Automated Storage and Retrieval Systems

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Automated Storage and Retrieval Systems

• AS/RS components
• storage racks
• storage/retrieval (S/R) machines
• input/output (I/O) or pickup/deposit (P/D)
  stations

• Travel times
• Single command
• Dual command

Simultaneous horizontal and vertical travel
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Rack Normalization
Scaling Factor (T) designates the longer (in
time) side of the rack max (th, tv)
Shape Factor (Q) designates the ratio of the
shorter (in time) side to the longer (in time)
side of the rack min (th/T, tv/T) 0 lt Q lt 1
Normalized rack is Q units long in one direction,
and one unit long in the other direction
Square in time (SIT) rack Q 1
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Cycle Time Calculations

• Assumptions
• Randomized storage
• End-of-aisle P/D station at the lower LH corner
  of the rack
• constant horizontal and vertical velocities (no
  acceleration/deceleration)
• Continuous approximation of the storage rack

Expected travel time for a single command cycle
E(SC) T1 Q2/3
Expected travel time from a storage location to a
retrieval location during a dual command cycle
E(TB) T/3010 5Q2 Q3
Expected travel time for a dual command cycle
E(DC) E(SC) E(TB) T/3040 15Q2 Q3
Expected single command cycle time TSC E(SC)
2TP/D
Expected dual command cycle time TDC E(DC)
4TP/D
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Conveyor Models
Conveyor analysis analysis of closed-loop,
irreversible, with discretely spaced carriers

• Three principles developed by Kwo (GE)
• Uniformity principle Material should be
  uniformly distributed
• Capacity principle The carrying capacity of the
  conveyor must be greater than or equal to the
  system throughput parameters
• Speed principle The speed of the conveyor ( of
  carriers/unit time) must be within permissible
  range, defined by loading and unloading station
  requirements

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Multi-station conveyor analysis
Proposed by Muth (1975)

• s stations (for loading and/or unloading) located
  around the conveyor (numbered in reverse sequence
  to the rotation of the conveyor)
• k carriers equally spaced around the conveyor
• Station 1 is used as reference point in defining
  time carrier n becomes carrier nk immediately
  after passing station 1
• The sequence of points in time at which a carrier
  passes station 1 is denoted by tn, where tn is
  the time at which carrier n passes station 1
• The amount of material loaded on carrier n as it
  passes station i is given by fi(n), for i
  1,2,,s (can be negative value denotes unload)
• The amount of material carried by carrier n
  immediately after passing station i is denoted by
  Hi(n)
• For steady state total amount of material loaded
  total amount of material unloaded

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Multi-station conveyor analysis (cont)

• Assumption Conveyor is operated over an infinite
  period of time ? the sequences fi(n) are
  assumed to be periodic with period p
• fi(n) fi(np)

• We use the following relation

• Muths results
• k/p cannot be integers for steady-state
  operations
• r k mod p, r/p must be a proper fraction for
  general sequences F1(n) to be accommodated
• It is desirable for p to be a prime number, as
  conveyor compatibility results for all admissible
  values of k

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Multi-station conveyor analysis (cont)

• The materials balance equation for carrier n
  H1(n) H1(n-r) F1(n)
• We need to find values of Hi(n)
• Method
• Let H1(n) be a particular solution to the above
  equation. Using recursion H1(n) H1(n-r)
  F1(n) by letting H1(1) 0.
• Given Hi(n), the value of Hi1(n) Hi(n)
  fi(n)
• Given Hi(n) for i 1,2,,s, let c min
  Hi(n)
• The desired solution is Hi(n) Hi(n) c
• The required capacity per carrier is B max
  Hi(n)