A Mathematical Modeling Approach to Floating Stock Policy

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• A Mathematical Modeling Approach to Floating
  Stock Policy
• Morteza Pourakbar, Andrei Sleptchenko, Rommert
  Dekker
• Erasmus Research Institute of Management, EUR
• October 2007

2
Outline

• Introduction Literature
• Motivation Research Idea
• Problem Description
• Conceptual Model Assumptions
• Mathematical Formulations
• Time Based Policy
• Quantity Based Policy
• Case Study
• Conclusion

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Motivation

• Inter-modal transport can be defined as the
  movement of goods in one and the same loading
  unit or vehicle by successive modes of transport
  without handling of the goods themselves during
  transfers between modes ( European Conference of
  Ministers of Transport (1993).
• It is highly advocated by governments. But on
  short distances, it is more costly than road
  transport.
• Transport studies typically make such comparisons
  between road transport and inter-modal transport,
  but in these studies inventories are left out of
  consideration.
• There are just a few studies that integrate
  transportation and inventory control, but they
  focus on the relation between either transport
  frequency or transit time reliability and
  inventory control (shipping scheduling problems).
• No study seems to exist which integrate
  inter-modal transport and inventory control,
  according to recent reviews on inter-modal
  research.

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Research Idea

• Floating Stock, a new distribution concept which
  was presented to exploit the opportunities that
  inter-modal transport offers to deploy
  inventories in the supply chain.
• The idea is that by advanced deployment with
  alternating transport modes we can
• reduce non-moving inventories
• shorten lead times

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Distribution Strategies

• For every full truck load unit, we have to decide
  whether
• to store it in a centralized or a decentralized
  location
• to use road or inter-modal transport.
• Distribution strategies can be categorized as
• Strategy CS Centralized storage and uni-modal
  transport
• Strategy DS Decentralized storage and
  inter-modal transport
• Strategy DS/CSS Decentralized storage,
  inter-modal transport, and centralized safety
  stock
• Strategy MS Mixed Strategy (Floating Stock)

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Distribution Strategies

• Strategy CS Centralized storage and uni-modal
  transport
• It is a just-in-time based strategy.
• Whole production batch and the safety stock are
  stored on-site at the factory storage.
• Upon order arrival, It is directly shipped from
  factory using road transport.
• In this strategy the emphasis is on fast
  transportations and easy coordination.

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Distribution Strategies

• Strategy DS Decentralized storage and
  inter-modal transport
• The complete production batch is shipped to
  regional terminals using inter-modal transport.
• Orders are delivered by truck from these
  terminals to the DCs.
• The safety stock is also stored in these regional
  terminals.
• The emphasis is on using inter-modal
  transportation and short order lead times .
• If the safety stocks are depleted at a terminal,
  lateral transshipments from other terminals are
  made.

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Distribution Strategies

• Strategy DS/CSS Decentralized storage,
  inter-modal transport, and centralized safety
  stock
• Safety Stock is stored at the factory storage,
  whereas the production batch is shipped to the
  terminal using inter-modal transport and stored
  there.
• The regular deliveries are fulfilled from the
  terminals, but in excess demand, first lateral
  transshipments from and then emergency deliveries
  are done from the factory storage.
• The safety stock storage costs will probably be
  lower in the DS/CSS strategy when compared to the
  DS strategy.
• Reliability increases if the safety stock is
  stored in a central location.

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Distribution Strategies

• Strategy MS Mixed storage (Floating Stock)
• Stores part of the production batch in the
  factory storage (centralized) and part of the
  production batch is stored in decentralized
  terminals. The safety stock is stored at the
  factory.
• All orders that are placed while the intermodal
  transport is in transit, are fulfilled from the
  on-site inventory at the factory using road
  transport.
• Once the products have arrived at the terminal,
  the orders are delivered from the terminal (with
  a shorter order lead time).
• Emergency orders in a period of excess demand are
  delivered using road transport from the safety
  stock stored at the factory.
• If the safety stock at the factory is depleted,
  lateral transshipments from other regional
  terminals are considered.
• A lower level of centralized inventory will
  either lower reliability or increase storage
  costs.

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Distribution Strategies
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Floating Stock

• In a paper by Ochtman, G., Dekker, R., van
  Asperen, E. Kusters, W. (2004) a real case is
  simulated for all four policies, they assumed
  that
• The products are containers including FMCG.
• Demand has triangular distribution function.
• In their model for the mixed strategy, the whole
  production batch is partitioned into two batches
  one is shipped to the terminal and the other one
  is hold at the factory.
• It is free of holding cost charge for a limited
  number of days in terminal and afterwards it is
  more expensive than factory.

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Floating Stock

• Conceptual Model

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Floating Stock

• The simulation results are
• According to Total Cost, MS strategy is cheaper
  than the other strategies, Even though, direct
  transport is cheaper, faster and more reliable.
• Total average inventory including pipeline and
  storage for the MS strategy is the least.
• Comparing the reliability of MS strategy with
  other ones reveals that it does not affect the
  reliability that much.

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Problem Description

• Assumptions
• It is a FMCG supply chain with two echelons.
• Cost structure is similar to the previous work.
• In case of stock-out in terminal demand is
  backlogged.
• Customers call off containers according to a
  renewal process with inter-demand distribution X
  
• Transshipment is not allowed.
• Production phase is left out of consideration.

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Formulation

• The following notations are used in these models
• hf constant holding cost at factory per time
• hi holding cost at inter-modal transit
• Cb backlogging cost or waiting cost.
• Tfi transportation time from factory to inter
  modal
• Tic transportation time from inter modal to
  customer
• Tnh number of days without incurring holding
  cost in inter-modal

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Formulation

• Issues
• What are the costs if container i is delivered
  too late.
• What are the costs if it is delivered too early
  In both cases, we may run into the costly period.
• Objective
• Minimizing total cost by shipping containers to
  inter modal terminal at the right time.

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Solution Time Based Policy

• Time Based Policy
• Description
• Determination of the rk for k1,2, ,m
• rk is the shipping time of container k
• Formulation

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Solution Time Based Policy

• Define arrival time to terminal as AkrkTfi
• Ak Tnh lt Dk
• container arrives to inter-modal, waits for the
  entire free of holding charge period and then
  demand is realized. This case leads to holding
  cost at inter-modal but it benefits from less
  holding cost at factory.
• Ak ltDk ltAj Tnh
• In this case, demand happens after arrival of the
  container and before the end of inter-modal free
  of holding cost period. Therefore, not only no
  holding cost incurred at inter-modal but also it
  benefits from less holding cost at factory.
• Dk ltAk
• In this circumstance, holding cost saving due to
  shipment to inter-modal does not happen. Moreover
  extra holding cost is paid at the factory. But
  since we assumed backlogging, demand waits for
  the container, therefore backlogging cost happens.

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Solution Time Based Policy

• Total cost function when backlogging occurs is
  defined as
• It is proved that the cost function is convex.

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Solution Time Based Policy

• Lemma.
• Given continuous distribution of the demand time
  Dt, optimal shipment moments rk, that minimize
  the expected cost for each container either solve
  equation
• or should be set equal to the production moment,
  when there is no optimal rk after the production
  moment.

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Solution Quantity Based Policy

• Quantity Based Policy
• - Description
• Determination of the rk and Sk k1,2, ,m
• rk release time of container k after inventory
  level drops down to Sk-1
• Sk total number of containers in pipeline and
  intermodal terminal
• Formulation
• Demand process
• Dt Dt-1 ?t

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Solution Quantity Based Policy

• Total cost function when backlogging occurs is
  defined as
• We assume that interdemand times are i.i.d and
  Erlang distributed, then it is easy to conclude
  that for each arrival Dt the optimal parameters
  St and rt will be identical such as S and r.

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Solution Quantity Based Policy

• Simplifying the last equation we obtain an
  equation for the time r(S) that is quite similar
  to the equation in the previous section
• and the optimal time r(S) is either equal to
  0 or solves this equation.

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Case Study Network Representation
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Case Study Specification

• The transit time for all railway routes is four
  days including handling time for in- and outbound
  in the on-site DC.
• The transit time from two terminal to all
  customer is one day.
• Costs Components are
• Centralized holding cost at factory storage is 8
  euro per container/day
• Decentralized holding cost in terminal 16 euro
  per container/day, but it is free for the first
  three days
• Backlogging cost at inter-modal 20 euro per day
• Direct road transport is 880 euro/container and
  intermodal transport is 900 euro/container

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Case Study Numerical Results, ?D2.5, ?M2 and
k3
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Case Study Numerical Results
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Conclusion

• In this study, we tried to find a mathematical
  model for floating stock policy.
• Two different approaches were proposed.
• Results of those approaches show that integration
  of transportation and inventory system in an
  intermodal environment using floating stock
  policy leads to less total cost even though
  intermodal transport is slower and more expensive
  but it does not affect fill rate dramatically.

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