One Step Ahead Optimal Control of a Single Echelon Supply Chain
using Mathematical Programming
Amit Bhaya
a
, Eugenius Kaszkurewicz
b
and Luiz Carlos de Barros Roth
c
COPPE, Federal University of Rio de Janeiro, PO Box 68504, Rio de Janeiro, Brazil
Keywords:
Supply Chain, Mathematical Programming, Optimal Control, Discrete-time Dynamical System.
Abstract:
A single echelon supply chain model problem, consisting of a store with known inventory and shipping ca-
pacities, a known delivery delay or lead time and a random demand for a product at the store is formulated
as an optimal control problem. In the practical case when only current and past demands are known, using
the concept of one step ahead optimal control, the problem is reformulated as the mathematical programming
problem of maximizing economic value added (EVA), subject to the dynamics and constraints, such as inven-
tory size. Illustrative examples are given and performance indices are proposed to evaluate the performance
of the proposed controller, which exhibits good efficiency and no bullwhip effect.
1 INTRODUCTION
(McGarvey and Hannon, 2004) describe a simple sup-
ply chain problem in the following terms: One of
our most difficult issues involves understanding the
complex way in which the world of the customer con-
nects with the world of retail business. The vagaries
of pleasing customers on the one hand while dealing
with suppliers on the other can sometimes seem over-
whelming. Charge too much or not have the article
in stock, and you may never see the customer again.
Order too much or too often, and you may run up the
inventory cost, causing profits to vanish. This is a bal-
ancing act if ever there was one. How can you walk
that tightrope and survive? Reliance on mental mod-
els of a process fraught with randomness, feedbacks,
and delays makes the most astute of mental manage-
ment models unreliable.
In this paper we will formulate the single echelon
supply chain control problem discussed in (McGar-
vey and Hannon, 2004, p.179-187) as an optimal
control problem, when the demand is known over the
entire planning horizon and then propose feedback
controllers for unknown demands, based on this
approach. There is a large and ever-growing literature
on control of supply chains and a brief review is given
a
https://orcid.org/0000-0002-3144-1242
b
https://orcid.org/0000-0001-5471-9596
c
https://orcid.org/0000-0002-0481-4728
in the following section, focusing only on the papers
most relevant to the approach proposed here.
1.1 Literature Review
(Ivanov et al., 2018) in their recent survey write
Modern production and logistics systems, supply
chains, and Industry 4.0 networks are challenged by
increased uncertainty and risks, multiple feedback cy-
cles, and dynamics. Control theory is an interesting
research avenue which contributes to further insights
concerning the management of the given challenges
in operations and supply chain management. They
also identify one of the main contributions of control
theory as being the application of dynamic feedback
control to production-inventory systems and point out
that a wide range of control-theoretic tools have been
used in this context, ranging from the classical PID
control to model predictive control. We also point
the reader to the recent survey (Lin et al., 2017) on
control-theoretic approaches to the inventory control
problem.
In the specific area of feedback control of a sup-
ply chain, based on a state model of its dynamics,
most of the literature has been devoted to models that
apply standard control ideas such as set point con-
trol, which involves choosing set points or targets for
system variables, such as inventory level. These ap-
proaches can be viewed as parametrized versions of
several standard algorithms from the SCM literature,
such as order-up-to-inventory control. A large class of
Bhaya, A., Kaszkurewicz, E. and Roth, L.
One Step Ahead Optimal Control of a Single Echelon Supply Chain using Mathematical Programming.
DOI: 10.5220/0009130702330240
In Proceedings of the 9th International Conference on Operations Research and Enterprise Systems (ICORES 2020), pages 233-240
ISBN: 978-989-758-396-4; ISSN: 2184-4372
Copyright
c
2022 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
233
such controllers is known by the acronym IOBPCS,
which stands for inventory and order based produc-
tion control system. Various prefixes, such as AP (for
automatic pipeline) and MP (for matched parameter)
identify variants of the basic IOBPCS scheme. Papers
in this area also assume linear models and study the
effect of the controller on the bullwhip effect and on
stability with regard to, for example, step changes in
the demand: see, for example, (Lin et al., 2019; Lin
and Naim, 2019; Hoberg et al., 2007).
The model predictive control (MPC) approach
uses a model of the real plant, as well as past and cur-
rent system data to successively predict on some hori-
zon; optimize some suitable performance index, on
a (possibly different) horizon, using the predictions;
and, finally, apply one or more of the computed op-
timal controls to the plant, before beginning a new
predict-optimize-control sequence. For examples of
this approach, see (Wang et al., 2007) and references
therein. If the performance index takes economic fac-
tors into account, the MPC approach is referred to as
economic MPC and a recent example in the supply
chain context is (Subramanian et al., 2014).
This paper uses an approach that is related to the
MPC approach; however, it uses only past values of
the demand and system state, does not use a predic-
tor, and uses a stage cost that is economic: it is the
so called Economic Value Added (EVA) performance
index, commonly used in the business and financial
world (McGarvey and Hannon, 2004). The main idea,
which can be described by the term greedy control
(Lavretsky, 2000), is to optimize EVA using past de-
mands, for just one step ahead of the current time, ap-
ply this optimal control to the system, update its state
and repeat the cycle of one step ahead optimization
of EVA, until the end of the horizon. Note that the
control scheme just described is set up to compute an
ordering decision (or control) in real time, although it
can also be used in a planning mode. The main differ-
ences between the approach of this paper and existing
results are as follows: (i) an economic stage cost func-
tion is used, similar to the one in (Subramanian et al.,
2014), but also including discounting of the profit to
its present value and not including any target or track-
ing costs, (ii) using only past demands and the current
demand to compute the present value of the control,
not involving any prediction, in contrast with (Wang
et al., 2007). The result is a real-time control that is
computationally cheap and, as will be shown, efficient
when compared with the globally optimal omniscient
control, which will be defined in the sequel.
2 DISCRETE-TIME SUPPLY
CHAIN DYNAMICS
The supply chain considered in this paper is that of a
store, the example being taken from (McGarvey and
Hannon, 2004). It has two main components: a de-
lay between ordering and receiving, corresponding to
a block named conveyor or pipeline and an accumu-
lator, which receives the ordered items after the stip-
ulated delay, corresponding to the store. Selling of
items in the store occurs in accordance with the de-
mand and it is assumed that all but w items (the ones
in the display windows) in the store can be sold. The
store manager’s objective is to choose an ordering se-
quence, in response to the demand, that maximizes
cumulative profit over a given time horizon, assuming
that there are costs associated to handling, shipping,
storage and shortage (being out-of-stock affects sales
negatively, since customers are turned away). Finally,
profits (=sales revenues minus costs) are discounted
using a fixed interest rate and then added over the
given horizon to obtain the cumulative profit, referred
to as economic value added (EVA). The following
notation will be used to formulate the mathematical
model.
k: discrete time counter.
x
i
(k): quantity of items in the ith stage of shipping,
i = 1,...,D, where i = 1 corresponds to receiving and
i = D + 1 to ordering.
D: delay between receiving and ordering.
s
t
(k): stock level in store at discrete time instant k.
d(k): customer demand at store at time k.
s
e
(k): sales (selling) at time k.
p: price per unit of material sold.
w: quantity of items to be displayed in the store win-
dow (and not sold), also called display stock.
c
os
(k): out-of-stock cost at time k.
α
os
(k): out-of-stock cost per unit of demand not met
at time k.
c
h
(k): handling cost at time k.
α
h
: handling cost per unit of material handled.
c
st
(k): cost of maintaining stock level in store at
time k.
α
st
: storage cost per unit of material stored.
c
sh
(k): shipping cost of material in transit.
α
sh
: shipping cost per unit of material being
shipped.
c
t
(k): total cost at time k.
d
p
(k): discounted profit at time k.
ρ: discount rate.
L(k): cumulative profit at time k.
K
f
: horizon over which cumulative profit is to be
maximized.
The delay between ordering and receiving is writ-
ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems
234
ten by introducing a string of D one unit delays as
follows:
x
i
(k + 1) = x
i+1
(k), i = 1,...,D (1)
x
D+1
(k) = o(k) (2)
Thus the quantity of material in transit (i.e., being
shipped) at time k, denoted by s
h
(k), can be written
as:
s
h
(k) =
D
i=1
x
i
(k), (3)
while the quantity of material being received at the
store is x
1
(k) and corresponds to an order placed D
time units in the past, so that the variable ordering,
denoted o(k) is just x
D+1
(k).
Evolution of the stock level in the store at time
k is determined by the quantity of material being re-
ceived (x
1
(k)) plus the quantity currently in the store
s
t
(k) (setting aside a quantity w to be put in the display
window and not sold) less the quantity sold (s
e
(k)) in
response to the demand. The corresponding equation
is as follows.
s
t
(k + 1) = s
t
(k) w + x
1
(k) s
e
(k) (4)
In order to describe the sales s
e
(k) at time k, observe
that sales can only occur if the store level s
t
(k) plus
the received material x
1
(k) less the quantity in the dis-
play window w is greater than the demand d(k). Thus
sales can be written as follows:
s
e
(k) = d(k) + min{0, s
t
(k) w + x
1
(k) d(k)}. (5)
Thus the quantity of sales matches the demand d(k) if
it is less than the effective available store level s
t
(k)
w + x
1
(k), otherwise it is just equal to the latter.
The costs are expressed as follows:
c
o
(k) = α
os
(d(k) s
e
(k)) (out of stock cost) (6)
c
st
(k) = α
st
s
t
(k) (storage cost) (7)
c
h
(k) = α
h
(x
1
(k) + o(k)) (handling cost) (8)
c
sh
(k) = α
sh
D
i=1
x
i
(k) (shipping cost) (9)
c
t
(k) = c
o
(k) + c
st
(k) + c
h
(k) + c
sh
(k) (total cost)
(10)
Discounted profit is the revenue from sales less the
total cost (all at time k), appropriately discounted:
d
p
(k) = (ps
e
(k) c
t
(k))e
ρk
. (11)
The cumulative profit L(k) accumulates the dis-
counted profit:
L(k + 1) = L(k) + d
p
(k). (12)
The objective function is simply the cumulative profit
at the end of the time horizon and thus, given the de-
mand d(k),k = 1,...,K
f
1 over the whole plan-
ning horizon, the optimization problem to be solved
is an optimal control problem, which will be referred
to as the omniscient optimal control problem (since
the demand over the entire planning horizon is as-
sumed to be known), and can be written as follows:
max
{o(k)}
K
f
1
k=1
L(K
f
) (13)
In optimal control terminology, this is a problem of
choosing an optimal ordering sequence that maxi-
mizes the final cumulative profit L(K
f
). More real-
istically, the following constraints are also needed:
o(k) 0, k = 1, ...,K
f
1 (14)
s
h
(k) C
max ship
, k = 1,...,K
f
1 (15)
s
t
(k) C
max store
, k = 1,...,K
f
1 (16)
For future reference, the state vector of the supply
chain at instant k is denoted as z(k) and defined as:
z(k) = [x
1
(k), ...,x
D
(k), s
t
(k)] (17)
3 ONE STEP AHEAD OPTIMAL
CONTROL
In practice, of course, only current and past demands
are known to the supply chain manager, who has to
make the ordering decision based on this information
and observation of the current variables, such as in-
ventory or store level, amount of goods in the ship-
ping pipeline, sales levels, etc. This section formu-
lates the so called one step ahead optimal control
(OSAOC) problem that respects this practical infor-
mation constraint.
Given the delay D between ordering and receiv-
ing, it is clear that, at time instant k, the value
of the objective function L(k + 1) (the cumulative
profit) at the next time instant k + 1 is dictated by
the ordering o(k D). This means that, since the
past demands and the current demand are assumed
known, one can optimize over the backward hori-
zon {k D, k D + 1,...,,k} (the past), to obtain the
optimal sequence {o
(i)}
k
i=kD
, assuming the state
z(k) = [x
1
(k), ...,x
D
(k), s
t
(k)] to be given. Note that,
from (1)-(2), the state z(k) can also be expressed as
z(k) = [o(k D),...,o(k 1),s
t
(k)]. Thus, after the
optimization step is carried out, the state is updated to
z(k + 1) = [o(k D + 1), ...,o
(k), s
t
(k + 1)], intro-
ducing only the current input and discarding the older
ones, which are given and cannot be changed. Now
a new optimization step can be carried out, from this
One Step Ahead Optimal Control of a Single Echelon Supply Chain using Mathematical Programming
235
new initial condition. In other words, at each step k,
although the entire vector {o
(i)}
k
i=kD
is determined,
only the first element of the sequence o
(k) is used: it
is injected into the sytem state and its effect on the ob-
jective function is only observed D instants later. The
OSAOC problem just described is denoted OS(z(k))
and written formally as follows:
max
{o(i)}
k
i=kD
L(k + 1)
subject to (1)-(12)
and z(k) = [x
1
(k), ...,x
D
(k), s
t
(k)] (specified i.c.)
(18)
Let the optimal control computed as the solution
to (18) be denoted as follows:
[o
(k D),...,o
(k)] := argmax
{o(i)}
k
i=kD
L(k + 1)
(19)
With this notation in place, the iterative one step
ahead control (OSAOC) scheme can be written as
shown (Algorithm 1):
Algorithm 1: One step Ahead Optimal Control (OSAOC).
1: Initial state z
0
:= z(k
0
), k
0
= D given
2: while D k K
f
1 do
3: z(k) := z
0
= [o(k D),..., o(k 1),s
t
(k)]
4: solve OS(z(k)) defined in (18)
5: to get [o
(k D);...;o
(k)]
6: z
0
[o(k D + 1),...,o
(k), s
t
(k + 1)]
7: k k + 1
8: end while
Remark: Algorithm 1 bears a resemblance to a model
predictive control (MPC) scheme. Note, however,
that it relies only on past values of the demand and
uses no prediction.
4 IMPLEMENTATION OF THE
ALGORITHM
The MATLAB implementation uses the Optimization
Toolbox, which is set up to carry out black box opti-
mization of a function, using interior-point, sequential
quadratic programming (SQP) and active set meth-
ods, amongst others. The black box in question is
the system model (1)-(5) of the supply chain dynam-
ics. In the examples presented in Sec. 5, the MAT-
LAB Optimization Toolbox fmincon command was
used to optimize, with the algorithm being chosen as
interior point.
5 SIMULATION EXAMPLES
In this section, an example of a simple demand curve
is first given to highlight the differences between om-
niscient and one step ahead optimal control. This is
followed by a more realistic example of a demand
generated from a Poisson distribution. In all cases,
the comparison is between the final value of the cu-
mulative profit attained by the OSAOC (L
OSA
) and
the omniscient control (L
omni
) for the same demand
sequence. The ratio between L
OSA
/L
omni
is referred
to as the efficiency η.
5.1 Demand with Step Change
The simple demand used to illustrate the differences
in behavior between the omniscient and OSAO con-
trol is constant, with one step change. The parame-
ters used in the simulation are as follows: α
os
= 20,
α
st
= 5, α
h
= 10, α
sh
= 5, p = 100, ρ = 0.005,
C
max store
= 30, C
max ship
= 50, s
t
(0) = 10, D = 7,
K
f
= 37, s
h
(0) = 0. Finally the demand is as fol-
lows: constant at value 4 from k = 0 to k = 20, after
which a step change to the value 6 occurs. The cumu-
lative profits L
OSA
and L
omni
attained by the OSAOC
and omniscient algorithms at the end of a 30 day
period are, respectively, 7288.7 and 8605.9, so that
the OSAOC achieves an efficiency η = 84.6%, with
respect to the globally optimum (omniscient) result.
Figures 1 to 3 show the main results for the demand
with step change and their captions provide additional
details and observations.
5.2 Demand with Poisson Distribution
Experiments in this subsection are carried out with
a demand d that has a Poisson distribution with
λ = 1 shifted by 1 (to eliminate demands equal to
zero), generated by the MATLAB command d =
poissrnd(1,37,1)+1. The parameters used in the
simulation are as follows: α
os
= 20, α
st
= 5, α
h
=
10, α
sh
= 5, p = 100, ρ = 0.005, C
max store
= 30,
C
max ship
= 50, s
t
(0) = 10, D = 7, K
f
= 37, s
h
(0) = 0.
The cumulative profits L
OSA
and L
omni
attained by the
OSAOC and omniscient algorithms at the end of a
30 day period are, respectively, 2107.6 and 2598.6,
so that the OSAOC achieves an efficiency η = 81.1%
with respect to the globally optimum (omniscient) re-
sult. Figures 4 to 9 show the main results for the Pois-
son demand and their captions provide additional de-
tails and observations.
ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems
236
5.3 Comparison of Average
Performance of OSAOC versus
Omniscient Control
Ten samples each of Poisson and uniformly dis-
tributed demands were generated and the average ef-
ficiency (resp. standard deviation) were found to be
78.25%(3.28%) (Poisson) and 73.52%(5.77%) (uni-
form).
The coefficient of variation is a measure of relative
variability, defined as the ratio of the standard devia-
tion to the mean of the variable under consideration.
Thus, a measure of amplification of demand variabil-
ity, also known as bullwhip, is the ratio of the coef-
ficients of variation of the ordering and the demand,
and is denoted B
x
od
, where the subscript od refers to
the ratio being calculated and the superscript x can be
os (resp. om) for one step (resp.omniscient). A related
measure is the ratio of the coefficients of variation of
the inventory (store) level and the demand, denoted
B
x
std
. For the Poisson demand of sec. 5.2, the calcu-
lations, carried out over the interval [D,K
f
D] so as
to exclude both initial transient due to pipeline ini-
tialization, and final transient due to the turnpike ef-
fect for the omniscient case, yield: B
os
od
= 0.43,B
os
std
=
0.32,B
om
od
= 0.70,B
om
std
= 0.28. Since all these mea-
sures are less than unity, the conclusion is that the
proposed one step ahead controller does not cause
amplification of the demand uncertainty. This should
be contrasted with the behavior of controllers of the
IOBPCS family (Lin and Naim, 2019) and the MPC
family with targets (Subramanian et al., 2014).
6 CONCLUDING REMARKS
This paper proposed a one step ahead optimal control
scheme for a single echelon supply chain that does
not require prediction but only the solution of a low-
dimensional (in terms of the number of decision vari-
ables) optimization problem at each time step, thus
being perfectly adequate for real time supply chain
control. This one step ahead or greedy control is
shown to be quite efficient, attaining, for Poisson or
uniformly distributed demands, at least 70% of the
(unattainable) omniscient global optimum. In addi-
tion, experiments show that the proposed OSAO con-
trol scheme does not cause amplification of the de-
mand uncertainty or bullwhip, which is attributable to
the fact that no target levels for inventory or pipeline
are used. Another important observation is that the
model (1)-(12), with the exception of (5), consists
of linear equations. It is easy to write (5) in piece-
wise linear form, which means that both the omni-
scient (13) and OSAOC optimization problems (18)
can be rewritten as linear programming problems,
which, from a computational viewpoint, implies that
the simple examples in Sec. 5, can easily be scaled to
much larger dimension. It should also be noted that a
large number of supply chain problems, including the
multi-echelon case, are described by linear or piece-
wise linear dynamics. Thus, the proposed scheme is
scalable and can: (i) be generalized to large multi-
echelon supply chains, which will be the subject of fu-
ture work, and (ii) used in a planning mode, to choose
inventory size, shipping capacity requirements and
prices that ensure profit margins, since these choices
can be recast as feasibility problems. This is also the
subject of ongoing work, to be reported in a future
version of this paper.
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One Step Ahead Optimal Control of a Single Echelon Supply Chain using Mathematical Programming
237
Figure 1: Showing ordering and selling over the horizon k = 0 to k = K
f
= 37, with a step increase in the constant demand at
k = 20: OSAOC (left), omniscient control (right). Note the turnpike effect for omniscient control over the horizon k
f
= 37,
with the ordering dropping to zero D instants before the end of the horizon is reached, because the demand will be met with
the orders in the pipeline. For this reason, all simulation examples are run till K
f
= 37, but the efficiency η is calculated
at k = 30 (in order to exclude the turnpike effect). Also note that, the omniscient control, with complete knowledge of the
demand, anticipates ordering in order to meet the demand perfectly, given a sufficiently large initial inventory, and maintains
the inventory at the minimum, which is the specified display stock of 1, thus operating in just-in-time mode.
Figure 2: Showing lost sales: OSAOC (left) and omniscient control (right), until k = 30. Observe that both controllers lead
to lost sales in the initial segment, between k = 9 and k = 14, because the pipeline is initially empty. However, the OSAO
controller also results in lost sales when the step increase in demand occurs, since it is causal (non-anticipative) and this, of
course, does not occur with the omniscient noncausal controller, which is able to anticipate the future demand.
Figure 3: Showing evolution of cumulative profit: OSAOC (left), omniscient control (right), with the OSAO controller
attaining an efficiency η = L
OSA
/L
omni
= 84.6%. Note that, after the end of the transient caused by the zero initial pipeline
(s
h
(0) = 0), the cumulative profit curve is monotonically increasing.
ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems
238
Figure 4: Showing the evolution of ordering, selling, store for a Poisson demand: OSAOC (left), omniscient (right) for the
horizon k = 0 to k = K
f
= 37, with the turnpike effect for the omniscient control starting from k = 30.
Figure 5: Showing the evolution of ordering, selling, store variables for a Poisson demand, with the graphs plotted from
instant k = 8 to k = 30, in order to start from the first computed ordering, assuming that demands from k = 1 to k = 7 are used
as the initial past demands, and also, for purposes of comparison, truncating the horizon at k = 30 to avoid the turnpike effect,
which occurs at the end of the horizon, for the omniscient case: OSAOC (left), omniscient (right).
Figure 6: Showing the evolution of the store and shipping variables, for a Poisson demand, for a 30 day period: OSAOC (left),
omniscient (right). Note that, after the initial transient (i.e., k 16), the OSAOC is not able keep the inventory level at the
minimum, which is the specified display stock of 1, unlike the anticipative omniscient controller.
One Step Ahead Optimal Control of a Single Echelon Supply Chain using Mathematical Programming
239
Figure 7: Showing the evolution of lost sales, for a Poisson demand: OSAOC (left), omniscient (right). Note that, lost sales
occur at k = 17, 19,24,26 only for OSAOC after the initial transient, but not for the omniscient controller. This means that,
in Figure 6, the OSAOC is at minimum store level at these instants, since it is unable to meet the demand. In contrast, the
omniscient controller, also at the minimum store level, is meeting demand with the orders being received, in the just-in-time
mode.
Figure 8: Showing the evolution of selling for a Poisson demand: OSAOC (left), omniscient (right). Note that, after the initial
transient, the omniscient selling matches the demand perfectly, operating, as explained in the caption to Fig. 7.
Figure 9: Showing the evolution of the cumulative discounted profit (EVA): OSAOC (left), omniscient (right), with the OSAO
controller attaining an efficiency η = L
OSA
/L
omni
= 81.1%. Despite the zero initial pipeline, the initial store level is able to
meet the initial demands during the lead time of D = 7.
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