Abstract
This paper presents stylized models for conducting performance analysis of the manufacturing supply chain network (SCN) in a stochastic setting for batch ordering. We use queueing models to capture the behavior of SCN. The analysis is clubbed with an inventory optimization model, which can be used for designing inventory policies . In the first case, we model one manufacturer with one warehouse, which supplies to various retailers. We determine the optimal inventory level at the warehouse that minimizes total expected cost of carrying inventory, back order cost associated with serving orders in the backlog queue, and ordering cost. In the second model we impose service level constraint in terms of fill rate (probability an order is filled from stock at warehouse), assuming that customers do not balk from the system. We present several numerical examples to illustrate the model and to illustrate its various features. In the third case, we extend the model to a three-echelon inventory model which explicitly considers the logistics process.
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Abbreviations
- Q :
-
number of units in one bucket
- K :
-
total number of buckets at warehouse
- Z :
-
maximum inventory at warehouse (KQ)
- λ :
-
demand arrival rate at warehouse
- A(t):
-
number of orders arrived up to time t at manufacturer
- μ :
-
service rate of manufacturing plant units/unit time
- I (t):
-
inventory at warehouse at time t
- N(t):
-
number of orders at manufacturing plant being processed at time t
- B(t):
-
number of back orders in the system at time t
- R(t):
-
number of orders arrived up to time t, but after the last batch was released for processing
- h :
-
inventory holding cost ($ per unit per unit time)
- b :
-
back order cost ($ per unit per unit time)
- C s :
-
order set up cost ($ per set up)
- α :
-
desired service level for orders at warehouse i.e. probability an order is filled from stock at warehouse
- ρ :
-
intensity of the system (λ/μ < 1)
- M :
-
number of retailers served by the warehouse
- λ m :
-
demand arrival are at retailer m units/time unit
- λ :
-
\({\sum_{m=1}^{M}\lambda_{m}}\), net demand rate at the warehouse units/time unit
- l m :
-
lead time for logistics for retailer m to receive items from warehouse, l 1 = l 2 = · · · = l M = l (Local haul assumed negligible), as exponential random variable
- X m :
-
demand during replenishment lead time, a Poisson random variable
- p m (a):
-
probability mass function of demand, P{ X m = a}
- F Xm (x):
-
cumulative distribution function of demand during lead time
- Γ:
-
expected number of orders in the queue M C/M/∞ in steady state
- ξ:
-
service rate of logistics process (exponentially distributed)
- ρ′:
-
intensity of the the logistics hub (λ/ξ < 1)
- W :
-
expected waiting time at warehouse due to back ordering alone
- L m :
-
mean lead time (including back ordering delay) for an order of items from retailer m to be filled from warehouse, L 1 = L 2 = · · · = L M = L
- θ m :
-
expected demand during replenishment lead time for item at retailer (θ m = λ m L m )
- β m :
-
probability of satisfying customer demand at retailer m from available stock
- r m :
-
reorder point for item at retailer m (decision variable)
- R m :
-
r m + 1, base stock level for item at retailer m
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Jain, S., Raghavan, N.R.S. A queuing approach for inventory planning with batch ordering in multi-echelon supply chains. Cent Eur J Oper Res 17, 95–110 (2009). https://doi.org/10.1007/s10100-008-0077-8
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DOI: https://doi.org/10.1007/s10100-008-0077-8