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Optimal inventory control policies of a two-stage push–pull production inventory system with lost sales under stochastic production, transportation, and external demand

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Abstract

An analytical model based on Markov processes is proposed for the analysis of a linear, horizontally integrated, two stages, push–pull inventory system. Uncertainty about both supply and demand is taken into consideration. Exponentially distributed lead times, compound Poisson external demand and lost sales are assumed. An algorithm that creates the infinitesimal generator matrix of the system is developed and an exact numerical solution of the system performance measures is also provided. The proposed model can be either used to evaluate what if scenarios exploring the behavior of the system or to optimize performance measures of the considered system. As an example, the model is used to analyze and get insights of the behavior of a supply–demand balanced system.

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Authors and Affiliations

  1. Department of Business Administration, University of the Aegean, Chios, Greece

    George Varlas & Michael Vidalis

  2. Department of Business Administration, University of Thessaly, Volos, Greece

    Stelios Koukoumialos

  3. Department of Economics, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Alexandros Diamantidis

Authors
  1. George Varlas
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  2. Michael Vidalis
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  3. Stelios Koukoumialos
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  4. Alexandros Diamantidis
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Correspondence to Alexandros Diamantidis.

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Appendices

Appendix A: Analytical example

An example of infinitesimal generator matrix for buffer capacity B = 2, Reorder point s = 1, Order quantity Q = 2, and maximum demand per customer n = 3 is examined. According to the above decision variables, the values of the system parameters are 0 ≤ Bt ≤ 3, 0 ≤ Tt ≤ 2, 0 ≤ It ≤ 3, while from (1), the possible states are 26. Βelow are given sequentially the possible states, the state transition diagram, and the infinitesimal generator matrix (Figs. A.1, A.2, A.3).

Fig. A.1
figure 24

Possible states for B = 2, s = 1, Q = 2, n = 3

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Fig. A.2
figure 25

State transition diagram for B = 2, s = 1, Q = 2, and n = 3

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Fig. A.3
figure 26

Infinitesimal generator transition matrix for B = 2, s = 1, Q = 2, and n = 3

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Appendix B: Table of notation

  • ALS: The average lost sales per external order at the retailer

  • B: The capacity of the finished goods buffer (FGB).

  • Bt: Inventory at buffet at time t

  • d: Amount of individual demand

  • E: The average demand per external customer

  • FR: Order Fill Rate

  • h1: Inventory holding cost per unit at buffer per time unit

  • h2: Inventory holding cost per unit on hand at the retailer per time unit

  • h3: Inventory holding cost per unit in transit to the retailer per time unit

  • h4: Cost incurred because of lost sales, per unit of lost sales

  • It: Inventory on hand at the retailer at time t

  • Lost_Sales: The average lost sales per lost order, i.e., the average lost sales per order partially met or not met at all from the inventory on hand at the retailer

  • n: The maximum demand per external customer, assuming a uniform distribution in the space [1,n].

  • Ns,Q,B: The dimension of the Markov Process states space

  • P: The infinitesimal generator matrix

  • pblock: The probability that S1 is blocked

  • Q: The number of orders requested by the retailer

  • ROR: The replenishment order rate, the number of replenishment orders from the buffer to the retailer per time unit

  • s: The reorder point at the retailer

  • SL2: The percentage of total external demand that is met from the inventory on hand at the retailer

  • SO: Stock-out probability. The probability of the retailer having zero inventory on hand

  • Tt: Inventory in transit toward the Retailer at time t

  • TC: Total cost per time unit as a function of the decision variables and demand variability

  • uP: The utilization of production station S1

  • uT: The utilization of transportation resource

  • WIPbuffer: The average inventory of buffer

  • WIPretailer: The average inventory of the retailer

  • WIPtotal: The average inventory of the system

  • WIPtransit: The average inventory in transit toward the retailer

  • X: The vector of the stationary probabilities

  • λ: The rate of external customers’ arrivals (Poisson process)

  • μ1: The production rate of the Station 1, (exponentially distributed production times)

  • μ2: The transfer rate of a replenishment order from the buffer to the retailer (exponentially distributed times).

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Varlas, G., Vidalis, M., Koukoumialos, S. et al. Optimal inventory control policies of a two-stage push–pull production inventory system with lost sales under stochastic production, transportation, and external demand. TOP 29, 799–832 (2021). https://doi.org/10.1007/s11750-021-00595-0

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