A new model of economic order quantity associated with a generalized conformable differential-difference operator

Abstract

The economic order quantity (EOQ) is the order quantity that minimizes the total holding costs and ordering costs. In this effort, we propose a design for EOQ by employing a conformable differential-difference operator, which accepts to minimize the EOQ.

1 Introduction

The economic order quantity (EOQ) was introduced by Harris [1] and modified by Woolsey [2] and Selen and Wood [3]. Ibrahim and Hadid [4] involved the differential term in its formula. Jaber et al. [5] presented a structure based on a physical scheme called entropy. Many studies have indicated new mathematical modeling of EOQ (see [6–9]).

Ibrahim et al. [10] suggested a model for EOQ by using a fractal idea (local fractional calculus) with cost functions corresponding to time at a returning time. They used the formula of Tsallis fractal entropy. The study presented an optimization relaxation problem for scheming a limited interval covering the optimal series distance. In addition, they introduced developments to calculate the optimal measure and the optimal order phase. The estimated EOQ by using a fractal definition did not recognize the error in the EOQ model. Therefore, one can think about a fractional calculus with a controller.

Anderson and Ulness introduced a new type of conformable calculus (CC) [11] involving a control term. Moreover, Ibrahim and Jahangiri [12] extended CC to the complex plane to study the analytic solution of a class of complex differential equations. Based on CC, we generalize a differential-difference operator, type Dunkl of the first rank (CDD). Consequently, we employ this CDD to generalize the univex function, which we will utilize to minimize the EOQ system.

2 Preparing

In this section, we deal with the definitions of the important concepts of this effort.

2.1 Conformable differential operator

In this place, we introduce the most resent definition of the conformable differential operator (CDO), which can be located in [11].

Definition 1

(Conformable differential operator)

A differential operator ${\mathcal{D}}^{\beta }$, $\beta \in [0,1]$ is conformable if and only if ${\mathcal{D}}^{0}$ is the identity operator and ${\mathcal{D}}^{1}$ is the ordinary differential operator. Particularly, the operator is conformable if and only if a differential function $\chi (t)$ satisfies

$$ {\mathcal{D}}^{0}\chi (t)=\chi (t)\quad \text{and}\quad { \mathcal{D}}^{1}\chi (t)=\frac{d}{dt}\chi (t)=\chi '(t). $$

Moreover, in the theory of control systems, a proportional-differential controller for controlling resultant υ at time t with two tuning criteria has the setting

$$ \upsilon (t)={\nu }_{p}\varSigma (t)+{\nu }_{d}\frac{d}{dt}\varSigma (t), $$

(1)

where ${\nu }_{p}$ is the proportional gain, ${\nu }_{d}$ is the derivative gain, and Σ is the error between the formal variable and the actual variable. Based on (1), Anderson and Ulness [11] presented the common idea of CDO.

Definition 2

(A special class of conformable differential operators)

For two continuous functions $\nu _{0}, \nu _{1}: [0,1]\times \mathbb{R}\rightarrow (0,\infty )$, we attain

$$ {\mathcal{D}}^{\beta }\chi (t) =\nu _{1}(\beta ,t) \chi (t)+\nu _{0} ( \beta ,t) \chi '(t) $$

(2)

such that

$$ \lim_{\beta \rightarrow 0}\nu _{1}(\beta ,t)=1, \qquad \lim _{\beta \rightarrow 1}\nu _{1}(\beta ,t)=0, \quad \nu _{1}( \beta ,t)\neq 0, \forall t, {\beta } \in (0,1), $$

and

$$ \lim_{\beta \rightarrow 0}\nu _{0}(\beta ,t)=0, \qquad \lim _{\beta \rightarrow 1}\nu _{0}(\beta ,t)=1, \quad \nu _{0}( \beta ,t)\neq 0, \forall t, {\beta } \in (0,1). $$

Definition 3

The integral operator corresponding to $\mathcal{D}^{\beta }$ is given by the following equality:

$$ \int \mathcal{D}^{\beta }\chi (t)\, d_{\beta }t=\chi (t)+\wp e_{0}(t,t_{0}), $$

(3)

where $\wp \in \mathbb{R}$, $d_{\beta }t=\frac{dt}{\nu _{0}(t)}$, $\nu \neq 0$, and

$$ e_{0}(t,\kappa )= \exp \biggl(- \int _{\kappa }^{t} \frac{\nu _{1}(\beta ,\varsigma )}{\nu _{0}(\beta ,\varsigma ))}\,d \varsigma \biggr). $$

(4)

In our investigation, we request one of the following formulas of $\nu _{1}$ and $\nu _{0}$:

$$\begin{aligned}& \nu _{1}(\beta ,t)= (1-\beta ) t^{\beta }, \qquad \nu _{0}(\beta ,t)= \beta t^{1-\beta }, \quad t \in (0, \infty ), \end{aligned}$$

(5)

$$\begin{aligned}& \nu _{1}(\beta ,t)= (1-\beta ) \vert t \vert ^{\beta }, \qquad \nu _{0}(\beta ,t)= \beta \vert t \vert ^{1-\beta }, \end{aligned}$$

(6)

$$\begin{aligned}& \nu _{1}(\beta ,t)= \cos \biggl(\frac{\beta \pi }{2}\biggr) t^{\beta }, \qquad \nu _{0}( \beta ,t)=\sin \frac{\beta \pi }{2} t^{1-\beta }, \quad t \in (0, \infty ), \end{aligned}$$

(7)

or

$$ \nu _{1}(\beta ,t)= \cos \biggl(\frac{\beta \pi }{2}\biggr) \vert t \vert ^{\beta }, \quad \nu _{0}(\beta ,t)=\sin \frac{\beta \pi }{2} \vert t \vert ^{1-\beta },\quad t \in \mathbb{R}\backslash \{0\}. $$

(8)

Finally, the conformable inner product between two continuous functions χ and φ is given in the following formula:

$$ \langle\chi ,\varphi \rangle = \int _{a}^{b}\chi (t)\varphi (t) e_{0}(b,t)\,d_{\beta }t. $$

2.2 Differential-difference Dunkl operator

Dunkl operator is an assembly of differential-difference operators taking the formula of first rank (see [13, 14])

$$ \Delta _{\kappa }\chi (t)= \chi ' (t) + \kappa \biggl( \frac{\chi (t)-\chi (-t)}{t} \biggr), \quad \kappa \geq 0. $$

(9)

It extended a set of special functions and integral transforms in numerous variables linked with reflection groups. This type of operators has established many other operators. It is employed in the analysis of quantum body schemes. Lately, this operator is formulated in terms of fractional calculus and fractal (see [15, 16]). In this place, we aim to introduce the conformable Dunkl operator (CDD) by using (2). Therefore, we have the following operator:

$$ \begin{aligned}[b] \Delta _{\kappa }^{\beta } \chi (t)&= {\mathcal{D}}^{\beta } \chi (t)+ \kappa \biggl( \frac{\chi (t)-\chi (-t)}{t} \biggr) \\ &= \nu _{1}(\beta ,t) \chi (t)+\nu _{0} (\beta ,t) \chi '(t) + \kappa \biggl( \frac{\chi (t)-\chi (-t)}{t} \biggr), \quad \kappa \geq 0. \end{aligned} $$

(10)

It is clear that when $\beta \rightarrow 1$, Eqs. (10) and (2) coincide. Moreover, the CDD has the following properties.

Proposition 2.1

Let the CDD be given as in (10), where$\beta \in [0,1]$. Then, for constantsa, b, cand differential functionsχandφ, we have:

• (i) $\Delta _{\kappa }^{\beta }[a\chi +b\varphi ](t)= a\Delta _{\kappa }^{\beta }[ \chi (t)]+b\Delta _{\kappa }^{\beta }[\varphi (t)]$;

• (ii) $\Delta _{\kappa }^{\beta }c= c \nu _{1}(\beta ,\cdot)$;

• (iii) $\Delta _{\kappa }^{\beta }[\chi \varphi ](t)= \chi (t)\Delta _{\kappa }^{\beta }\varphi (t)+ \varphi (t)\Delta _{\kappa }^{\beta }\chi (t) - \chi (t) \varphi (t) \nu _{1}(\beta ,t)+\kappa ( \frac{\varPhi (t)-\varPhi (-t)}{t} )$, where

  $$ \varPhi (t):= \chi \biggl(\frac{\varphi }{2}-1\biggr)+\varphi \biggl( \frac{\chi }{2}-1\biggr); $$

• (iv) $\Delta _{\kappa }^{\beta }[\frac{\chi }{\varphi }](t)= \frac{\varphi (t) \Delta _{\kappa }^{\beta }\chi (t)-\chi (t) \Delta _{\kappa }^{\beta }\varphi (t)}{\varphi ^{2}(t)}+ \frac{\chi (t)}{\varphi (t)}\nu _{1}(\beta ,t)+ K_{\kappa }(t)$, $\varphi \neq 0$, where

  $$ K_{\kappa }(t):= \kappa \biggl( \frac{[\frac{\chi }{\varphi }](t)-[\frac{\chi }{\varphi }](-t)}{t} \biggr)- \frac{\kappa }{\varphi ^{2}(t)} \biggl( \frac{\chi (t)-\chi (-t)}{t} \biggr)-\frac{\kappa }{\varphi ^{2}(t)} \biggl( \frac{\varphi (t)-\varphi (-t)}{t} \biggr). $$

Proof

By using the definition of $\Delta _{\kappa }^{\beta }$, we have

$$ \begin{aligned} &\Delta _{\kappa }^{\beta }[a\chi +b\varphi ](t)\\ &\quad = { \mathcal{D}}^{\beta }[a\chi +b\varphi ](t)+ \kappa \biggl( \frac{[a\chi +b\varphi ](t)-[a\chi +b\varphi ](-t)}{t} \biggr) \\ &\quad = \nu _{1}(\beta ,t) [a\chi +b\varphi ](t)+\nu _{0} (\beta ,t) [a \chi +b\varphi ]'(t) + \kappa \biggl( \frac{[a\chi +b\varphi ](t)-[a\chi +b\varphi ](-t)}{t} \biggr) \\ &\quad = a \biggl[ \nu _{1}(\beta ,t) \chi (t)+\nu _{0} (\beta ,t) \chi '(t) + \kappa \biggl( \frac{\chi (t)-\chi (-t)}{t} \biggr)\biggr] \\ &\qquad{}+b\biggl[ \nu _{1}(\beta ,t)\varphi (t)+\nu _{0} ( \beta ,t) \varphi '(t) + \kappa \biggl( \frac{\varphi (t)-\varphi (-t)}{t} \biggr) \biggr] \\ &\quad = a\Delta _{\kappa }^{\beta }\chi (t)+b \Delta _{\kappa }^{\beta } \varphi (t). \end{aligned} $$

This completes the first part. For the second part, a calculation implies that

$$ \begin{aligned} \Delta _{\kappa }^{\beta }c &= { \mathcal{D}}^{\beta } c= \nu _{1}(\beta ,t) c. \end{aligned} $$

Now, we proceed to proving the multiplication rule under $\Delta _{\kappa }^{\beta }$. We have

$$ \begin{aligned} &\Delta _{\kappa }^{\beta }[\chi \varphi ](t) \\ &\quad = \nu _{1}(\beta ,t) [\chi \varphi ](t)+\nu _{0} (\beta ,t) \bigl[\chi \varphi '+\varphi \chi '\bigr](t) + \kappa \biggl( \frac{[\chi \varphi ](t)-[\chi \varphi ](-t)}{t} \biggr) \\ &\quad =\chi (t) \biggl( \nu _{1}(\beta ,t)\varphi (t)+\nu _{0} ( \beta ,t) \varphi '(t) + \kappa \biggl( \frac{\varphi (t)-\varphi (-t)}{t} \biggr) \biggr) \\ &\qquad{}+\varphi (t) \biggl( \nu _{1}(\beta ,t)\chi (t)+\nu _{0} (\beta ,t) \chi '(t) + \kappa \biggl( \frac{\chi (t)-\chi (-t)}{t} \biggr) \biggr)- \nu _{1}(\beta ,t)\chi (t)\varphi (t) \\ &\qquad{}+\kappa \biggl( \frac{[\chi (\frac{\varphi }{2}-1)+\varphi (\frac{\chi }{2}-1)](t)-[\chi (\frac{\varphi }{2}-1)+\varphi (\frac{\chi }{2}-1)](-t)}{t} \biggr) \\ &\quad := \chi (t) \Delta _{\kappa }^{\beta }\varphi (t)+\varphi (t)\Delta _{\kappa }^{\beta }\chi (t)-\nu _{1}(\beta ,t)\chi (t)\varphi (t)+\kappa \biggl( \frac{\varPhi (t)-\varPhi (-t)}{t} \biggr). \end{aligned} $$

Finally, we impose

$$\begin{aligned} &\Delta _{\kappa }^{\beta }\biggl[ \frac{\chi }{\varphi }\biggr](t) \\ &\quad = \nu _{1}(\beta ,t) \biggl[\frac{\chi }{\varphi }\biggr](t)+\nu _{0} (\beta ,t) \biggl[ \frac{\varphi \chi '-\chi \varphi '}{\varphi ^{2}}\biggr](t) + \kappa \biggl( \frac{[\frac{\chi }{\varphi }](t)-[\frac{\chi }{\varphi }](-t)}{t} \biggr) \\ &\quad = \frac{ \nu _{1}(\beta ,t) [\varphi \chi ](t)+\nu _{0} (\beta ,t) [\varphi \chi '-\chi \varphi '](t)}{\varphi ^{2}(t)}+ \kappa \biggl( \frac{[\frac{\chi }{\varphi }](t)-[\frac{\chi }{\varphi }](-t)}{t} \biggr) \\ &\quad = \frac{ \varphi (t) [ \nu _{1}(\beta ,t) \chi (t)+\nu _{0} (\beta ,t) \chi '(t)+ \kappa ( \frac{\chi (t)-\chi (-t)}{t} )] }{\varphi ^{2}(t)} \\ &\qquad{}- \frac{ \chi (t) [ \nu _{1}(\beta ,t) \varphi (t)+\nu _{0} (\beta ,t) \varphi '(t)+ \kappa ( \frac{\varphi (t)-\varphi (-t)}{t} )]}{\varphi ^{2}(t)} \\ &\qquad{}+ \kappa \biggl( \frac{[\frac{\chi }{\varphi }](t)-[\frac{\chi }{\varphi }](-t)}{t} \biggr)-\frac{\kappa }{\varphi ^{2}(t)} \biggl( \frac{\chi (t)-\chi (-t)}{t} \biggr)-\frac{\kappa }{\varphi ^{2}(t)} \biggl( \frac{\varphi (t)-\varphi (-t)}{t} \biggr) \\ &\qquad{}+ \nu _{1}(\beta ,t) \biggl(\frac{\chi }{\varphi }\biggr) (t) \\ &\quad := \frac{\varphi (t) \Delta _{\kappa }^{\beta }\chi (t)-\chi (t) \Delta _{\kappa }^{\beta }\varphi (t)}{\varphi ^{2}(t)}+ \frac{\chi (t)}{\varphi (t)}\nu _{1}(\beta ,t)+K_{\kappa }(t). \end{aligned}$$

 □

Proposition 2.2

The integration by parts of$\Delta _{\kappa }^{\beta }$satisfies the following integral:

$$ \begin{aligned}[b] \int _{a}^{b}\varphi (\varsigma ) \Delta _{\kappa }^{\beta }\chi (\varsigma )e_{0}(b,\varsigma )\,d_{\beta }\varsigma &=\varphi ( \varsigma ) \chi (\varsigma )e_{0}(b,\varsigma )|^{b}_{a}- \int _{a}^{b} \chi (\varsigma ) \bigl[{ \mathcal{D}}^{\beta } \varphi (\varsigma ) \\ &\quad{}-\nu _{1}(\beta ,\varsigma ) \varphi (\varsigma )\bigr] e_{0}(b, \varsigma )\,d_{\beta }\varsigma +\kappa \langle\varphi , \varLambda \rangle, \end{aligned} $$

(11)

where$\varLambda (\varsigma )= \frac{\chi (\varsigma )-\chi (-\varsigma )}{\varsigma }$.

Proof

By the definition of $\Delta _{\kappa }^{\beta }$, we have

$$ \begin{aligned} &\int _{a}^{b}\varphi (\varsigma ) \Delta _{\kappa }^{\beta }\chi (\varsigma )e_{0}(b,\varsigma )\,d_{\beta }\varsigma\\ &\quad = \int _{a}^{b} \varphi (\varsigma ) \biggl[{ \mathcal{D}}^{\beta }\chi (\varsigma )+ \kappa \biggl( \frac{\chi (\varsigma )-\chi (-\varsigma )}{\varsigma } \biggr)\biggr]e_{0}(b,\varsigma )\,d_{\beta }\varsigma \\ &\quad = \int _{a}^{b}\varphi (\varsigma ) \bigl[{ \mathcal{D}}^{\beta }\chi ( \varsigma )\bigr]e_{0}(b,\varsigma )\,d_{\beta }\varsigma + \int _{a}^{b} \varphi (\varsigma ) \kappa \biggl( \frac{\chi (\varsigma )-\chi (-\varsigma )}{\varsigma } \biggr)e_{0}(b, \varsigma )\,d_{\beta }\varsigma \\ &\quad := \int _{a}^{b}\varphi (\varsigma ) \bigl[{ \mathcal{D}}^{\beta }\chi ( \varsigma )\bigr]e_{0}(b,\varsigma )\,d_{\beta }\varsigma +\kappa \int _{a}^{b} \varphi (\varsigma ) \varLambda ( \varsigma )e_{0}(b,\varsigma )\,d_{\beta }\varsigma . \end{aligned} $$

By using Lemma 1.9 in [11] and the definition of the conformable inner product between two continuous functions, we have the desired conclusion. □

In this place, we note that the CDD is not commute operator, because the conformable operator ${\mathcal{D}}^{\beta }$ is not commute operator [11] for a real case, while it is commute operator in a complex domain [12] for some classes of conformal function (has no zero derivative). Moreover, CDD is not invariant because the conformable operator ${\mathcal{D}}^{\beta }$ is not invariant for a real case, while for the set of conformal functions in the open unit disk, the operator ${\mathcal{D}}^{\beta }$ is invariant [12]. In the next subsection, we shall use the CDD to generalize the definition of the univex functions. These functions are useful in optimization analysis to minimize the EOQ.

2.3 Generalized univex function

A class of univex functions is a generalized class of convex functions [17]. This type of functions is usually used to solve multi-objective problems, especially in control theory. Multi-objective difficulties are reflected in control of space constrictions, aeronautical control scheme, and industrial development control, control of production, impulsive control problems and inventory, mechanics, economics, and mechanical engineering difficulties. Here, we aim to employ the CDD to introduce the conformable univex function (CUF).

Definition 4

For a real-valued function, χ is a univex function if and only if there are real-valued functions Ψ, f, and g satisfying the differential inequality

$$ g(t,\tau ) \varPsi \bigl[\chi (t)-\chi (\tau )\bigr]\geq f(t,\tau ) \chi '(t), \quad g\neq 0. $$

(12)

Now, by using the conformable operator $\mathcal{D}^{\beta }$ defined in (3), we have the following conformable univex function (CUF).

Definition 5

For a real-valued function, χ is called a β-conformable univex function (CUF of order $\beta \in [0,1]$) if and only if there are real-valued functions Ψ, f, and g satisfying the differential inequality

$$ g(t,\tau ) \varPsi \bigl[\chi (t)-\chi (\tau )\bigr]\geq f(t,\tau ) \mathcal{D}^{ \beta }\chi (t), \quad g\neq 0. $$

(13)

Consequently, by utilizing the CDD ($\Delta _{\kappa }^{\beta }$), we have the generalized conformable univex function as follows.

Definition 6

For a real-valued function, χ is called a $(\beta ,\kappa )$-conformable univex function of order $\beta \in [0,1]$ and index $\kappa \geq 0$ if and only if there are real-valued functions Ψ, f, and g satisfying the differential inequality

$$ g(t,\tau ) \varPsi \bigl[\chi (t)-\chi (\tau )\bigr]\geq f(t,\tau ) \Delta _{\kappa }^{\beta }\chi (t), \quad g\neq 0. $$

(14)

2.4 Economic order quantity

The pattern of EOQ for the first time was produced by Ford W. Harris [1] as follows:

$$ Q= \sqrt{\frac{2 A B}{ C}}, $$

(15)

where A indicates the annual demand quantity, B represents the fixed cost for each unit, and C takes the annual holding cost for each unit. In [4], the authors studied the marginal ordinary continuous case taking the form

$$ \begin{aligned} Q'(t) = \varTheta \bigl( Q(a t), Q(b t), Q(c t) \bigr), \end{aligned} $$

(16)

where a, b, and c are nonnegative EOQ model constants indicating to the regular variations in A, B, and C, while the fractal case of EOQ was studied in [10]. Here, we consider the CDD in Eq. (10) to generalize (16) as follows:

$$ \begin{aligned}[b] \Delta _{\kappa }^{\beta }Q(t)& =\nu _{1}(\beta ,t) Q(t)+ \nu _{0} (\beta ,t) Q'(t)+ \kappa \biggl( \frac{Q(t)-Q(-t)}{t} \biggr) \\ &= \nu _{1}(\beta ,t) Q(t)+\nu _{0} (\beta ,t) \varTheta \bigl( Q(a t), Q(b t), Q(c t) \bigr)+ \kappa \biggl( \frac{Q(t)-Q(-t)}{t} \biggr). \end{aligned} $$

(17)

Because of a state in which the consumer uses an EOQ system to control the account of a failing item, when the shortage is not allowed (see [18]), we consider the following modification:

$$ \begin{aligned}[b] \Delta _{\kappa }^{\beta }Q(t)& = \nu _{1}(\beta ,t) Q(t)+ \nu _{0} (\beta ,t) \varTheta \bigl( Q(a t), Q(b t), Q(c t) \bigr) \\ &\quad{}+ \kappa \biggl( \frac{Q(t)-Q(\tau )}{t-\tau } \biggr), \quad t>\tau . \end{aligned} $$

(18)

We need to estimate the optimal solution of units according to (17). That is, we minimize the total cost connected with the purchase, distribution, and storing of the manufactured goods. In the existence of a planned customer, who reacts optimally to discount program, the strategy of optimal quantity discount structure by the provider is complex and has to be done sensibly. This is principally happening when the request of the customer is itself unclear. An exciting result called the “reverse bullwhip” takes place where a rise in consumer request uncertainty really decreases order quantity indecision at the provider. There are different methods and theorems to minimize the total cost functions (minimize EOQ) such as the planning horizon theorem [19], stochastic processing [20], and fixed point theorem [4].

From the above information, one can reduce the EOQ model in (18) into the following optimization problem:

$$ \begin{aligned} & \text{Minimize} \quad \Delta _{\kappa }^{\beta }Q(t), \quad t\in J=[0,T] \\ & \text{subject to} \quad Q_{0}=q \leq 0, \end{aligned} $$

(19)

where q is the initial value of EOQ and Q is continuous for all $t \in J$. For this purpose, we need the following definitions.

Definition 7

For any solution $Q(t)$ for (19), a point $\eta \in \mathfrak{J}:=\{ \eta \in J: Q(\eta ) \rightarrow 0 \}$ is called an efficient point of (19) if there occurs no other point satisfying the inequality $Q(t) \leq Q(\eta )$. Furthermore, it is a weak efficient point if $Q(t)< Q(\eta )$, $\forall t \in J$.

Definition 8

A couple $(\beta ,\kappa )-(\chi ,\varUpsilon )$ is called conformable univex of order β if and only if there are real-valued functions $f_{1}$, $f_{2}$, $g_{1}$, and $g_{2}$ such that

$$ g_{1}\bigl[\chi (t)-\chi (\tau )\bigr]\geq f_{1}(t,\tau ) \Delta _{\kappa }^{\beta }\chi (t), \quad g_{1}\neq 0, $$

and

$$ -g_{2}\bigl[\varUpsilon (t)-\varUpsilon (\tau )\bigr]\geq f_{2}(t,\tau ) \Delta _{\kappa }^{\beta }\varUpsilon (t), \quad g_{2}\neq 0. $$

Remark 1

The EOQ model accepts stable demand of a business, product, and immediate availability of items to be re-stocked. It does not account for regular or economic fluctuations. It accepts fixed costs of inventory units, ordering charges, and holding charges. Therefore, this term refers to the contraction in the economy. Consequently, the suggested model is the first model describing this issue by using Dunkl operator.

3 Results

In this section, we present some sufficient optimal hypotheses for a point to be an efficient solution of (19) under the conformable $(\beta ,\kappa )$-univexity.

Theorem 3.1

Letηbe an initial point of the solution of the multi-objective problem (19) and$\epsilon _{1}$and$\epsilon _{2}$be two nonnegative constants achieving the following assumptions:

• (A) $Q(\eta )=0$ (the initial value of EOQ);

• (B) $\epsilon _{1} f_{1}(t,\tau ) \Delta _{\kappa }^{\beta }Q(t)+ \epsilon _{2}f_{2}(t, \tau ) \Delta _{\kappa }^{\beta }q(t)> \epsilon _{1} \|g_{1}\|+\epsilon _{2} \|g_{2}\|$;

• (C) A couple$(\beta ,\kappa )-(\chi ,\varUpsilon )$is a conformable univex of orderβ;

Thenηis an efficient solution to minimize (19).

Proof

Assume that η is not an efficient solution of (19). Therefore, there occurs $x \in \mathfrak{J}$ such that $Q(x) \leq Q(\eta )$. By utilizing assumption (C), we obtain

$$ \epsilon _{1} \bigl(f_{1}(x,\eta ) \Delta _{\kappa }^{\beta } Q(x) \bigr) < \epsilon _{1} \Vert g_{1} \Vert $$

(20)

and

$$ \epsilon _{2} \bigl(f_{2}(x,\xi ) \Delta _{\kappa }^{\beta } q(x) \bigr) \leq \epsilon _{2} \Vert g_{2} \Vert . $$

(21)

By taking the summation of (20) and (21), we have

$$ \begin{aligned} &\epsilon _{1} \bigl(f_{1}(x,\eta ) \Delta _{\kappa }^{\beta } Q(x) \bigr)+\epsilon _{2} \bigl(f_{2}(x,\eta ) \Delta _{\kappa }^{\beta } q(x) \bigr) \leq \epsilon _{1} \Vert g_{1} \Vert +\epsilon _{2} \Vert g_{2} \Vert , \end{aligned} $$

which contradicts assumption (B). Hence, η is an efficient solution which minimizes problem (19). This completes the proof. □

Theorem 3.2

Suppose that the following hypotheses are achieved:

• (A) ηis a weakly efficient point of the solution$Q(t)$of the optimal problem (19);

• (B) A couple$(\beta ,\kappa )-(Q,q)$is conformable univex of order$\beta \in [0,1]$in the direction of$\eta \in \mathfrak{J}$. Moreover, for some$\bar{x}\in \mathfrak{J}$with$q(\bar{x})<0$,

there occur two constants$\epsilon _{1}\geq 0$and$\epsilon _{2}\geq 0$satisfying

$$ \epsilon _{1} \bigl(f_{1}(x,\eta )\Delta _{\kappa }^{\beta } Q(x) \bigr)+ \epsilon _{2} \bigl(f_{2}(x,\eta ) \Delta _{\kappa }^{\beta } q(x) \bigr) \geq 0. $$

Proof

We aim to show that the system

$$ f_{1}(x,\eta ) \Delta _{\kappa }^{\beta } Q(x) < 0, \qquad f_{2}(x,\eta ) \Delta _{\kappa }^{\beta } q(x) < 0 $$

has no solution for all $x \in \mathfrak{J}$. If the system has at least one solution $y \in \mathfrak{J}$, then in view of Definition 8, we get

$$ Q(\eta +\rho _{1} y)< Q(\eta )\quad \text{and} \quad q(\eta +\rho _{2}y)< q( \eta ) $$

for sufficient small arbitrary constants $\rho _{1}, \rho _{2}>0$. By letting $\bar{x}:=\eta +\rho _{2}y$, which leads to $\bar{x}\in \mathfrak{J} \cap N_{\rho _{2}}(\eta )$, in view of assumption (B), one can get

$$ q(\eta +\rho _{2}y)=q(\bar{x})< 0, $$

which is a contradiction with assumption (A), where η is a weak solution ($q(\eta )\geq 0$). Hence, the above inequalities are nonnegative. Consequently, there exist two nonnegative constants $\epsilon _{1}$ and $\epsilon _{2}$ achieving the conclusion

$$ \epsilon _{1} \bigl(f_{1}(x,\eta ) \Delta _{\kappa }^{\beta } Q(x) \bigr)+ \epsilon _{2} \bigl(f_{2}(x,\eta )\Delta _{\kappa }^{\beta } q(x) \bigr) \geq 0, $$

with the property $\epsilon _{2} q(\eta )\geq 0$.

This completes the proof. □

Remark 2

• In Theorem 3.2, x̄ is called a feasible point, which appears in the solution space. This space represents the initial set of candidate solutions to the optimal problem;

• The optimal problem (19) is a generalization of recent works such as [21–23];

• Minimizing problem (19) is equivalent to minimizing the problem

  $$ \begin{aligned} & \text{Minimize} \quad \mathcal{D}^{\beta } Q(t), \quad t\in J=[0,T] \\ & \text{subject to} \quad Q_{0}=q\leq 0; \end{aligned} $$

  (22)

• Theorems 3.1 and 3.2 can be generalized to $\mathbb{R}^{n}$.

3.1 Numerical examples

Consider the following data: $\varPsi (\upsilon )=\upsilon$, $f(t,\tau )=t-\tau$, $g(t,\tau )=1$, $\beta = \kappa =0.5$ and $\nu _{1}(\beta ,t)= (1-\beta ) t^{\beta }$, $\nu _{0}(\beta ,t)= \beta t^{1-\beta }$, $t \in [0,5]$. To minimize problem (19), we have the following calculation:

$$ \Delta _{\kappa }^{\beta }Q(t)= \Delta _{\kappa }^{\beta } \sqrt{t^{3}} \approx t^{4}+0.7t^{2}+0.7\sqrt{t}, \quad t \in [0,5], $$

where $A=B=C=1$. It is clear that $Q(0)=0$, and for arbitrary constants $\epsilon _{1}>0$ and $\epsilon _{2} >0$, we get

$$ f(t,\tau ) \Delta _{\kappa }^{\beta }Q(t) = 5 *t^{4}+0.7*t^{2}+0.7*t^{0.5}>1. $$

Thus, in view of Theorem 3.1, the minimization can be recognized when $t\rightarrow 0$ (see Fig. 1).

Figure 1

Minimization of 1D and 2D- problem (19) for $f(t,\tau )=t-\tau$. It is clear that minimization appears when $t\rightarrow 0$

Suppose that $f(t,\tau )= (e^{t-\tau }-1)$, $t-\tau \geq 1$. Then we obtain

$$ f(t,\tau ) \Delta _{\kappa }^{\beta }Q(t) = 1.5 *t^{4}+0.7*t^{2}+0.7*t^{0.5}>1. $$

Hence, in view of Theorem 3.1, the minimization can be seen in Fig. 2. Similarly, one can optimize problem (19) by using Theorem 3.2.

Figure 2

Minimization of 1D and 2D-problem (19) for $f(t,\tau )=(e^{t-\tau }-1)$ respectively. It is clear that minimization appears when $t\in (0,1]$