Applied Mathematical Modelling 56 (2018) 275–288

Contents lists available at ScienceDirect

Applied Mathematical Modelling

journal homepage: www.elsevier.com/locate/apm

Uncertain dynamical system-based decision making with

application to production-inventory problems

Linxue Sheng

a , b , Yuanguo Zhu

b , ∗, Kai Wang

c

a State Key Laboratory of Air Traffic Management System and Technology, The 28th Research Institute of China Electronics Technology

Group Corporation, Nanjing, Jiangsu 210014, China b School of Science, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China c Science and Technology on Information Systems Engineering Laboratory, The 28th Research Institute of China Electronics Technology

Group Corporation, Nanjing, Jiangsu 210 0 07, China

a r t i c l e i n f o

Article history:

Received 8 April 2016

Revised 29 November 2017

Accepted 5 December 2017

Available online 8 December 2017

Keywords:

Uncertainty theory

Production-inventory model

Optimal control

Control constraint

critical value

a b s t r a c t

Uncertainty theory is a new methodology to describe subjective indeterminacy. This pa-

per discusses an infinite-horizon production-inventory problem with constraints that the

production rates shall be restricted to an appropriate interval, and the indeterminacy of

the decision system is extracted and analyzed through uncertainty theory. To solve such a

problem, an uncertain linear quadratic control model is investigated with a critical value

criterion. Applying the equation of optimality, it is shown that an optimal feedback control

exists. In addition, the behavior of the value function that is characterized benefits from

numerical illustrations. Furthermore, we compare these optimal solutions with the solu-

tions of an unconstrained problem. Then some economic interpretations are provided to

reveal the significance of the model to practical applications.

© 2017 Elsevier Inc. All rights reserved.

1. Introduction

Analysis of dynamical systems is a classical work in system sciences, mathematics, and cybernetics. In many real-life

decision-making situations, optimizing a utility for a dynamical system is classified as an optimal control problem. However,

the implementations of many control systems are always intractable because, in general, real system dynamics will involve

some degree of data indeterminacy, and possibly some parameters or external disturbances as well. In fact, indeterminacy

is absolute whereas determinacy is relative, thus how to deal with indeterminacy is an essential research subject not only

in mathematics, but also in data science and engineering. In probability theory, normally indeterminate or imprecise data

could be described as random variables and, hence, when the dynamic control system is considered, the state of the system

over time would be a stochastic process, and we will surely be faced with the stochastic optimal control problem.

In 1965, Zadeh [1] initiated the concept of the fuzzy set via a membership function. The proposed fuzzy set theory is, in

contrast, used to depict vagueness especially when there is data ambiguity as a result of subjective or linguistic judgments.

As for fuzzy optimization, fuzziness in train timetable problems [2] has been studied through a fuzzy programming model,

and an effective branch-and-bound algorithm has been proposed to obtain robust solutions. The idea of applying fuzzy sets

to control problems was explicitly presented by Chang and Zadeh [3] for the first time. Consequently, the most impor-

tant research on fuzzy controllers by Mamdani and Assilian [4] and Mamdani [5] marked the start of fuzzy control theory.

∗ Corresponding author.

E-mail addresses: ygzhu@mail.njust.edu.cn , ygzhu@njust.edu.cn (Y. Zhu).

https://doi.org/10.1016/j.apm.2017.12.006

0307-904X/© 2017 Elsevier Inc. All rights reserved.

276 L. Sheng et al. / Applied Mathematical Modelling 56 (2018) 275–288

Recently, some research [6–8] has begun to handle dynamic optimization problems via the expected-value-based fuzzy op-

timal control model.

In fact, the sources of data indeterminacy may be inexhaustible and more complex. While solving practical problems, we

often lack samples or cannot afford numerous experiments to obtain statistical data due to economic reasons or technical

difficulties, leading to failure in building a probability distribution. Instead, some domain experts are invited to evaluate a

belief degree measuring such an indeterminate quantity. A belief degree represents the strength with which we believe the

event will happen. However, people tend to put too much weight on unlikely events. Thus, subjective probability sometimes

fails to model the belief degree unless some observed data are obtained to revise the belief degree. Some theories have

been proposed to deal with the belief degree such as possibility theory [9] and Dempster–Shafer theory [10,11] . To deal

with belief degrees rationally, an uncertainty theory was constructed by Liu [12,13] based on the uncertain measure in

2007, and then became a branch of axiomatic mathematics systems for modeling the indeterminacy. As counterparts of

stochastic processes and Brownian motion, Liu [13] proposed uncertain processes and canonical processes, respectively. The

notion of an uncertain differential equation driven by a canonical process was presented in 2008 [13] . After that, Yao and

Li [14] proposed an uncertain alternating renewal process Gao and Yao [15] studied continuous dependence theorems in

uncertain differential equations. Nowadays, uncertainty theory has become much more mature in a variety of aspects such as

uncertain programming, uncertain statistics, and uncertain propositional logic. In practical applications, Gao et al. [16] were

the first to introduce the uncertainty theory into rail transportation problems.

When a system is affected by uncertain factors, it can be characterized by an uncertain differential equation. To inves-

tigate an uncertain control system that is guided by the uncertain differential equation, Zhu [17] was the first to study

an uncertain optimal control problem in which the expected value operator was used to optimize the uncertain objec-

tive function. The solution methodology was dynamic programming and an equation of optimality as a counterpart of the

Hamilton–Jacobi–Bellman (HJB) equation was provided and subsequently studied by some researchers [18–22] .

From another aspect, with the greater use of methods and results in mathematics and computer science, optimal con-

trol theory has achieved great progress and a variety of practical areas have increased tremendously such as engineering,

medicine, and management science. In particular, production planning problems have emerged as central to controlling the

production management systems in practice. Among them, the quadratic production control models are very important, as

made famous by Holt et al. in 1960 [23] . They considered both production costs and inventory holding costs over time,

and the analytical study was based on a calculus of variations method. Since then, a great number of production-inventory

problems have been investigated by many researchers. A model was proposed and analyzed by applying the maximum prin-

ciple in the work of Thompson and Sethi [24] . In their model, they determined production rates over time to minimize an

integral of a discounted quadratic loss function. The turnpike points were further obtained when the horizon is infinite. In

[25,26] , different types of production planning problems with some constraints have been developed. In addition, Kleindor-

fer et al. [27] solved the discrete version of production planning problems using optimal control theory. Data indeterminacy

is also pervasive in production management issues. For instance, some research into the stochastic version of production-

inventory problems was presented by Sethi and Thompson [28] , Sethi and Zhang [29] , Bensoussan et al. [30] , and van der

Laan and Salomon [31] . An infinite-horizon stochastic production planning problem [30] was considered in which the ad-

missible control must be non-negative. From [29] , it is specified that production planning problems where production is

done by machines that are unreliable or failure-prone could be formulated as stochastic optimal control problems involving

jump Markov processes.

In this article, we consider a production-inventory problem involving uncertain data, with constraints that the production

rates are restricted to an appropriate interval. The infinite-horizon case is primarily treated. As mentioned above, usually the

best suggestion is to maximize or minimize the expected value of the indeterminate objective function. However, owing to

the complexity of the control systems, optimality problems under the customary criterion are not always favorable owing

to the different requirements from people. Parameters of the objective function are not easy to measure so that the object

is difficult to achieve in reality. On the other hand, decision-makers may have different preferences in that some may be

cautious while others may be adventurous. Therefore, in the model formulation, we make use of the critical-value-based

criterion, which was investigated in our previous work, Sheng et al. [32,33] and used successfully in saddle point prob-

lems [34] , switched linear systems [35] , and stability analysis [36] .

It is worth noting that the preceding works on uncertain optimal control are reasonably well behaved with unrestricted

admissible control, except for the bang–bang control problem whose objective function is linear in the state variable. The

problem introduced in this paper could be mathematically abstracted by an uncertain linear quadratic optimal control model

with bounded control. Borrowing ideas from dynamic programming techniques, this kind of production-inventory problem

under uncertainty is solved along with some economic interpretations, including optimal feedback control and the charac-

terization of the value function. Furthermore, some numerical illustrations are offered to demonstrate the efficiency of the

proposed model and the solution procedure.

2. Theoretical background

For reference, we summarize some basic knowledge of uncertainty theory and uncertain optimal control models that will

be frequently used in the following.

L. Sheng et al. / Applied Mathematical Modelling 56 (2018) 275–288 277

2.1. Uncertainty theory

Mathematically, let � be a nonempty set, let L be a σ -algebra over �, and let M be an uncertain measure. Then the

triplet ( �, L , M ) is called an uncertainty space [12] . An uncertain variable is a measurable function from an uncertainty

space ( �, L , M ) to the set of real numbers. The distribution �: R → [0, 1] of an uncertain variable ξ is defined by �(x ) =M{ γ ∈ � | ξ (γ ) ≤ x } for any real number x ; in many cases, it is sufficient to know the uncertainty distribution rather than

the uncertain variable itself.

For an uncertain variable ξ , the expected value of ξ is defined by

E [ ξ ] =

∫ + ∞

0

M { ξ ≥ r } d r −∫ 0

−∞

M { ξ ≤ r } d r

provided that at least one of the two integrals is finite. This expected value represents the size of an uncertain variable.

When dealing with control problems with nondeterministic interference, the expected value is usually used as an effective

measurement to optimize the objective function. In reality, people tend to take into account the possibility of reliability

or risk, favorable or adverse events. For example, for the same investment, there are two return options, 1 million yuan

with reliability 1 or 10 million yuan with reliability 0.1. At this point, investors have a choice: risk-averse investors will

choose low-risk investments, i.e. the first option; risk takers will largely choose the latter option; but for people who do

not care about risk, the options will appear the same. That is, the expectation model is not always applicable in practical

problems. Correspondingly, to handle the uncertain optimal control problems in an alternative manner, critical values have

been designed by Liu based on the uncertain measure. Let β ∈ (0, 1]. Then

ξsup (β) = sup { r|M { ξ ≥ r } ≥ β} , ξinf (β) = inf { r|M { ξ ≤ r } ≥ β} are called the β-optimistic value to ξ and the β-pessimistic value to ξ , respectively. From the definition, we can immediately

infer that ξsup (β) = �−1 (1 − β) and ξinf (β) = �−1 (β) . For instance, assume that ξ is an uncertain income giving a high

confidence level 0.8. Then for each scheme, ξsup (0 . 8) is the same as saying that the uncertain return can achieve up to

ξsup (0 . 8) , i.e. its reliability can reach at least 80%; ξinf (0 . 8) = ξsup (0 . 2) is the same as saying that the uncertain return

can achieve up to ξsup (0 . 2) , but its reliability is only 20%. Thus, when the reliability is high, the optimistic value gains

denote that the decision maker is cautious in the assessment of income to obtain a conservative result; on the other hand,

pessimistic values indicate that the decision maker is willing to take risks in predicting earnings, and may be able to take

greater risks with a high return at the same time.

The uncertain variables ξ1 , . . . , ξn are said to be independent if

M

{

n ⋂

i =1

{ ξ ∈ B i } }

=

n ∧

i =1

M{ ξ ∈ B i }

for any Borel sets B 1 , B 2 , . . . , B n . This means that some uncertain variables are independent if they can be separately defined

on different uncertainty spaces. If ξ and η are independent uncertain variables with finite expected values, then for any real

numbers a and b , we have E [ aξ + bη] = a E [ ξ ] + bE [ η] . Meanwhile, the following properties of critical values hold:

( a ) if a ≥ 0, then ( aξ ) sup (β) = aξsup (β) , and ( aξ ) inf (β) = aξinf (β) ;

( b ) if a < 0, then ( aξ ) sup (β) = aξinf (β) , and ( aξ ) inf (β) = aξsup (β) ;

( c ) ( ξ + η) sup (β) = ξsup (β) + ηsup (β) , ( ξ + η) inf (β) = ξinf (β) + ηinf (β) .

Further, the variation of uncertain phenomena over time is studied by uncertain calculus using the concept of an un-

certain differential equation as a type of differential equation driven by a canonical process. An uncertain process C t is said

to be a canonical Liu process if: (i) C 0 = 0 and almost all sample paths are Lipschitz continuous; (ii) C t has stationary and

independent increments; (iii) every increment C s + t − C s is a normal uncertain variable with expected value 0 and variance

t 2 , whose uncertainty distribution is

�(x ) =

(1 + exp

(−πx √

3 t

))−1

, x ∈ R.

For each γ ∈ �, the function C t ( γ ) is called a sample path of canonical process C t . If f and g are some given functions,

d X t = f (t, X t )d t + g(t, X t )d C t

is called an uncertain differential equation. A solution is a Liu process X t that satisfies the equation identically in t . We

let α be a number with 0 < α < 1. The former uncertain differential equation is said to have an α-path X αt if it solves the

corresponding ordinary differential equation

d X

αt = f (t, X

αt )d t + | g(t, X

αt ) | �−1 (α)d t,

278 L. Sheng et al. / Applied Mathematical Modelling 56 (2018) 275–288

where �−1 (α) is the inverse standard normal uncertainty distribution, i.e. �−1 (α) =

√

3 π ln

α1 −α . Furthermore, we have

M { X t ≤ X

αt , ∀ t } = α

M { X t > X

αt , ∀ t } = 1 − α,

which could be called the Yao–Chen Formula. Then the solution X t has an inverse uncertainty distribution: �−1 t (α) = X αt .

The Yao–Chen formula relates uncertain differential equations and ordinary differential equations, in the same way that the

Feynman–Kac formula relates stochastic differential equations and partial differential equations.

For more detailed properties and interpretations of uncertainty theory, such as critical values or uncertain differential

equations, the interested reader is referred to Liu [12,13,37] .

2.2. Critical value model of uncertain optimal control

Inspired by the Hurwicz decision criterion, which was designed by Hurwicz [38] in 1951, we composited the optimistic

value criterion and the pessimistic value criterion to generate a new model of uncertain optimal control problems in [33] :⎧ ⎨

⎩

J(t, x ) ≡ sup

u ∈ U { ρF sup (α) + (1 − ρ) F inf (β) }

subject to

dX s = b(s, X s , u ) ds + σ(s, X s , u ) dC s and X t = x

(1)

where F =

∫ T t f (s, X s , u ) ds + h (T , X T ) , and J ( t , x ) is called the value function. A selected coefficient ρ ∈ (0, 1) denotes the

optimism degree and we have a predetermined confidence level β ∈ (0, 1). The vector X s and the vector u ∈ U are the state

variable and the control variable, respectively. Assume that C t = (C t1 , C t2 , . . . , C tk ) τ, where C t1 , C t2 , . . . , C tk are independent

canonical processes, and the symbol v τ denotes the transpose of a vector or a matrix v .

To find the best control such that the given objective functional is optimized, we presented the following equation of

optimality.

Theorem 2.1. Suppose J(t, x ) ∈ C 2 ([0 , T ] × R n ) . Then we have

−J t (t, x ) = sup

u ∈ U

{f (t, x , u ) + ∇ x J(t , x ) τ b(t , x , u ) + (2 ρ − 1)

(√

3

πln

1 − β

β

)‖

∇ x J(t , x ) τσ(t , x , u ) ‖ 1

}, (2)

where J t ( t , x ) is the partial derivative of the function J ( t , x ) in t , ∇ x J ( t , x ) is the gradient of J ( t , x ) in x , and ‖ · ‖ 1 is the 1-norm

for vectors, that is, ‖ p‖ 1 =

∑ n i =1 | p i | for p = (p 1 , p 2 , . . . , p n ) .

3. Problem specification

This section is devoted to the model assumptions. We introduce the mathematical description of the production-

inventory system with nondeterminacy items, and we consider the lack of historical data when optimizing production level

and inventories of new products. This case study offers the dual purpose of conducting some verifications of our proposed

uncertain optimal control theory and presenting an important avenue to advance its own extension.

Consider a factory producing homogeneous products that it stores in an inventory warehouse: guided by the previous

work and the practical circumstances, we plug uncertain variables into this dynamic system. Let us define the following

quantities:

• X ( t ), the inventory level at time t (state variable); • P ( t ), the production rate at time t (control variable); • S , the constant demand rate at time t, S > 0; • ˆ X , the factory-optimal inventory level, which can be interpreted as a safety stock; • ˆ P , the factory-optimal production rate, that denotes the most favorable level to run the factory; • h , the inventory holding cost coefficient, h > 0; • c , the production cost coefficient, c > 0; • δ, the constant positive discount rate; • C t , the canonical processes (uncertain variable); • σ , the constant disturbance coefficient.

Note that the control constraint is required, which keeps the production rate P ( t ) between a specified lower bound 0 and

a specified upper bound M . Note further that the inventory level is not always nonnegative, i.e. X ( t ) < 0 cannot denote the

situation such as backlogging of demand. We now present the dynamics of the inventory level described by an uncertain

difference equation

d X (t) = (P (t) − S)d t + σd C t , X (0) = x, (3)

where x denotes the initial inventory level. The expression σd C t is considered as a sale sudden fluctuation of the new

product, such as the sale return, product defect, or inventory spoilage, that can be modeled by some uncertain variables.

L. Sheng et al. / Applied Mathematical Modelling 56 (2018) 275–288 279

Fig. 1. Frequency, probability, and uncertainty [45] .

Next, if the controller has to balance one’s desire to minimize the cost due to the difference between the factory-optimal

levels and the real state values and hold the system under stable control, we adopt a conceptual quadratic objective:

min

0 ≤P(t) ≤M

F x,P(t) = min

0 ≤P(t) ≤M

∫ ∞

0

e −δt [

h

(X (t) − ˆ X

)2 + c (P (t) − ˆ P

)2 ]

d t. (4)

Because the dynamic system operation involves uncertain variables, the crux is how to measure the object converting the

uncertain objective into its crisp equivalent. As many methods are discussed in [39] involving this scheme, we plan to

take advantage of the critical value criterion. The underlying philosophy is that for two uncertain variables ξ and η, we

say that ξ > η if and only if ξsup ( inf ) (β) > ηsup ( inf ) (β) for some predetermined confidence level β . Since there are so many

indeterminate factors in the real world, no decisions can guarantee 100% success rate. Thus, we can make decisions under

some given success rate. Let β ∈ (0, 1), we formulate the objective function

min

0 ≤P(t) ≤M

H

λβ

{F x,P(t)

}= min

0 ≤P(t) ≤M

{λF x,P(t) sup

(β) + (1 − λ) F x,P(t) inf (β)

}, (5)

where λ∈ [0, 1] specifies a selected coefficient that means the optimism degree 1 − λ surely represents a measure of the

controller’s pessimism. With the confidence level β , the uncertain variable F x, P ( t ) will reach a lower bound F x,P(t) inf (β) ;

meanwhile, F x, P ( t ) could reach an upper bound F x,P(t) sup (β) . Then their weighted average need to be minimized in the

presence of an appropriate control policy. Of course, according to the properties of critical values, we have min H

λβ

{F x,P(t)

}=

− max H

1 −λβ

{−F x,P(t)

}.

Here, we emphasize why the uncertainty theory arises in this situation. Several researchers have argued that nonde-

terminacy in production-inventory problems could be modeled by random variables based on probability theory. Recently,

Widyadana and Wee [40] analyzed the effect of machine breakdown in a production-inventory system with stochastic repair

time. Pan and Li [41] employed stochastic optimal control theory to model the deteriorating items in a production-inventory

system with environmental constraints. Other stochastic models could be found, for example, in [42,43] . However, a funda-

mental premise of applying probability theory is that the estimated probability is close enough to the long-run frequency,

no matter whether the probability is interpreted as subjective or objective. Otherwise, the law of large numbers is no longer

valid and probability theory is no longer applicable.

The reasons for a lack of observed data about the unknown state of nature may not only be economic, but also technical.

How can we deal with this case? We have to invite some domain experts to evaluate their belief degree that each event will

occur. Since humans usually give undue weightage to unlikely events [44] , the belief degree may have much larger variance

than the real frequency, therefore we should deal with it by uncertainty theory.

Fig. 1 shows that when the sample size is large enough, the estimated probability (left curve) is close enough to the

cumulative frequency (left histogram) and probability theory is applicable. When the sample size is too small or in the event

there are no samples, the belief degree (right curve) usually has much larger variance than the cumulative frequency (right

histogram: it is actually unknown); in this case, uncertainty theory is the more appropriate approach. For more detailed

interpretations, the interested reader is referred to Liu [45] .

4. Uncertain optimal control approach

The above model is solved via the equation of optimality presented by Sheng and Zhu [33] , which is satisfied by a certain

value function. To simplify the mathematics, we suppose that ˆ X =

ˆ P = 0 and h = c = 1 . The assumption is without loss of

generality as the following analysis can be extended in a parallel manner for the case without this assumption. We can

rewrite F x, P ( t ) in (4) as ∫ ∞

e −δt [X

2 (t) + P 2 (t) ]d t. (6)

0

280 L. Sheng et al. / Applied Mathematical Modelling 56 (2018) 275–288

Define V = V (x ) to represent the current value of the linear quadratic optimal control problem (5), (6) , subject to the

state Eq. (3) . First, inspired by the preceding work and the linearity of the system, we characterize the behavior of V ( x ) as

follows.

Theorem 4.1. The value function V ( x ) is strictly convex.

The proof can be found in the Appendix.

Subsequently, we choose V ( x ) in strictly convex form with quadratic growth. According to the equation of optimality, V ( x )

satisfies

SV x (x ) + δV (x ) + σ (1 − 2 λ)

(√

3

πln

1 − β

β

)| V x (x ) | = x 2 + min

0 ≤P≤M

(P 2 + P V x (x )) . (7)

In [33] , we called this flexible criterion the Hurwicz criterion. In particular, by setting the coefficient λ to 1 and 0, the

criterion reduces to the optimistic value criterion and the pessimistic value criterion, respectively; if λ = 0 . 5 , the above

equation becomes the same as the result in the case of the expected model [17] .

First, we give the unconstrained solutions P ∗, V

∗. By an argument similar to the proof of Theorem 4.1 , V

∗ is strictly convex,

and we have to solve

SV

∗x (x ) + δV

∗(x ) + ρ| V

∗x (x ) | = x 2 + min (P 2 + P V

∗x (x )) ,

where ρ = σ (1 − 2 λ)( √

3 π ln

1 −ββ

) , and let ρ < S . Setting d(P 2 + P V ∗x ) / d P = 0 , we obtain P ∗ = −V ∗x / 2 . It follows from the con-

vexity that there exists a unique x ′ such that V ∗x (x ′ ) = 0 , and V

∗( x ) satisfies

(S + ρ) V

∗x + δV

∗ = x 2 − V

∗2 x

4

, x ≥ x ′ and V

∗x ≥ 0 ; (8a)

(S − ρ) V

∗x + δV

∗ = x 2 − V

∗2 x

4

, x ≤ x ′ and V

∗x ≤ 0 . (8b)

For these two equations, taking derivatives in x on both sides, we obtain (S + ρ +

V

∗x

2

)V

∗xx = 2 x − δV

∗x , x ≥ x ′ ;(

S − ρ +

V

∗x

2

)V

∗xx = 2 x − δV

∗x , x ≤ x ′ .

Since V xx is positive, we infer that x ′ ≥ 0 and V

∗ is quadratic growth for x ≥ x ′ . However, towards the case x ≤ x ′ , the

general solutions of the differential equation cannot be guaranteed to represent quadratic growth. Here, we could suppose

that V ∗(x ) = ax 2 + bx + c, where a, b , and c are real numbers. Substituting the expression into (8b) and collecting the terms,

we obtain

[ a 2 + aδ − 1] x 2 + [2 a (S − ρ) + ab + bδ] x + [ b(S − ρ) + b 2 / 4 + cδ] = 0 ,

which must hold for any value of x ≤ x ′ , so we must have

a 2 + aδ − 1 = 0 ,

2 a (S − ρ) + ab + bδ = 0 ,

b(S − ρ) + b 2 / 4 + cδ = 0 .

We obtain

a =

√

δ2 + 4 − δ

2

> 0 , b = −2 a 2 (S − ρ) , c =

−b 2 − 4 b(S − ρ)

4 δ,

and 2 ax ′ + b = 0 , which yields x ′ = (S − ρ)( √

δ2 + 4 − δ) / 2 . Further, V

∗ is quadratic growth. For x ≥ x ′ , V

∗ satisfies an Abelian

equation that only has an implicit solution. Fortunately, it can be solved numerically to express its manner. For numerical

illustration, we set δ = 0 . 2 , S = 2 , β = 0 . 98 , σ = 0 . 7 , and λ = 0 . 6 to generate Fig. 2 .

Note that, in this section, we do not restrict the production rate to be nonnegative or below some upper bound. In other

words, we permit disposal and unlimited production capacity, for mathematical expedience.

In the next section, we discuss the solutions of the optimal control model (3) –(6) . As defined previously, V ( x ) is the value

function for the uncertain optimal control problem. We restate it as

SV x + δV + ρ| V x | = x 2 + min

0 ≤P≤M

(P 2 + P V x ) .

Because of the strict convexity of V , the optimal feedback control (denoted by P x ) is given by

P x =

{

0 , if V x ≥ 0 ;−V x / 2 , if − 2 M ≤ V x ≤ 0 ;

M, if V x ≤ −2 M,

or P x =

{

0 , if x ≥ x 0 ;−V x / 2 , if x m

≤ x ≤ x 0 ;M, if x ≤ x m

.

(10)

L. Sheng et al. / Applied Mathematical Modelling 56 (2018) 275–288 281

Fig. 2. Behavior of V ∗ .

In the above formulation, x 0 and x m

are separately the unique roots of V x (x 0 ) = 0 and V x (x m

) = −2 M. We emphasize that

the switching points x 0 and x m

and the optimal objective value V are all M -dependent.

(I) We first verify the conclusion under the case that M > (S − ρ) .

For x ≥ x 0 , V satisfies the differential equation (S + ρ) V x + δV = x 2 , and its general quadratic growth solution is

V (x ) =

1

δx 2 − 2(S + ρ)

δ2 x +

2(S + ρ) 2

δ3 + c̄ · e −

δx (S+ ρ) , (11)

since V x (x 0 ) = 0 , c̄ is determined as

c̄ =

[2(S + ρ)

δ2 x 0 − 2(S + ρ) 2

δ3

]e

δx 0 (S+ ρ) .

Similarly, for x ≤ x m

we have (S − ρ − M) V x + δV = x 2 + M

2 . Then

V (x ) =

1

δx 2 − 2(S − ρ − M)

δ2 x +

2(S − ρ − M) 2

δ3 +

M

2

δ+

¯̄c · e −δx

(S−ρ−M) , (12)

and

¯̄c =

[2(S − ρ − M)

δ2 x m

− 2(S − ρ − M) 2

δ3 +

2 M(S − ρ − M)

δ

]e

δx m (S−ρ−M) .

Following from the condition V xx > 0, we can infer that x 0 > 0 and x m

< −δM. When x takes the value between the switching

points x m

and x 0 , the solution of (7) is also considered as follows,

V (x ) =

˜ a x 2 + ̃

b x +

˜ c , (13)

where ˜ a , ˜ b , and ˜ c are real numbers. Then V x = 2 ̃ a x + ̃

b , x 0 = −˜ b / 2 ̃ a , and x m

= (−2 M − ˜ b ) / ̃ a . We have calculated that ˜ a = a,˜ b = b, and ˜ c = c. In addition, it is easily verified that

x 0 = (S − ρ)

√

δ2 + 4 − δ

2

> 0 , x m

=

(S − ρ − M)( √

δ2 + 4 − δ)

2

− δM < −δM.

To summarize the points that we have just made, a strictly convex solution V ( x ) with quadratic growth has been obtained

in a piecewise manner.

To visualize the results, we adopt the parameters that have been specified in the unconstrained case. In addition, we give

M = 3 . Fig. 3 shows different initial state levels and their corresponding optimal objective values.

As displayed in Fig. 3 , the solid line denotes the value function that meets Eq. (7) ; the dashed line V

∗ represents the

unconstrained value function; the dashed lines V and V m

, respectively, indicate the value functions for production-inventory

0

282 L. Sheng et al. / Applied Mathematical Modelling 56 (2018) 275–288

Fig. 3. Behavior of V , with M = 3 .

problem with P ≡ 0 and P ≡ M = 3 . When x takes values between x 0 and x m

, we can note that V ∗(x ) = V (x ) . Further, from

the quadratic growth and the observation of the behavior of V , we may infer that

V → V 0 as x → + ∞; V → V m

as x → −∞ .

Intuitively, the justification of this result is that for large inventory levels, it is naturally optimal to produce nothing, i.e. set

P = 0 . Similarly, if there are large-scale shortages, it is optimal to produce at the highest possible rate, i.e. set P = M.

(II) When M ≤ (S − ρ) is required, we perform the following analysis.

For x ≥ x 0 and x ≤ x m

, the value functions have the same structures as (11) and (12) , respectively. However, since (S − ρ −M) ≥ 0 , if the quadratic growth of V ( x ) is to be guaranteed, we have to set ¯̄c = 0 . Therefore, for x ≤ x m

,

V (x ) =

1

δx 2 − 2(S − ρ − M)

δ2 x +

2(S − ρ − M) 2

δ3 +

M

2

δ, (14)

and x m

= (S − ρ − M) /δ − δM. For x m

≤ x ≤ x 0 , the function V ( x ) satisfies

(S − ρ) V x + δV = x 2 − V

2

4

, V x (x m

) = −2 M,

and x 0 results from V x (x 0 ) = 0 . Yet this differential equation only has an implicit solution. This situation looks similar to

the unconstrained case in that we must resort to numerical execution for the problem. Here, we still take the previous

parameters and set M = 1 , and we obtain the value function presented in Fig. 4 .

Referring to the computation illustrated in Fig. 4 , x m

= 3 . 3 can be calculated directly, and via the numerical execution

we find that x 0 takes a value of about 3.639. Further, the value-function whose dependent variable acts as x m

≤ x ≤ x 0 (the

dash–dotted line) can be bound by the quadratic growth functions.

At the end of this section, we present the results of some numerical experiments to illustrate the advantage of adopting

the critical value criterion, rather than the expected value criterion. In this case, the parameters are initialized as: M = 3 ,

δ = 0 . 2 , S = 2 , β = 0 . 9 , σ = 0 . 02 , λ = 0 (that means the controller chooses the pessimistic criterion), and X(0) = 1 . Since

the uncertain variable is equipped with an uncertainty space (�, L , M ) , in which some sample paths can be realized due

to several γ ∈ �.

As displayed in Fig. 5 , we compare these realization points of uncertain cost (based on the sample path of uncertain

variables) with the pessimistic value of 0.9 and the realization points of uncertain cost and its expected value. It can be

observed that the minimal total abatement cost is larger with the pessimistic criterion than with the expected criterion.

Although the minimal expected cost is optimal, the effective cost realized can be far from this value,whereas the pessimistic

cost may be easily reached. Minimizing the pessimistic cost is cautious to some extent: it actually provides the least bad

cost with a belief degree of 0.90. However, in this problem, the target is not only to reduce the cost, but also to stabilize

the inventory level to satisfy the demand and other suddenly incomplete predicted situations. From this perspective, using

the critical value criterion may be more flexible and realistic.

L. Sheng et al. / Applied Mathematical Modelling 56 (2018) 275–288 283

Fig. 4. Behavior of V , with M = 1 .

Fig. 5. Discounted costs realizations.

Remark 4.1. Since the optimal control gives a piecewise continuous form, it is difficult to substitute the optimal control in

Eq. (3) to obtain the optimal state and the closed-form expressions for the optimal value functions. In this case, we decided

to discretize the dynamical system in time. We consider a given time interval �t , and define a recursive equation

X (i · �t) = X ((i − 1) · �t) + [ P ((i − 1) · �t) − S]�t + σηi , (15)

where i = 1 , 2 , . . . denotes an index, ηi is the realization of the uncertain variable �C t , following from the definition of a

canonical process, �C t is a normal uncertain variable with expected value 0 and variance �t 2 , and the uncertainty distribu-

tion is denoted by �. Thus, ηi can be produced by ηi = �−1 (r i ) for an arbitrarily generated number r i from the interval [0,

1]. In addition, the selected planning period N ·�t can ensure the difference (between the optimal value in N ·�t and the

optimal value in (N − 1) · �t approaching zero.

284 L. Sheng et al. / Applied Mathematical Modelling 56 (2018) 275–288

5. Discussion and extension

5.1. Discussion

Recall the figures that were presented in the previous section. For the value function V , we now rewrite it as V

M . From

the previous figures, for the same given x we may infer the following result.

Theorem 5.1. Let M > N ≥ 0 . Then 0 ≤ V

∗( x ) ≤ V

M ( x ) ≤ V

N ( x ) ≤ V 0 ( x ) .

Proof. In this section, only the case M > (S − ρ) ≥ N ≥ 0 is considered. Other cases can be easily verified in an analogous

way.

(I) 0 ≤ V

∗ ≤ V

M . From the previous section, we have x ′ = x 0 and V ∗(x ) = V M (x ) if x 0 ≤ x ≤ x m

. Let ˜ V M = V M − V ∗, clearly ˜ V M

is quadratic growth. For x ≥ x 0 , there exists

(S + ρ) ̃ V

M

x + δ ˜ V

M = h (x ) ≥ 0 , ˜ V

M

x (x 0 ) = 0 ,

where h (x ) = − min −∞≤P≤+ ∞

(P 2 + P V ∗x ) . The solution of this initial-value problem can be expressed as

˜ V

M = e −δx

(S+ ρ)

[∫ x

x 0

h (z)

(S + ρ) e

δz (S+ ρ) d z

].

Because of h (z) ≥ 0 ⇒

h (z) (S+ ρ)

e δz

(S+ ρ) ≥ 0 , we have that ∫ h (z)

(S+ ρ) e

δz (S+ ρ) d z is an increasing function with respect to z ( x ≥ x 0 ).

Hence, ˜ V M ≥ 0 ⇒ V M ≥ V ∗. Likewise, when x ≤ x m

, V

M ≥ V

∗ could also be derived. Eventually, we should point out that in x ′ (i.e. x 0 ), V

∗ has the minimum value x ′ 2 / δ ≥ 0.

(II) V

M ≤ V

N . In this section, the minimal points for V

M and V

N are denoted by x M

0 and x N

0 , respectively. Optimal feedback

control is separated as P M

and P N . Here x M

0 ≤ x N

0 ( x M

0 > x N

0 can be analyzed in the same manner). Define ˜ V N = V N − V M , there

are three cases: for x ≥ x N 0 , we have

(S + ρ) ̃ V

N x + δ ˜ V

N = 0 , ˜ V

N x (x N 0 ) ≤ 0 ,

the general solution of which is given by ˜ V N = ̃

k e − δx

(S+ ρ) . Following the initial-value condition, ˜ k ≥ 0 . Then

˜ V N ≥ 0 ⇒ V N ≥ V M .

For x M

0 ≤ x ≤ x N

0 , we have

(S − ρ) ̃ V

N x + δ ˜ V

N ≥ −ρ(V

N x + V

M

x ) + S(V

N x − V

M

x ) + δ(V

N − V

M )

= min

0 ≤P≤N (P 2 + P V

N x ) − min

0 ≤P≤M

(P 2 + P V

M

x )

≥ P 2 N + P N V

N x − P 2 N − P N V

M

x

= P N ̃ V

N x .

Therefore, (S − ρ − P N ) ̃ V N x + δ ˜ V N ≥ 0 with (S − ρ − P N ) > 0 since P N ≤ N ≤ (S − ρ) . Returning to the analysis in (I), we obtain

V

N ≥ V

M . Where x ≤ x M

0 , the situation is similar to the case of x M

0 ≤ x ≤ x N

0 .

(III) V

N ≤ V 0 . For this comparison, we summarize that with the methods used in (I) and (II), the result is easy to prove.

We do not repeat the procedure. �

5.2. Extension

Here we have studied an infinite-horizon production planning problem in an uncertain environment. If this problem

needs to be optimized over a fixed period of time T , such a complex issue will not be easy to analyze with the same

method. Since the optimal production rate depends on not only the level of inventory x , but also the distance from the

current time s to the end time T , the value function would be indicated by V ( s, x ). However, an extension to the infinite case

might be natural. Consider

V (s, x ) = min

0 ≤P(t) ≤M

{λF T sup (β) + (1 − λ) F T inf (β)

},

subject to dynamic system (1), where F T =

∫ T s

[X(t) 2 + P (t) 2

]d t and M > (S − ρ) . The value function V ( s, x ) obeys the equa-

tion of optimality

−V s (s, x ) + SV x (s, x ) + ρ| V x (s, x ) | = x 2 + min

0 ≤P≤M

(P 2 + P V x (s, x )) .

Suppose −2 M ≤ V x (s, x ) ≤ 0 , we have optimal control P s,x = −V x (s, x ) / 2 . For a finite T , in the plane ( s, x ) there are two loci

(separately denoted by x m

( s ) and x 0 ( s )), along which the optimal control will switch from the boundary value M to values in

the interior of the interval [0, M ], and then change from the interior, i.e. −V x / 2 , to the other boundary 0. When the optimal

control takes values between 0 and M , the equation of optimality becomes

−V s (s, x ) + (S − ρ) V x (s, x ) = x 2 − V

2 x (s, x )

4

L. Sheng et al. / Applied Mathematical Modelling 56 (2018) 275–288 285

and the expressions of those loci can be determined as

V x (s, x m

) = −2 M and V x (s, x 0 ) = 0 .

Let V (s, x ) = A (s ) x 2 + B (s ) x + D (s ) , where A ( s ), B ( s ) and D ( s ) are deterministic functions of s . Following from this form, we

have V s =

d A (s ) d s

x 2 +

d B (s ) d s

x +

d D (s ) d s

and V x = 2 A (s ) x + B (s ) . Substituting these into the equation of optimality above, we calcu-

late functions A ( s ), B ( s ), and D ( s ). Then

x m

=

2e T + s (S − ρ) + (e 2 T + e 2 s )(M − S + ρ)

(e 2 T − e 2 s ) , x 0 =

(e T − e s )(S − ρ)

(e T + e s ) .

6. Conclusion

In this paper, we have explored a production-inventory problem in an uncertain environment with bounded production

rates. An uncertain optimal control model with Hurwicz criterion has been formulated. Drawing support from the equa-

tion of optimality, we have provided the optimal feedback policy P x and characterized the value function V ( x ), which we

then simulated successfully. The complexity of the real world makes the control systems that we face take numerous dif-

ferent forms. The essential difference between uncertain optimal control and stochastic optimal control is that the former

is concerned with the study of dynamic uncertain phenomena while the latter is about the study of dynamic stochastic

phenomena. Uncertainty is regarded as subjective indeterminacy, and it is estimated by experts when there is no or very

limited historical data on hand. This means that our production-inventory model will be useful when there is a lack of

historical data, such as the problems of production-inventory management for new products. On the contrary, stochastic

planning model may be applied when the managers have enough data. In other words, the two types are, respectively, suit-

able for different cases, and it may be unreasonable to evaluate stochastic model and uncertain model by using the same

criteria because they are proposed for different purposes.

We now discuss the limitations of the proposed approach: (1) this paper discusses an infinite-horizon production-

inventory problem. If this problem needs to be optimized over a fixed period of time, such a complex issue will not be

easy to be analyzed with the same method. Even though we give an extended result of such case, it is necessary to further

develop the complete process of calculation and proof; (2) With respect to the uncertainty distribution of an uncertain pro-

cess, we note that, for a fixed time moment, it must be estimated by domain experts. In our work, uncertainty distributions

are assumed to be known just as probability distributions are assumed to be known in a stochastic production-inventory

model. This presents the practitioner with the challenge to estimate the uncertain difference equation, which describes the

dynamics of the inventory level.

In future research, the authors are intending to continue investigating optimal control for uncertain systems and more

practical application aspects, such as: (1) an uncertain production planning problems with the constraints on both the inven-

tory level and the production rate; (2) Uncertain optimal control for partially observable nonlinear systems; (3) Maximum

principle approach of uncertain optimal control theory.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 61673011 ).

Appendix

Let us give a proof of Theorem 4.1 .

Proof. Firstly, we prove that H

λβ

{F x,P(t)

}is strictly convex in x and P ( · ). Assume x 1 and x 2 , P 1 ( · ) and P 2 ( · ) be arbitrarily and

differently chosen. Let θ ∈ (0, 1). Define

x θ = θx 1 + (1 − θ ) x 2 , and P θ = θP 1 + (1 − θ ) P 2 ,

thus the solutions of (1) with different initial conditions

X 1 (t) = x 1 +

∫ t

0

(P 1 − S)d s +

∫ t

0

σd C 1 s ,

X 2 (t) = x 2 +

∫ t

0

(P 2 − S)d s +

∫ t

0

σd C 2 s ,

X θ (t) = x θ +

∫ t

0

(P θ − S)d s +

∫ t

0

σd C θs ,

286 L. Sheng et al. / Applied Mathematical Modelling 56 (2018) 275–288

where C 1 s , C 2 s and C θs are treated as identically distributed independence canonical processes. It follows from the properties

of α-path, we obtain

X

α1 (t) = x 1 +

∫ t

0

(P 1 − S)d s + σ

∫ t

0

�−1 (α)d s,

X

α2 (t) = x 2 +

∫ t

0

(P 2 − S)d s + σ

∫ t

0

�−1 (α)d s,

X

αθ (t) = x θ +

∫ t

0

(P θ − S)d s + σ

∫ t

0

�−1 (α)d s.

Then, the α-path X αθ(t) can be rewritten as

X

αθ (t) = θX

α1 (t) + (1 − θ ) X

α2 (t) = [ θX 1 (t) + (1 − θ ) X 2 (t) ]

α.

It means that X θ ( t ) and [ θX 1 (t) + (1 − θ ) X 2 (t) ] have the same uncertainty distribution. Further, if we respectively denote

uncertainty distributions of X 2 θ(t) and [ θX 1 (t) + (1 − θ ) X 2 (t) ]

2 by � t and � ′

t , we have �t = � ′ t . The detail calculations can

be found in Zhu [46] . Consider the critical values of uncertain variables introduced in the second section, we obtain [X

2 θ (t)

]inf

(α) =

[[ θX 1 (t) + (1 − θ ) X 2 (t) ]

2 ]

inf (α) .

Since

[ θX 1 (t) + (1 − θ ) X 2 (t) ] 2 = θ2 X

2 1 (t) + (1 − θ )

2 X

2 2 (t) + 2 θ (1 − θ ) X 1 (t) X 2 (t)

<

[θ2 + θ (1 − θ )

]X

2 1 (t) +

[(1 − θ )

2 + θ (1 − θ ) ]X

2 2 (t)

= θX

2 1 (t) + (1 − θ ) X

2 2 (t) ,

similarly, we also obtain P 2 θ

< θP 2 1

+ (1 − θ ) P 2 2

.

Drawing support from the critical value criterion, we have [θX

2 1 (t)

]inf

(α) +

[(1 − θ ) X

2 2 (t)

]inf

(α) >

[[ θX 1 (t) + (1 − θ ) X 2 (t) ]

2 ]

inf (α) =

[X

2 θ (t)

]inf

(α) .

If we use ϕ1, t and ϕ2, t to represent the uncertainty distribution of X 2 1 (t) and X 2

2 (t) , respectively. Then

θϕ

−1 1 ,t (α) + (1 − θ ) ϕ

−1 2 ,t (α) > ϕ

−1 θ,t

(α) .

For {F x θ ,P θ (t)

}inf

(β) =

{

∫ ∞

0

e −δt [X

2 θ (t) + P 2 θ (t)

]d t

}

inf

(β) ,

due to {

∫ ∞

0

e −δt [X

2 θ (t) + P 2 θ (t)

]d t ≤

∫ ∞

0

e −δt [ϕ

−1 θ,t

(β) + P 2 θ (t) ]d t

}

⊃{

X

2 θ (t) ≤ ϕ

−1 θ,t

(β) },

we can obtain

M

{

∫ ∞

0

e −δt [X

2 θ (t) + P 2 θ (t)

]d t ≤

∫ ∞

0

e −δt [ϕ

−1 θ,t

(β) + P 2 θ (t) ]d t

}

≥ M

{X

2 θ (t) ≤ ϕ

−1 θ,t

(β) }

= β.

Similarly,

M

{

∫ ∞

0

e −δt [X

2 θ (t) + P 2 θ (t)

]d t >

∫ ∞

0

e −δt [ϕ

−1 θ,t

(β) + P 2 θ (t) ]d t

}

≥ M

{X

2 θ (t) > ϕ

−1 θ,t

(β) }

= 1 − β.

Following the axiom of self-duality, we have

M

{

∫ ∞

0

e −δt [X

2 θ (t) + P 2 θ (t)

]d t ≤

∫ ∞

0

e −δt [ϕ

−1 θ,t

(β) + P 2 θ (t) ]d t

}

= β.

Therefore, the time integral

∫ ∞

0 e −δt

[X 2 θ (t) + P 2 θ (t)

]d t has an inverse uncertainty distribution

�−1 (β) =

∫ ∞

e −δt [ϕ

−1 θ,t

(β) + P 2 θ (t) ]d t,

0

L. Sheng et al. / Applied Mathematical Modelling 56 (2018) 275–288 287

and then according to the property of the critical value, which has been mentioned in Section 2 , we obtain {F x θ ,P θ (t)

}inf

(β) = �−1 (β) =

{

∫ ∞

0

e −δt [ϕ

−1 θ,t

(β) + P 2 θ (t) ]d t

}

.

Similarly, {F x 1 ,P 1 (t)

}inf

(β) =

{

∫ ∞

0

e −δt [ϕ

−1 1 ,t (β) + P 2 1 (t)

]d t

}

,

{F x 2 ,P 2 (t)

}inf

(β) =

{

∫ ∞

0

e −δt [ϕ

−1 2 ,t (β) + P 2 2 (t)

]d t

}

.

From the above, {F x θ ,P θ (t)

}inf

(β)

=

{

∫ ∞

0

e −δt [ϕ

−1 θ,t

(β) + P 2 θ (t) ]d t

}

<

{

∫ ∞

0

e −δt [θϕ

−1 1 ,t (β) + (1 − θ ) ϕ

−1 2 ,t (β) + P 2 θ (t)

]d t

}

< θ{

∫ ∞

0

e −δt [ϕ

−1 1 ,t (β) + P 2 1 (t)

]d t

}

+ (1 − θ ) {

∫ ∞

0

e −δt [ϕ

−1 2 ,t (β) + P 2 2 (t)

]d t

}

= θ{

F x 1 ,P 1 (t)

}inf

(β) + (1 − θ ) {

F x 2 ,P 2 (t)

}inf

(β) . (A.1)

With the same derivation method, we can also get {F x θ ,P θ (t)

}sup

(β) < θ{

F x 1 ,P 1 (t)

}sup

(β) + (1 − θ ) {

F x 2 ,P 2 (t)

}sup

(β) . (A.2)

Now, synthesize Eq. (A.1) and Eq. (A.2) , we obtain

H

λβ

{F x θ ,P θ (t)

}= λ

{F x θ ,P θ (t)

}sup

(β) + (1 − λ) {

F x θ ,P θ (t)

}inf

(β)

<

{

λθ{

F x 1 ,P 1 (t)

}sup

(β) + (1 − λ) θ{

F x 1 ,P 1 (t)

}inf

(β)) }

+

{

λ(1 − θ ) {

F x 2 ,P 2 (t)

}sup

(β) + (1 − λ)(1 − θ ) {

F x 2 ,P 2 (t)

}inf

(β)) }

= θH

λβ

{F x 1 ,P 1 (t)

}+ (1 − θ ) H

λβ

{F x 2 ,P 2 (t)

}.

Next, let P ∗1 (t) and P ∗

2 (t) denote the optimal controls for initial states x 1 and x 2 , respectively. Therefore

V (x θ ) = V (θx 1 + (1 − θ ) x 2 ) = min

0 ≤P(t) ≤M

H

λβ

{F x θ ,P(t)

}≤ H

λβ

{F x θ ,θP ∗

1 (t)+(1 −θ ) P ∗

2 (t)

}< θH

λβ

{F x 1 ,P ∗1 (t)

}+ (1 − θ ) H

λβ

{F x 2 ,P ∗2 (t)

}= θV (x 1 ) + (1 − θ ) V (x 2 ) .

This proves the strict convexity of V ( x ). �

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