A Robust Optimization Approach to Resources Allocation in Maintained Systems

A. H. Shokouhi & H. Shahriari

Amir H. Shokouhi, PhD student, Industrial Engineering Department, K.N. Toosi University of Technology
Hamid Shahriari, Assistance professor, Industrial Engineering Department, K.N. Toosi University of Technology

Keywords ABSTRACT

Robust Optimization,
Availability;
Redundancy Optimization;
Maintenance Resources Optimization of maintenance resources to maximize the system availability is a major concern in different manufacturing systems. Therefore, a lot of effort is put to construct optimization models to reach the maximum availability level and to reduce the costs of lack of availability. However, despite these efforts, data uncertainty in the real world problems was neglected in proposed models which results models with sophisticated nonlinear constraints. In this article the budget uncertainty model of optimization is considered and its counterpart in resources allocation for maintained systems is presented. The proposed model takes into account the degrees of conservativeness to introduce a robust model of optimization. This model is capable of maximizing the availability for the maintained system under certain conditions. The capability of the model in controlling the effects of uncertainty is investigated by means of simulation studies and reasonable results concluded.

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1Reliability
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ƽƵŶºƴƷŵƱŚƄºƳ y i ƾůLjƇřšřźǀưƘţƖŝŚƴƯƱřżǀƯreci ƱōŹŵƶĩ ƾůLjºƇřźºǀưƘţƖŝŚƴƯƽƶƴƿżƷcreci  i ƮŤƀǀſźƿŻŌřżūřŵřŶƘţ
qi T  ƹ i ƮŤƀ ǀſź ƿŻŹŵƾůLj Ƈřšřź ǀưƘţƽƶ ƴƿżƷ cmi
ƱŵƺºŝƾƿŚºưƳƉźºƟŚºŝƶĩŢſř i ƮŤƀǀſźƿŻŌřżūřƾŝřźųƩŚưŤůř
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AS  i1 11 I 1reci 1d1i  
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ƾºƟźƘƯëŢƿŵƹŶŰƯƢŞƏřŹ T ŢŝŚŧƱŚƯŻƽřźŝƮŤƀǀſƾůLjƇř
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ŶƴĩƮưƿżĩŚƯŢƫŚů
ƾſźŤſŵŮƐſŹřƺŤſřƽŻŚſƶƴǀƸŝƩŶƯè
ƶƿŚěƩŶƯƽřźŝŹřƺŤſřśřƺūĨƿƶŝƾŝŚǀŤſŵŹƺƔƴƯƶŝƶƫŚƤƯƲƿřŹŵ ƶºƿřŹřŵƺƃƾƯƶŤƠĭÏRMROƶĩƱōƩŵŚƘƯŹřƺŤſřƩŶƯÎMRO ŹƺƌůŚŝƝŶƷƖŝŚţŹřŶƤƯƱŵźĩƶƴǀƄǀŝƩŚŞƳŵƶŝƩŶƯƲƿřŵŵźĭƾƯ
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ƽřƵŻŚŝŢǀƘƐƣƭŶƗƽřźŝŹřƺŤſřƩŶƯèæ
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ƶƿŚě ƩŶƯ ƽŚƷźŤƯřŹŚě Ʊŵƺŝ ƾƘƐƣ ƭŶƗ ŵƺƃƾƯ ƽŹřŸĭƶƿŚě Ʊō źŝ
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ƭŶƗ Ʋƿř ƕƺƋƺƯ šŚǀŝŵř Źŵ ŶƳƺƃƾƯ ƶŤųŚƴƃ ŹřƺŤſř ƽŚƷśřƺū
ŢǀƘƐƣƭŶƗƹƽřƵŻŚŝŢǀƘƐƣƭŶƗšŹƺƇƹŵƶŝƾƬĩŹƺƏƶŝŢǀƘƐƣ
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ƵŵřŵƱŚƄ Ƴ creciƹ cmi  vi  wi  ci Ś ŝŜ ǀţźţƶ ŝƶ ĩŶƴŤƀ Ʒ ci,ci ciƽƵŻŚ ŝŹŵŜ ǀţźţƶ ŝŚ ƷƱōŻřƭřŶºĩź Ʒƹŵƺ ƃƾº Ư
creci,creci creci  ƹcmi,cmi cmi vi,vi viwi,wi wi
řŹ J crec ƹ J cm  J v  J w  J c ŶƴŤƀºƷƱŹŚƤŤƯƖƿŻƺţĨƿƽřŹřŵ ƽřźŝ creci ƹ cmi  vi  wi  ci ŜǀţźţƶŝƶĩŢſřƾƿŚƷ i ƶƗƺưŬƯ
ƽŚºƷźǀƜŤƯĨƿźƷƱřżǀƯƱřƺţƾƯƲƿřźŝŚƴŝŶƃŚŝƾƯƾƘƐƣźǀƛŚƷƱō ŵřŵƱŚƄƳźƿŻƪĪƃƶŝřŹơƺƟ
ci ci ciic ic 0,1 iææ
wi wi wiiw iw 0,1 iæç
vi vi viiv iv 0,1 iæè
cmi cmi cmiicm icm 0,1 iæé
creci creci creciicrec icrec 0,1 iæê
ƾƟŵŚƈ ºţƽŚ ºƷźǀƜŤƯicrec ƹicm iv iw ic ƱōŹŵƶ ºĩ
ƱŚƄºƳřŹŚºƷƵŵřŵŹŵƱŚºſƺƳƶºĩŶƴŤƀºƷĨºƿƹźƠºƇƲǀºŝƪƤŤƀƯ
ƱŚºſƺƳŹřŵźºŝźºƇŚƴƗƶºưƷƱōŹŵƶºĩŢºƫŚůƲºƿřƶŝŶƴƷŵƾƯ
ƽřƵŻŚºŝŢºǀƘƐƣƭŶºƗƩŶºƯŶºƴƴĩźǀǀƜţ[0,1] ƵŻŚŝŹŵŶƴƳřƺţƾƯ ŶƴƿƺĭƾƯ
1Maintenance Resources Optimization (MRO)
2 Robust Maintenance Resources Optimization (RMRO)
çîƽŹŚƿźƸƃŶǀưůƹƾƷƺĪƃƲǀƀůźǀƯřƽŚƷƮŤƀǀſŹŵƖŝŚƴƯƆǀƈŴţŹŵŹřƺŤſřƽŻŚſƶƴǀƸŝŵźĪƿƹŹ

ŢǀƘƐƣ ƭŶƗ ƩŶƯ Żř ƵŵŚƠŤſř Śŝ ŹřƺŤſř ƽŻŚſƶƴǀƸŝ èç ƽřƶūŵƺŝ
ƶƳřŹŚĩƶƔƟŚŰƯšŶƃƶŝƶĩ[æë]źŤƀƿƺſƁƹŹŜǀƗŚŝƶƬŝŚƤƯƽřźŝ
çå@ƽƹŚƣƩřƹ>æîƹæìæí@ƾĪƀƟƹźǀưƳƹƩŚţƲŝŶƴĩƾƯƪưƗ ŦƗŚŝŵƺųŚƷƱōƽŵŚƸƴƄǀěƽŚƷƁƹŹŶƳŵźĩƶƿřŹřƾƿŚƷƵƺǀƃ>çæƹ
ƪƿŚƀƯƽƹŹźŝƱōƩŚưƗřŢǀƬŝŚƣƭŶƗƹŹřƺŤſřƶƫŚƀƯƱŶƃźţŢŴſ
ŵ,ƁƾƯƶŤƀƀĭƽŚƷźǀƜŤƯŚŝƽŻŚſƶƴǀƸŝ ƅŚºųŹƺƏƶŝƶŤƀƀĭƽŻŚſƶƴǀƸŝƪƿŚƀƯƽřźŝ>çç@ƺƿƹžǀƫƹƺƧ ƩŚºŞƳŵƶŝƶƧŶƳřƵŵźƧƶƿřŹřŹřƺŤſřƶŤƀƀĭƽŻŚſƶƴǀƸŝśƺģŹŚģƦƿ
ƽřƶºƗƺưŬƯŢºŰţřŹŢǀƘƋƹƲƿźţŶŝŵźƨƬưƗƶƧŢſřƾŝřƺūƲŤƟŚƿ ŚºƷƱōŵźºƨƿƹŹŢŰţƶƳŚƠſŚŤƯŶƴƧƶƴǀưƧŚƷƵŵřŵƽřźŝŚƷƺƿŹŚƴſŻř ƪůƪŝŚƣƶŤƀƀĭƽŻŚſƶƴǀƸŝƪƿŚƀƯŻřƽŹŚǀƀŝƩŵŚƘƯŹřƺŤſřƩŶƯ ƽŻŚºſƶƴǀƸŝƁƹŹĨƿźĮƿŵƽƺſŻřŶƴŤƀǀƳƽřƶƬưūŶƴģƱŚƯŻŹŵ ƹçè@ƮǀſƹžưǀƀţźŝƎſƺţƾƐųƽżƿŹƶƯŚƳźŝƪƿŚƀƯƽřźŝŹřƺŤſř ƮǀºƔƴţƪºŝŚƣƱōƽŹŚºĩƶºƔƟŚŰƯƱřżǀƯƺſĨƿŻřƶĩŶƃƶƿřŹř>çé ƁƹŹƲƿřŶƿřżƟřƾưƳƶƫŚƀƯƾŤŴſƶūŹŵźŝźĮƿŵƽƺſŻřƹŶƃŚŝƾƯ
ƲǀºƴĤưƷƹƶŤƀºƀĭƽŻŚºſƶºƴǀƸŝƪƿŚƀƯƽƹŹźŝƱōƩŚưƗřŢǀƬŝŚƣ
ŶƃŚŝƾƯżǀƳƾŞǀĩźţƽŻŚſƶƴǀƸŝ ƹŢºſřƽřƶūŵƺŝ>çéƹçè@ƮǀſƹžưǀƀţźŝƁƹŹŹŵŢǀƘƐƣƭŶƗ ŚºƿŚƷŢƿŵƹŶŰƯŜƿřźƋŻřŚţ  ŵřŶƘţƶŝźŨĩřŶůƶĩŵƺƃƾƯƉźƟ ŵŶƗĨƿƹŶƴƿƺĭƾƯŢƓŚƠůŮƐſ  ƶŝŶƴĩƾƯźǀǀƜţƝŶƷƖŝŚţ ŹŵƾºƘƐƣźºǀƛƽŚºƷƵŵřŵŵřŶºƘţƽƹŚƀºƯŚƿźŤĪģƺĩƾƠƴƯŚƳŮǀŰƇ ƶĩƾƯŚĮƴƷŜǀţźţƲƿŶŝŶƃŚŝƾƯƶƏƺŝźƯƝŶƷƖŝŚţŚƿŚƷŢƿŵƹŶŰƯ ŢſŵƶŝŹřƺŤſřƶƴǀƸŝśřƺūŶƴĩźǀǀƜţŜƿřźƋƲƿřŻřŚţ  źŨĩřŶů ŻřƂǀºŝƶºĩƾƘƣřƺƯŹŵźĮƿŵƽƺſŻřŶƳŚƯŶƷřƺųƶƴǀƸŝŚƘƐƣƵŶƯō
ŚºŝŹřƺŤºſřƶºƴǀƸŝśřƺºūŻŚºŝŶƴĩźǀǀƜţƝŶƷƖŝŚţŜƿřźƋŻřŚţ 
ŶƳŚƯƾƯƾƣŚŝƶƴǀƸŝƶŝĨƿŵżƳƹƶūƺƯƾƿLJŚŝƩŚưŤůř ƽřƵŻŚŝŢǀƘƐƣƭŶƗŚŝŹřƺŤſřƽŻŚſƶƴǀƸŝŻřƾŞƿźƤţƶĩƁƹŹƲƿřŹŵ ƱōƩŚưƗřƹæîƹæíæìæëƎŝřƹŹƲŤƟźĭźƔƳŹŵŚŝŢſř
ŢºſŵƶºŝźƿŻšŹƺƇƶŝ RMRO2 ƩŶƯMROƶƫŚƀƯƽƹŹźŝ
ŶƿōƾƯ
n  y i 
ƹ cmi  vi  wi  ci ƽřźŝŜǀţźţƲƿŶŝŶƴƴĩŹŚǀŤųřřŹŵƺųŹřŶƤƯ
 vi vi  wi  wi  ci ci źƿŵŚƤƯŜǀţźţƶŝ creci
ƽƶƫŚƀƯƹŵƺƃƾƯƶŤƟźĭƾƯźƔƳŹŵ creci creci ƹ cmi cmi
ŵŵźĭƾƯƪůMRO ƱōŻřƶĩçæƎŝřƹŹƕƺưŬƯƽřƵŻŚŝŢǀƘƐƣƭŶƗŵźĪƿƹŹƽŚƴŞƯźŝ
ƶƫŚƀƯ ŹřƺŤſř ƽŻŚſƶƴǀƸŝ ƩŶƯ ŵƺƃƾƯ ŵŚƿ RMRO1 ƭŚƳ Śŝ
ŶƷŵƾƯƶŗřŹřřŹæåƎŝřƹŹƕƺưŬƯŹŵMRO
max
sto. :
 
58802025336

AS in111I11reci 1d1i yi 
  

Cn ci ciyi
i1
n wi wiyi W
i1
n çæ
vi viyi V
i1
n reci creci crecin qi Tyi cmi cmiCM
i1i1
n reci 1
i1
yi N i reci 0 i
Ţſř ŹřƺŤſř ƽŻŚſƶƴǀƸŝ ƁƹŹ ƲƿźţƶƳřŹŚĩƶƔƟŚŰƯ ƶĩ ŵźĪƿƹŹ Ʋƿř ƶƫŚƀƯ ƪů ƽřźŝ ƽŹŚĪŤŝřřźƟ ƽŚƷƮŤƿŹƺĮƫř ƹ ŶƃŚŝƾƯ ƵŵŚſ ŹŚǀƀŝ ƲƿźţŶŝƶĩŚŬƳōŻřŚƯřžŵźĩƵŵŚƠŤſřƱōƪůƽřźŝƱřƺţƾƯřŹMRO ƝŶƷ ƖŝŚţ ŹřŶƤƯ Żř ƽŵŚƿŻ ƱřżǀƯ ƶŝ ŵƺƃƾƯ ƶŤƟźĭ źƔƳ Źŵ ŢƫŚů ƱřżǀƯ ƶĩ ŹřƺŤſř ƽŻŚſƶƴǀƸŝ ƁƹŹ Ĩƿ ƶƯřŵř Źŵ ŵŵźĭƾƯ ƶŤſŚĩ
ƵŵřŵŮǀƋƺţMROƶƫŚƀƯƽřźŝŢſřƮǀƔƴţƪŝŚƣƱōƽŹŚĩƶƔƟŚŰƯ
ŵƺƃƾƯ
1
1422823-3553

maxAS  i 1 1  1 I  1 reci  1d1i  
s.to : n ci y i c  c  C
i1
n wi y i w  w  Wçç
i1
n vi y i v  v  V
i1
n creci reci n qi T  y i cm i  cmi crec  crec cm  cm  CM
i1i 1
13710209574

ƽŹŚƿźƸƃŶǀưůƹƾƷƺĪƃƲǀƀůźǀƯřƽŚƷƮŤƀǀſŹŵƖŝŚƴƯƆǀƈŴţŹŵŹřƺŤſřƽŻŚſƶƴǀƸŝŵźĪƿƹŹèå
135579574354


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قیمت: تومان


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