Cycle Time Calculations

1
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
2
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

3
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

4
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

5
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)

6
Conveyor Power Requirements

• Cost of trolley conveyor depends on
• length of conveyor
• number and size of carriers
• Installation cost
• Operation cost

• Horsepower requirement depends on
• conveyor speed
• weight of conveyor
• material being transported

7
HP calculations (Belt Conveyor)
Quick and dirty model

• Belt Conveyors Transporting packages, toteboxes,
  pallet

• S conveyor speed (fpm)
• L load to be carried by conveyor (lb)
• TL total length of conveyor (ft)
• RC roller spacing (in)
• WBR width between rails (in) (belt width)
• angle of incline (degrees)
• LLI live load on incline (lb) (weight of
  material on the conveyor section that is
  inclined)
• BV base value ( 2/3 WBR)
• FF friction factor ( 0.05 for roller
  supported 0.3 for slider bed supported)
• LF length factor (given in table 12.11)
• HP BV LF(TL) FF(L) LLI(sin ?)(S)/14,000

8
HP Calculations (Roller Conveyors)

• Similar to Belt Conveyor
• Different values of FF, BV, LF
• BV 4.60 0.445(WBR)
• FF 0.10 (for flat belt to drive the rollers)
  0.085 (for a zero pressure accumulating belt
  to drive the rollers 0.075 (for a v-belt
  to drive the rollers) 0.05 (chain driven
  rollers)
• LF is given in table 12.12