶƗƲƿřƲŤƟźƿŸěŢſř
ŢſřƵŶƃƵŵŚƸƳŚƴŝŽŚſřƲƿřźŝżǀƳƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƶĩŢſř
ƩŶƯƽřźŝƱřƺţƾƯřŹƽŹřŶƿŚěƕƺƳƹŵơƺƟƵŶƃŭźƐƯŜƫŚƐƯƶŝ ŵƺưƳƞƿźƘţƢǀƤŰţƲƿřƾƟŵŚƈţƽżƿŹƶƯŚƳźŝ
ŲſŚěƽŹřŶƿŚěĨƿƕƺƳƽŹřŶƿŚě
ƎƿřźƃƶŝƶūƺţŚŝƶĩŢſřŲſŚěƽŹřŶƿŚěƽřŹřŵƽřƶƴǀƸŝśřƺū
ƶƴǀƸŝƶŝĨƿŵżƳmM ƹsS ƽŚƷƺƿŹŚƴſƶǀƬĩƽřźŝƾĮƴǀƸŝ
ŶƃŚŝ
ƩŶƯƽŹřŶƿŚěƹŵƕƺƳƽŹřŶƿŚě
ƎƿřźƃƶŝƶūƺţŚŝƶĩŢſřƩŶƯƽŹřŶƿŚěƽřŹřŵƽřƶƴǀƸŝśřƺū
ŚŞƿźƤţ mM ƹ sS ƽŚƷƺƿŹŚƴſƶǀƬĩƽřźŝƱŵƺŝƶūƺƯ
ŶƃŚŝƶūƺƯ
ƹŚƋŚƤţƽřźŝƱŚƴǀưƏřƭŶƗƱŚǀŝŢƸūŶƃƱŚǀŝżǀƳLjŞƣƶĩŹƺƐƳŚưƷ ƹŚƋŚƤţƽřźŝƾƠƬŤŴƯƽŚƷƺƿŹŚƴſƶĩŢſřƵŶƃƉźƟŢǀƟźƓ
T
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Downloaded from ijiepm.iust.ac.ir at 16:20 IRST on Saturday November 4th 2017

 xkj æjk1
ŚŝŢſřźŝřźŝŶƳƺƃƾƯũŹŚųƭřkŻƹŹŹŵƶĩƾƳřźƟŚƀƯƪĩŵřŶƘţƹ
k1
 xik çi0
ƶƿŚěŢƫŚůŹŵƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƩŶƯè
řŹƱřźƟŚƀƯũƹźųƹŵƹŹƹƵŶƃƞƿźƘţƽŚƷźǀƜŤƯƶŝƶūƺţŚŝ źƔƳŹŵƶĪŞƃŹŵƵźĭĨƿƾūƹźųƹƽŵƹŹƹƱŚƿźūšŹƺƈŝƱřƺţƾƯ
ŵřŶƘţ k,k 1,….,k 1,ƆŴƄƯŻƹŹƽřźŝæŹřŵƺưƳŢƟźĭ ŶƴŤƀƷƶŞſŚŰƯƪŝŚƣźƿŻšŹƺƈŝƪŤƷŹŵźƋŚůƱřźƟŚƀƯ

ƭřkŻƹŹŹŵũƹźųƹŵƹŹƹƱŚƿźūæŹřŵƺưƳ
ƭřKŻƹŹŹŵƪŤƷŹŵźƋŚůƱřźƟŚƀƯŵřŶƘţ è
k1 TTk1kT
 xij   xkj  xik  xij
i0 jkjk1i0i0 jk1
šŹƺƇƾūƹźųĢǀƷźƠƇŻƹŹŹŵšŚǀƋźƟŽŚſřźŝźĮƿŵƝźƏŻř ŵŹřƹƲƿźƟŚƀƯřŸƫŢſřźƿŸěƱŚĪƯřƱřźƟŚƀƯŵƹŹƹŚƸƴţƹŵźǀĭƾưƳ
ŶƳƺƃƾƯũŹŚųƪŤƷŻřTŚţæƽŚƷŻƹŹŻřƾĪƿŹŵŻƹŹƲƿřŹŵƵŶƃ ŚŝŢſřźŝřźŝŻƹŹƲƿřŹŵƽŵƹŹƹƱřźƟŚƀƯŵřŶƘţƲƿřźŝŚƴŝ
T
x0j éj1
ŭźƃƶŝƶƿŚěŢƫŚůŹŵƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƩŶƯŜǀţźţƲƿřƶŝ >æ@ŹŵƵŶƃƶŗřŹřƩŶƯƶǀŞƃŢſřźƿŻ
T1T
Max Rij xij ê
i0 ji1
S.T: ë
kT xij  C
i0 jk1
T
x0 j  C ìj1
æêƾƠŬƳƽŶƸƯƹŽŹŶƯŶưŰƯƪŤƷŶƯōŹŵŹřŶƿŚěƽŻŚſƶƴǀƸŝƾƟŵŚƈţƽżƿŹƶƯŚƳźŝ

Ţſř
ŢưƀƣƹŵƽřŹřŵæçƩƹřƝŶƷƖŝŚţŵƺƃƾƯƵŶƷŚƄƯƶĩŹƺƐƳŚưƷ ƹƱřźƟŚƀƯƶŝƾƷŵžƿƹźſŻřƪƇŚůŵƺſźĮƳŚǀŝƩƹřŢưƀƣŢſř ƽřźŝŶƯōŹŵƎſƺŤƯŻřƝřźŰƳřƢƬƐƯŹŶƣƎſƺŤƯźĮƳŚǀŝƭƹŵŢưƀƣ ŶƯōŹŵƝLjŤųřƶĤƳŚƴģƶĩŢſřƾƸƿŶŝŢſřƞƬŤŴƯƽŚƷƺƿŹŚƴſ ŹřŶƤƯ ƶưƿźū ƱřżǀƯ ŶƃŚŝ ŵŚƿŻ ƞƬŤŴƯ ƽŚƷƺƿŹŚƴſ Żř ƶƬƇŚů
ƩŚŞƳŵƶŝƩŶƯƹźƴƿřŻřŶŝŚƿƾƯƂƿřżƟřŶƯōŹŵƎſƺŤƯŻřƶŤƟŚƿƂƷŚĩ
ŜƿźƋ ƱřƺƴƗƶŝ żǀƳŜƿźƋ Ţſř ƝLjŤųř Ʋƿř Ʊŵźĩ ƶƴǀưĩ ŶƃŚŝźŤƄǀŝŢƿźƿŶƯƽźƿŸěƦƀƿŹƱřżǀƯƶģźƷŢſřƽżƿźĭĨƀƿŹ
ƝŶƷƖŝŚţƲƿřŜǀţźţƲƿřƶŝŵƺƃƾƯśŚŴŤƳřźŤƨģƺƧŹřŶƤƯ ƭƹŵƝŶƷƖŝŚţŢſřŲſŚěƽŹřŶƿŚěƱřżǀƯƱŵźĩźŨĩřŶůƩŚŞƳŵƶŝ ŻřƝřźŰƳřƱřżǀƯźĮƳŚǀŝƩƹřŢưƀƣŢſřŢưƀƣƹŵƽřŹřŵżǀƳæèźĮƳŚǀŝżǀƳƭƹŵŢưƀƣƹƵŵƺŝsS ƞƬŤŴƯƽŚƷƺƿŹŚƴſƽŚƋŚƤţ
ŜƿźƋŻřŢſř mM ŚƷƺƿŹŚƴſƲƿřŢǀƟźƓŻřƝřźŰƳřƱřżǀƯ
ŵƺƃƾƯƵŵŚƠŤſřƝřźŰƳřƹŵƲƿřƱŵźĩƱŚƀĪƿŢƸūżǀƳ ƆŴƄƯƕƺƣƹƩŚưŤůřƽřŹřŵŚƷƱōŻřĨƿźƷƶĩŵŹřŵŵƺūƹŢǀƟźƓ
ƶĪƿŹƺƐŝŶƴŤƀƷ
S Ps 1æås1
M m 1ææ
m1
ƪŤƷŶƯōŹŵŢƿźƿŶƯƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƩŶƯŢƠĭƱřƺţƾƯƹźƴƿřŻř ƩŶƯƽŹřŶƿŚěƹŲſŚěƽŹřŶƿŚěŻřŶƴţŹŚŞƗƶĩŵŹřŵƾěŹŵřŹƝŶƷƹŵ ƞƬŤŴƯƽŚƷƺƿŹŚƴſƽřźŝŚƷŢưǀƣƽƹŚƀţƉźƟƶŝƶūƺţŚŝƩŚů
šLJŵŚƘƯšŹƺƈŝƪŤƷŢƿźƿŶƯƾƟŵŚƈţƽżƿŹƶƯŚƳźŝƩŶƯŚƋŚƤţ
ƆŴƄƯƽŚƋŚƤţŻřƝřźŰƳřƶưƿźūƹƽżƿźĭĨƀƿŹƱřżǀƯźĮƳŚǀŝ
2094627-119249P

P

4253235-119249

2798421493899

3849543493899

Max Z1  S Ps  T1T Rij xS T1TS T1Tsæçs sRijs xij PsRij xij
ij
s1i0 ji1s1i0 ji1 s1i0 ji1
1604103-18493

Min Z2  S Ps T1 T wijUijMm M mC m  æè
s1i0 ji1s  xij  m1mC m1
ŢſřæìŚţæçŜǀţźţƶŝƶĩƵŵƺŝƾƠƴƯźǀƛƾƳŻƹƽŚƷźŤƯřŹŚěwij ƹƽŚƷźŤƯřŹŚě
ƾƳŚƯŹōƽżƿŹƶƯŚƳźŝæƶǀƌƣ
MinZ 



قیمت: تومان


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