Here, we consider a fuzzy demand Δtρ of the manufacture ρ at the month t as symmetric triangle, fuzzy-valued by the following: (see Fig. 1)

(1) Δtρ={(d,η(d)):η(d)=(d,(1−δd)d,(1+δd)d)tρ}

where d indicates the shrinking middle point (a halfway point), Ldtρ:=(1−δd)d represents to the left side of d, and Rdtρ:=(1+δd)d indicates the right side of d.

δd denotes the spread function formulating as a variance function in terms of selling price (the impulse response point):

δd(∁tρ)=a1+a2∁tρ

(for some constants a1 and a2 in R ). Moreover, the symbol η is equivalent to ηλ(d)≡(d,Ldtρ,Rdtρ) representing to the membership grade of the element d in the set Δtρ and is known as the membership function. By the definition of the spread function, we may describe any value in the above space as follows:

η(α)={0,α≤Ldtρα−Ldtρd−Ldtρ,Ldtρ<α≤dRdtρ−αRdtρ−d,d<α≤Rdtρ

We proceed to define another item in the parametric space, which is the fuzzy buying cost, as follows:

(2) B∁tȷ={(♭,ηλ(♭)):η(♭)=(♭,(1−δ♭)♭,1+δ♭)♭)},

where ♭ is the middle price of the type oil ȷ , L♭tȷ:=(1−δ♭)♭ is the left part of ♭ and R♭tȷ:=(1+δ♭)♭ is the right part to ♭ of the type oil ȷ at month t. Now, we introduce the uncertain production price as follows:

(3) P∁t,φρ={(℘,η(℘)):η(℘)=(℘,(1−δ℘)℘,(1+δ℘)℘)},

where ℘ is the rough middle value, L℘t,φρ:=(1−δ℘)℘ is the left part of ℘ and R℘t,φρ:=(1+δ℘)℘ is the right part to ℘ of the type oil ρ at month t from refinery φ. Similarly, we define the rest of parametric spaces, fuzzy inventory price, fuzzy transportation and fuzzy management value, respectively as follows:

(4) I∁t,φρ={(ι,ηλ(ι)):η(ι)=(ι,(1−δι)ι,(1+δι)ι)},

(5) ⊤∁t,φρ={(τ,ηλ(τ)):η(τ)=(τ,(1−δτ)τ,(1+δτ)τ)},

(6) M∁t,φρ={(μ,ηλ(μ)):η(μ)=(μ,(1−δμ)μ,(1+δμ)μ)}.

Yang presented the local fractional derivative (fractal) of the function g(x) of order (0 < λ < 1) at the fixed value x0 as [10]:

(7) υλg(x)=limx→x0Dλ(g(x)−g(x0))(x−x0)λ,

where Dλ(g(x)−g(x0))≈Γ(λ+1)D(g(x)−g(x0)) and the forward difference D formulated as D(g(x)−g(x0))=g(x)−g(x0) , and Γ represents the gamma function Γ(n+1)=n!.

Now, by using the concept of a fractal, we introduce a generalization of the spread function δd(∁tρ)=a1+a2∁tρ as in the following fractal difference equation

(8) δdλ(∁tρ):=Dλ(δd(∁tρ)−δd(∁0ρ))≈Γ(λ+1)D(δd(∁tρ)−a1)=Γ(λ+1)(δd(∁tρ)−a1)=a2Γ(λ+1)∁tρ,:=λΓ(λ+1)∁tρ,

where a2=λ,δd(∁tρ)>0,δd(∁0ρ)=a1 and the fractal λ∈(0,1). Note that if λ∈(0,1), then we have Γ(λ+1)∈(0,1). In this place, we note that λ plays a critical role in improving the classical system when λ≈1 (see [11]). In our investigation, we suppose that λ=0.5 to get a good result for maximization. We shall use the fractal difference operator δdλ in all spaces given in Eqs. (1)–(6). For example, a fuzzy fractal demand Δtρ of the manufacture ρ at the month t becomes

(9) Δtρ={(d,ηλ(d)):ηλ(d)=(d,(1−δdλ)d,(1+δdλ)d)tρ}

where d indicates the shriveled middle point (i.e., halfway point), Ldtρ:=(1−δdλ)d represents to the left side of d and Rdtρ:=(1+δdλ)d indicates the right side of d . δdλ is given in Eq. (8). Similarly, for the other parametric spaces B∁tȷ,P∁t,φρ,I∁t,φρ,⊤∁t,φρ and M∁t,φρ .

The aim is to maximize the Πt which is formulated by

(10) Maximize∑t‍(Πt−B_∁t−P_∁t−I_∁t−T_∁t−M_∁t),

Πt=(∑φ‍∑ρ‍Θφ,tρ.∁tρ),

B_∁t=∑ȷ‍(B∁tȷ.χtȷ)=∑ȷ‍((♭,(1−δ♭λ)♭,(1+δ♭λ)♭)tȷ.χtȷ)

P_∁t=∑φ‍∑ρ‍(P∁φ,tρ.Θφ,tρ)=∑φ‍∑ρ‍((℘,(1−δ℘λ)℘,(1+δ℘λ)℘)φ,tρ.Θφ,tρ)

I_∁t=∑φ‍∑ρ‍(I∁φ,tρ.Iφ,tρ)=∑φ‍∑ρ‍((ι,(1−διλ)ι,(1+διλ)ι)φ,tρ.Iφ,tρ)

T_∁t=∑φ‍∑ρ‍(T∁φ,tρ.Tφ,tρ)=∑φ‍∑ρ‍((τ,(1−δτλ)τ,(1+δτλ)τ)φ,tρ.Tφ,tρ),

M_∁t=∑φ‍∑ρ‍(M∁φ,tρ.Θφ,tρ)=∑φ‍∑ρ‍((μ,(1−δμλ)μ,(1+δμλ)μ)φ,tρ.Θφ,tρ),

where χtȷ,Θφ,tρ,Iφ,tρ,Tφ,tρ are variables.

We have the following list of constraints inequalities:

• Invention of manufacturing in a month plus the previous manufacturing inventory must be larger than or equal to the fluctuating demand (variation demand) for the manufacturing. Therefore, we suggest the average by using λ as follows:

(11) (1Φ∑φ?(Θφ,tρ+Iφ,t−1ρ)1/λ)λ≥(d,(1−δdλ)d,(1+δdλ)d)tρ,

(t=1,…,T,λ∈(0,1),ρ=1,…,P);

• The invention of every manufacturing at all refineries should be no less than the economical manufacture quantity of every manufacture at all refineries. Consequently, we obtain the following inequality

(12) Θφ,tρ≥M_φ,tρ,(t=1,…,T,ρ=1,…,P,φ=1,…,Φ);

• The manufacturing at all refineries is controlled by the maximum production capability. Thus, we conclude the following inequality

(13) Θφ,tρ≤M_φ,tρ,(t=1,…,T,ρ=1,…,P,φ=1,…,Φ);

• The quantity of every category of oil produced should be greater than or equal to the minimum monthly buying amount of every category of oil by agreements. Hence, we obtain the next inequality

(14) χtȷ≥B_tȷ,(ȷ=1,…,J;t=1,…,T);

• The total sum of oil purchases rendering to agreements is at least a positive fraction β (balance parameter) of the total sum of basic oils by the refinery manufacturing. Then as a conclusion, we have the constrained inequality

(15) ∑ȷ‍B_tȷ≥β∑ȷ‍χtȷ,,(ȷ=1,…,J;t=1,…,T);

• The total of manufactured material transported from a refinery by any type of transportation (trains, pipelines or trucks) is less than or equal to the maximum permissible amount of the transportation

(16) Tφ,tρ≤T_φ,tρ(t=1,…,T,ρ=1,…,P,φ=1,…,Φ);

• The overall sum of manufactured material transported from refineries ought to be less than or equal to the refineries’ overall manufacturing output. Hence, we get the inequality

(17) ∑φ?Tφ,tρ≤∑φ?Θφ,tρ,(t=1,…,T,ρ=1,…,P,φ=1,…,Φ);

• The overall sum of manufactured material transported from refineries has to be greater than or equal to the fluctuating (changing) demand for the manufacturing process. Connected to what has been mentioned above, we have the following constrain

(18) ∑φ?Tφ,tρ≥(d,(1−δdλ)d,(1+δdλ)d)tρ,t=1,…,T,λ∈(0,1),ρ=1,…,P,φ=1,…,Φ);

• Overall, the variables must be non-negative

(19) χtȷ,Θφ,tρ,Iφ,tρ,Tφ,tρ≥0,(ȷ=1,…,J,ρ=1,…,P;λ∈(0,1),t=1,…,T,φ=1,…,Φ).

We aim to maximize the objective function Λ=cχ, where c indicates symmetric coefficients. By utilizing the symmetric coefficient terms:

(c,(1−δcλ)c,(1+δcλ)c):=(c,lλc,rλc).

Based on the 3-D parametric space, we represent the objective function by the 3-multi-objective system, as follows:

(20) maxΛ1=c1χ1+…+cnχnmaxΛ2=lλc1χ1+…+lλcnχnmaxΛ3=rλc1χ1+…+rλcnχn,

where lλ=1−δcλ and rλ=1+δcλ. The next step is to compute the upper and lower bound of Λ, ( ΛU,ΛL ) . Accordingly, we have the following system for the lower bound

(21) Λ1L=min(c1χ1+…+cnχn)Λ2L=min(lλc1χ1+…+lλcnχn)Λ3L=min(rλc1χ1+…+rλcnχn),

and the following system for the upper bound

(22) Λ1U=max(c1χ1+…+cnχn)Λ2U=max(lλc1χ1+…+lλcnχn)Λ3U=max(rλc1χ1+…+rλcnχn).

Following the membership function of Λ, we employ:

(23) ηλ(Λ)={0,Λ≤ΛLΛ−ΛLΛU−ΛL,ΛL<Λ≤ΛU1,Λ>ΛU

where we aim to maximize ηλ(Λ); or maximize ⋎=min(⋎1,⋎2,⋎3) where ηλ(Λ)≥⋎. This solves the following issue:

(24) max⋎,subjectto(c1χ1+…+cnχn)−⋎1(Λ1U−Λ1L)≥Λ1L(lλc1χ1+…+lλcnχn)−⋎2(Λ2U−Λ2L)≥Λ2L(rλc1χ1+…+rλcnχn)−⋎3(Λ3U−Λ3L)≥Λ3Lχi≥0,i=1,…,n⋎∈[0,1],

where ΛkL,k=1,2,3 is given in Eq. (21).

We formulate the multiple objective functions (Λ1,Λ2,Λ3) based on Eq. (10) as follows:

(25) maxΛ1=∑t‍(Πt−B∁t−P∁t−I∁t−T∁t−M∁t)maxΛ2=∑t‍(Πt−(1−δ∁λ)(B∁t+P∁t+I∁t+T∁t−M∁t))maxΛ3=∑t‍(Πt−(1+δ∁λ)(B∁t+P∁t+I∁t+T∁t−M∁t)),

where δ∁=a1+a2∁,a1,a2∈R. Next, by utilizing Mathematica 11.2, we compute the upper and lower bounds. Accordingly, the system becomes as follows:

(26) ∑t‍(Πt−B∁t−P∁t−I∁t−T∁t−M∁t)−⋎1(Λ1U−Λ1L)≥Λ1L∑t‍(Πt−(1−δ∁λ)(B∁t+P∁t+I∁t+T∁t−M∁t))−⋎2(Λ1U−Λ1L)≥Λ2L∑t‍(Πt−(1+δ∁λ)(B∁t+P∁t+I∁t+T∁t−M∁t))−⋎3(Λ1U−Λ1L)≥Λ3L.

Next, we proceed to convert the fuzzy inequality constraints as it is implied by Eqs. (11) and (18):

(27) (1Φ∑φ‍(Θφ,tρ+Iφ,t−1ρ)1/λ)λ−⋎d(1−(1−δdλ))tρ≥d(1−δdλ)tρ,λ∈(0,1),

and

(28) ∑φ‍Tφ,tρ−⋎d(1−(1−δdλ))tρ≥d(1−δdλ)tρ.

Finally, we combine the multi-objective functions to maximize the following:

(29) maxγ, subjectto∑t(Πt−B∁t−P∁t−I∁t−T∁t−M∁t )−γ1(Λ1U−Λ1L ) ≥Λ1L∑t(Πt−(1−δ∁λ )(B∁t+P∁t+I∁t+T∁t−M∁t ))−γ2(Λ1U−Λ1L ) ≥Λ2L∑t(Πt−(1+δ∁λ)(B∁t+P∁t+I∁t+T∁t−M∁t ))−γ3(Λ1U−Λ1L ) ≥Λ3L(1Φ∑φ(Θφ,tρ+Iφ,t−1ρ)1/λ )λ−γ d(1−(1−δdλ ))tρ ≥d(1−δdλ)tρ∑φTφ,tρ −γ d(1−(1−δdλ ))tρ≥d(1−δdλ)tργ∈[0,1],

taking in account that inequalities (12)-(17) and (19) are all achieved.

To find the optimal solution, we collected our data as in Appendix A from its sources. Fig. 3 provides symmetric triangle values based on the parameter values 0<λ<1. The tables involve three values of λ:0.2,0.5,0.8; which implies three different values of ⋎ . It is evident that there is a relation between ⋎ and λ which can be recognized by the equation (Λ1,Λ2,Λ3) . Each product has its own parameter fuzzy value λ. For example, to determine (Λ1,Λ2,Λ3) for the production costs, we search for the maximum value in all refineries (which is 28.8 in the Al-Samawa refinery). Then by setting the three values λ:0.2,0.5,0.8, we determine the parametric fuzzy number of the product by using (8) as follows:

For λ=0.2 , we have λΓ(1+0.2)=0.2×0.91≈0.2→λ_=1−0.2=0.8 . Therefore, we obtain two parametric fuzzy numbers corresponding to 28.8:(23.04,51.84) (see Fig. 3). This new method provides many advantages, such as high accuracy and stability of the data. Using the information in Section 3 and Section 4, we follow the steps:

• Step 1: we obtained the upper bounds Λ3 and lower bounds Λ2 of the objective function (see the second and third column of the first matrix in Fig. 3, respectively)

• Step 2: we determine the vector (GaTkGoFo) by employing the values in step one. This vector represents the interval or the best value of ⋎; for instance, the value ⋎=0.5 implies the proper evaluation for the left and right amounts of the triangle.

• Step 3: Using the vector in step two, we estimate the interval or the value of ⋎∈[0,1] to maximize the problem in (24). As we realize that the value of ⋎=λ (the fractal parameter). Based on the analysis shown in Fig. 3, the most accurate analysis is given when λ=0.5.

• Step 4: By employing the vector in step two, we calculate the interval or the value of ⋎∈[0,1] to maximize the problem in (24). In this place, we confirm that by using Mathematica 11.2, solving system (24) implies that the exact value ⋎≈45 maximizes the system.

Since the hypotheses and incomes are time irregular, several mathematical tests can describe the impact of uncertainty of the recent model. By shifting both demand and the price factors, as well as the estimate of the systems (see Fig. 3) with various symmetric triangular fuzzy number, the demand D and the price factors can be examined. The graphs in Fig. 3 indicate the date of the corresponding matrix (Λ1,Λ2,Λ3); The data show that the manufacturers’ order conferring to income action is unaffected by rises in the selling cost.