库存误差下单阶段生产系统稳健补货与生产控制策略

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本文是一篇深入研究的"具有库存不准确性的单阶段生产/库存系统的稳健补货与生产控制策略"的研究论文，发表在《IEEE Transactions on Systems, Man, and Cybernetics: Systems》上。作者是郑王和费利克斯·T·S·陈。文章主要关注的是实际生产/库存系统中普遍存在的库存准确性问题，这往往对原材料补货和生产控制的性能造成显著负面影响。研究的核心议题在于设计一种稳健的策略，旨在在一个单阶段生产/库存系统中最小化平均库存和生产成本。该系统的特点包括库存数据的不准确性和两个关键的外部因素：原材料订购的提前期（lead-time）以及机器的不可靠性。这两个因素对于任何生产计划来说都是不容忽视的挑战。论文首先针对这样一个系统提出问题：如何在存在订单提前期和不确定生产能力的情况下，有效地管理库存，并考虑到库存误差的影响。解决这个问题的方法被分为了两个步骤： 1. 在没有库存不准确性的前提下，研究者首先探讨了单阶段生产/库存系统中的补货和生产控制问题，这里主要考虑的是提前期和生产资源的不确定性。通过数学模型和优化技术，他们可能提出了一个近似解决方案，该方案在处理这类复杂动态环境时提供了一种基础框架。 2. 随后，当库存准确性成为另一个要考虑的因素时，研究者进一步扩展了先前的模型。他们可能引入了概率建模或者鲁棒优化方法来处理库存误差带来的随机性，确保补货决策在面临真实世界中的不确定性和波动时仍能保持系统稳定和高效。整个研究不仅关注理论分析，还可能包含了实际案例研究或仿真结果，以验证策略的有效性和实用性。通过这种方法，本文旨在为解决单阶段生产/库存系统中的库存不准确性问题提供一套实用且适应性强的管理策略，这对于任何寻求提升生产效率并减少库存成本的企业都具有重要的参考价值。

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

WANG AND CHAN: ROBUST REPLENISHMENT AND PRODUCTION CONTROL POLICY 3

an optimal state-dependent base-stock control policy based on

a continuous time Markov decision model.

To deal with the uncertainties in manufacturing systems

(e.g., variable processing time, failure of machines, or ﬂuc-

tuation of demands), robust production scheduling is often

applied. Kouvelis et al.[27] measured the uncertainty of pro-

cessing time by the range of its ﬂuctuation, and modeled the

robust scheduling problem by an integer programming model

to minimize the gap between the scheduling performance in

the worst case and that of the optimal scheduling. For the

uncertainty from machine failure, Yellig and Mackulak [28]

developed a robust scheduling policy to avoid frequent

rescheduling. For uncertainties from both processing time vari-

ability and machine breakdown, Goren and Sabuncuoglu [29]

deﬁned three robustness and two stability measures and pro-

posed a branch and bound algorithm and a beam search

heuristic to solve the problem. Robust optimization is an

often-used technique for solving robust production scheduling

problems, where uncertain parameters in production schedul-

ing are usually captured by variables in a known interval [30],

or random variables with a known probability distribution [31].

This technique has been applied to robust production schedul-

ing problems in process industry [32], [33]. It is also applied

to solve the production planning and scheduling problem for

complex manufacturing and supply chain systems with ran-

dom market demand, uncertain yield or inaccurate production

data [34]–[36].

C. Summary

From the above survey on the related work, we ﬁnd as

follows.

1) Existing research on inventory inaccuracy pays more

attention to inventory control policy for hedging against

the negative impact caused by this inaccuracy. However,

little attention has been paid to the inventory inaccuracy

at the production stage of a supply chain and robust pro-

duction control policies again this inaccuracy. The robust

replenishment and production control problem of a pro-

duction/inventory system with inventory inaccuracy has

not been sufﬁciently studied.

2) Existing research on the control of a produc-

tion/inventory system has not taken inventory inaccuracy

into consideration, although many other uncertainties

(e.g., stochastic processing time, random demand, ﬂuc-

tuation of production capacity, or variable lead-time)

have been investigated, and many control policies (e.g.,

the policies based on classical control theory, opti-

mal production and inventory control policies, or robust

production scheduling theory) were developed.

Due to the reasons given from the above two perspec-

tives, the robust replenishment and production control problem

for a production/inventory system with inventory inaccuracy

becomes the main issue of this research.

III. P

ROBLEM DESCRIPTION

In this paper, we try to solve the robust replenishment

and production control problem for a single-stage produc-

tion/inventory system with inventory inaccuracy. This system

Fig. 1. Single-stage production/inventory system.

is composed of a replenishment process, a production process,

and a demand satisfying process. In the replenishment pro-

cess, the system orders raw materials from outside to replenish

its raw materials stock, which is located between the replen-

ishment process and the production process and has inﬁnite

capacity. In the production process, the system consumes raw

materials and produces ﬁnal goods, which are stored in the

ﬁnal goods storage with inﬁnite capacity. In the demand satis-

fying process, the ﬁnal goods are delivered to customers. The

structure of this production inventory system is depicted in

Fig. 1.

The dynamics of this single-stage production/inventory

system can be captured by two differential equations: one is

for the replenishment process, and the other is for the produc-

tion and demand satisfying process. The state variables are

the raw materials inventory level and the ﬁnal goods inven-

tory level. It is necessary to note that the inventories of raw

materials and ﬁnal goods are inaccurate. We assume that the

errors between observed inventory level and actual inventory

level for the raw materials and the ﬁnal goods are normally

distributed with known means and variances. The control vari-

ables are the replenishment rate and the production rate. The

lead-time between placing an order for raw materials and the

arrival of raw materials are taken into consideration. However,

we assume that the demand can be satisﬁed immediately when

the ﬁnal goods stock is not empty, namely, there is no time-

delay between production and ﬁnal goods delivery. We also

assume that the demand rate is constant, the production capac-

ity is ﬁnite, and the machines are failure-prone. We assume

that the time to failure and the time to repair of the machine

are exponentially distributed. The objective of this problem is

to minimize the average inventory and production cost, which

includes the holding cost of raw materials, the holding cost of

ﬁnal goods, and the backlog cost of demand.

The mathematical model of this problem can be constructed

as follows:

Minimize lim

T→∞







(t) + c

−

(t)





(1)

Subject to

(t)



−u(t), if 0 ≤ t < L

r(t − L) − u(t), if t ≥ L

(2)

(t)

= u(t) − d(t) (3)

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