Robust optimization is a new modeling method to study uncertain optimization problems, which is to find a solution with good performance for all implementations of uncertain input. This paper studies the optimal location allocation of processing plants and distribution centers in uncertain supply chain networks under the worst case. Considering the uncertainty of the supply chain and the risk brought by the uncertainty, a data-driven two-stage sparse distributionally robust risk mixed integer optimization model is established. Based on the complexity of the model, a distribution-separation hybrid particle swarm optimization algorithm (DS-HPSO) is proposed to solve the model, so as to obtain the optimal location allocation scheme and the maximum expected return under the worst case. Then, taking the fresh-food supply chain under the COVID-19 as an example, the impact of uncertainty on location allocation is studied. This paper compares the data-driven two-stage sparse distributionally robust risk mixed integer optimization model with the two-stage sparse risk optimization model, and the data results show the robustness of this model. Moreover, this paper also discusses the impact of different risk weight on decision-making. Different decision makers can choose different risk weight and obtain corresponding benefits and optimal decisions. In addition, the DS-HPSO is compared with distribution-separation hybrid genetic algorithm and distributionally robust L-shaped method to verify the effectiveness of the algorithm.
Citation: Zhimin Liu. Data-driven two-stage sparse distributionally robust risk optimization model for location allocation problems under uncertain environment[J]. AIMS Mathematics, 2023, 8(2): 2910-2939. doi: 10.3934/math.2023152
[1] | Zhimin Liu, Ripeng Huang, Songtao Shao . Data-driven two-stage fuzzy random mixed integer optimization model for facility location problems under uncertain environment. AIMS Mathematics, 2022, 7(7): 13292-13312. doi: 10.3934/math.2022734 |
[2] | Shuai Huang, Youwu Lin, Jing Zhang, Pei Wang . Chance-constrained approach for decentralized supply chain network under uncertain cost. AIMS Mathematics, 2023, 8(5): 12217-12238. doi: 10.3934/math.2023616 |
[3] | Chang-Jun Wang, Zi-Jian Gao . Two-stage stochastic programming with imperfect information update: Value evaluation and information acquisition game. AIMS Mathematics, 2023, 8(2): 4524-4550. doi: 10.3934/math.2023224 |
[4] | Ruiping Wen, Wenwei Li . An accelerated alternating directional method with non-monotone technique for matrix recovery. AIMS Mathematics, 2023, 8(6): 14047-14063. doi: 10.3934/math.2023718 |
[5] | Ebenezer Fiifi Emire Atta Mills . The worst-case scenario: robust portfolio optimization with discrete distributions and transaction costs. AIMS Mathematics, 2024, 9(8): 20919-20938. doi: 10.3934/math.20241018 |
[6] | Massoumeh Nazari, Mahmoud Dehghan Nayeri, Kiamars Fathi Hafshjani . Developing mathematical models and intelligent sustainable supply chains by uncertain parameters and algorithms. AIMS Mathematics, 2024, 9(3): 5204-5233. doi: 10.3934/math.2024252 |
[7] | Yang Chang, Guangyang Liu, Hongyan Yan . Bang-bang control for uncertain random continuous-time switched systems. AIMS Mathematics, 2025, 10(1): 1645-1674. doi: 10.3934/math.2025076 |
[8] | Zongqi Sun, Peng Yang, Jing Wu, Yunpeng Fan . The defined contribution pension plan after retirement under the criterion of a revised loss considering the economic situation. AIMS Mathematics, 2024, 9(2): 4749-4761. doi: 10.3934/math.2024229 |
[9] | Yanfei Chai . Robust strong duality for nonconvex optimization problem under data uncertainty in constraint. AIMS Mathematics, 2021, 6(11): 12321-12338. doi: 10.3934/math.2021713 |
[10] | Zongqi Sun, Peng Yang, Ying Wang, Jing Lu . Research on the wealth management fees of defined contribution pensions during the pre-retirement stage. AIMS Mathematics, 2024, 9(12): 36102-36115. doi: 10.3934/math.20241713 |
Robust optimization is a new modeling method to study uncertain optimization problems, which is to find a solution with good performance for all implementations of uncertain input. This paper studies the optimal location allocation of processing plants and distribution centers in uncertain supply chain networks under the worst case. Considering the uncertainty of the supply chain and the risk brought by the uncertainty, a data-driven two-stage sparse distributionally robust risk mixed integer optimization model is established. Based on the complexity of the model, a distribution-separation hybrid particle swarm optimization algorithm (DS-HPSO) is proposed to solve the model, so as to obtain the optimal location allocation scheme and the maximum expected return under the worst case. Then, taking the fresh-food supply chain under the COVID-19 as an example, the impact of uncertainty on location allocation is studied. This paper compares the data-driven two-stage sparse distributionally robust risk mixed integer optimization model with the two-stage sparse risk optimization model, and the data results show the robustness of this model. Moreover, this paper also discusses the impact of different risk weight on decision-making. Different decision makers can choose different risk weight and obtain corresponding benefits and optimal decisions. In addition, the DS-HPSO is compared with distribution-separation hybrid genetic algorithm and distributionally robust L-shaped method to verify the effectiveness of the algorithm.
The algebra of derivations and generalized derivations play a crucial role in the study of functional identities and their applications. There are many generalizations of derivations viz., generalized derivations, multiplicative generalized derivations, skew generalized derivations, b−generalized derivations, etc. The notion of b−generalized derivation was first introduced by Koșan and Lee [17]. The most important and systematic research on the b−generalized derivations have been accomplished in [11,17,22,26] and references therein. In this manuscript, we characterize b−generalized derivations which are strong commutative preserving (SCP) on R. Moreover, we also discuss and characterize b−generalized derivations involving certain ∗−differential/functional identities on rings possessing involution.
In the early nineties, after a memorable work on the structure theory of rings, a tremendous work has been established by Brešar considering centralizing mappings, commuting mappings, commutativity preserving (CP) mappings and strong commutativity preserving (SCP) mappings on some appropriate subset of rings. Since then it became a tempting research idea in the matrix theory/operator theory/ring theory for algebraists. Commutativity preserving (CP) maps were introduced and studied by Watkins [32] and further extended to SCP by Bell and Mason [6]. Inspired by the concept of SCP maps, Bell and Daif [5] demonstrated the commutativity of (semi-)prime rings possessing derivations and endomorphisms on right ideals (see also [27] and references therein). In [12], Deng and Ashraf studied strong commutativity preserving maps in more general context as follows: Let R be a semiprime ring. If R admits a mapping ψ and a derivation δ on R such that [ψ(r),δ(v)]=[r,v] for every r,v∈R, then R is commutative. In 1994, Brešar and Miers [7] characterized an additive map f:R→R satisfying SCP on a semiprime ring R and showed that f is of the form f(r)=λr+μ(r) for every r∈R, where λ∈C, λ2=1 and μ:R→R is an additive map. Later, Lin and Liu [23] extended this result to Lie ideals of prime rings. Chasing to this, several techniques have been developed to investigate the behavior of strong commutativity preserving maps (SCP) using restrictions on polynomials, invoking derivations, generalized derivations, etcetera. An account of work has been done in the literature [3,10,12,13,15,20,21,23,24,25,27,30,31].
On the other hand, the study of additive maps on rings possessing involution was initiated by Brešar et al. [7] and they characterized the additive centralizing mappings on the skew-symmetric elements of prime rings possessing involution. In the same vein, Lin and Liu [24] describe SCP additive maps on skew-symmetric elements of prime rings possessing involution. Later, Liau et al. [21] improved the above mentioned result for non-additive SCP mappings. Interestingly, in 2015 Ali et al. [1] studied the SCP maps in different way on rings possessing involution. They established the commutativity of prime ring of charateristic not two possessing second kind of involution satisfying [δ(r),δ(r∗)]−[r,r∗]=0 for every r∈R, where δ is a nonzero derivation of R. Recently, Khan and Dar [8] improved this result by studying the case of generalized derivations.
Motivated by the above presented work, in this manuscript we have characterized b−generalized derivations on prime rings possessing involution. Moreover, we also present some examples in support of our main theorems.
Throughout the manuscript unless otherwise stated, R is a prime ring with center Z(R), Q is the maximal right ring of quotients, C=Z(Q) is the center of Q usually known as the extended centroid of R and is a field. "A ring R is said to be 2− torsion free if 2r=0 (where r∈R) implies r=0". The characteristic of R is represented by char(R). "A ring R is called a prime ring if rRv=(0) (where r,v∈R) implies r=0 or v=0 and is called a semiprime ring in case rRr=(0) implies r=0". "An additive map r↦r∗ of R into itself is called an involution if (i) (rv)∗=v∗r∗ and (ii) (r∗)∗=r hold for all r,v∈R. A ring equipped with an involution is known as a ring with involution or a ∗−ring. An element r in a ring with involution ∗ is said to be hermitian/symmetric if r∗=r and skew-hermitian/skew-symmetric if r∗=−r". The sets of all hermitian and skew-hermitian elements of R will be denoted by W(R) and K(R), respectively. "If R is 2−torsion free, then every r∈R can be uniquely represented in the form 2r=h+k where h∈W(R) and k∈K(R). Note that in this case r is normal, i.e., rr∗=r∗r, if and only if h and k commute. If all elements in R are normal, then R is called a normal ring". An example is the ring of quaternions. "The involution is said to be of the first kind if Z(R)⊆W(R), otherwise it is said to be of the second kind. In the later case it is worthwhile to see that K(R)∩Z(R)≠(0)". We refer the reader to [4,14] for justification and amplification for the above mentioned notations and key definitions.
For r,v∈R, the commutator of r and v is defined as [r,v]=rv−vr. We say that a map f:R→R preserves commutativity if [f(r),f(v)]=0 whenever [r,v]=0 for all r,v∈R. Following [7], "let S be a subset of R, a map f:R→S is said to be strong commutativity preserving (SCP) on S if [f(r),f(v)]=[r,v] for all r,v∈S". Following [33], "an additive mapping T:R→R is said to be a left (respectively right) centralizer (multiplier) of R if T(rv)=T(r)v (respectively T(rv)=rT(v)) for all r,v∈R. An additive mapping T is called a centralizer in case T is a left and a right centralizer of R". In ring theory it is more common to work with module homomorphisms. Ring theorists would write that T:R→R is a homomorphism of a ring module R into itself. For a prime ring R all such homomorphisms are of the form T(r)=qr for all r∈R, where q∈Q (see Chapter 2 in [4]). "An additive mapping δ:R→R is said to be a derivation on R if δ(rv)=δ(r)v+rδ(v) for all r,v∈R". It is well-known that every derivation of R can be uniquely extended to a derivation of Q. "A derivation δ is said to be Q−inner if there exists α∈Q such that δ(r)=αr−rα for all r∈R. Otherwise, it is called Q−outer (δ is not inner)". "An additive map H:R→R is called a generalized derivation of R if there exists a derivation δ of R such that H(rv)=H(r)v+rδ(v) for all r,v∈R". The derivation δ is uniquely determined by H and is called the associated derivation of H. The concept of generalized derivations covers the both the concepts of a derivation and a left centralizer. We would like to point out that in [19] Lee proved that "every generalized derivation can be uniquely extended to a generalized derivation of Q and thus all generalized derivations of R will be implicitly assumed to be defined on the whole Q".
The recent concept of generalized derivations were introduced by Koșan and Lee [17], namely, b−generalized derivations which was defined as follows: "An additive mapping H:R→Q is called a (left) b−generalized derivation of R associated with δ, an additive map from R to Q, if H(rv)=H(r)v+brδ(v) for all r,v∈R, where b∈Q". Also they proved that if "R is a prime ring and 0≠b∈Q, then the associated map δ is a derivation i.e., δ(rv)=δ(r)v+rδ(v) for all r,v∈R". It is easy to see that every generalized derivation is a 1−generalized derivation. Also, the mapping r∈R→αr+brc∈Q for a,b,c∈Q is a b−generalized derivation of R, which is known as inner b−generalized derivation of R. In spite of this, they also characterized b−generalized derivation. That is, every b−generalized derivation H on a semiprime ring R is of the form H(r)=αr+bδ(r) for all r∈R, where a,b∈Q. Following important facts are frequently used in the proof of our results:
Fact 2.1 ([1,Lemma 2.1]). "Let R be a prime ring with involution such that char(R)≠2. If K(R)∩Z(R)≠(0) and R is normal, then R is commutative."
Fact 2.2. "Let R be a ring with involution such that char(R)≠2. Then every r∈R can uniquely represented as 2r=w+s, where w∈W(R) and s∈K(R)."
Fact 2.3 ([17,Theorem 2.3]). "Let R be a semiprime ring, b∈Q, and let H:R→Q be a b−generalized derivation associated with δ. Then δ is a derivation and there exists α∈Q such that H(r)=αr+bδ(r) for every r∈R."
Fact 2.4 ([8,Lemma 2.2]). "Let R be a non-commutative prime ring with involution of the second kind such that char(R)≠2. If R admits a derivation δ:R→R such that [δ(w),w]=0 for every w∈W(R), then δ(Z(R))=(0)."
We need a well-known lemma due to Martindale [28], stated below in a form, convenient for our purpose.
Lemma 2.1 ([28,Theorem 2(a)]). "Let R be a prime ring and ai,bi,cj,dj∈Q. Suppose that ∑mi=1airbi+∑nj=1cjrdj=0 for all r∈R. If a1,⋯,am are C−independent, then each bi is a C−linear combination of d1,⋯,dn. If b1,⋯,bm are C−independent, then each ai is a C−linear combination of c1,⋯,cn."
We need Kharchenko's theorem for differential identities to prime rings [16]. The lemma below is a special case of [16,Lemma 2].
Lemma 2.2 ([16,Lemma 2]). "Let R be a prime ring and ai,bi,cj,dj∈Q and δ a Q−outer derivation of R. Suppose that ∑mi=1airbi+∑nj=1cjδ(r)dj=0 for all r∈R. Then ∑mi=1airbi=0=∑nj=1cjrdj for all r∈R."
We begin with the following Lemma which is needed for developing the proof of our theorems:
Lemma 3.1. Let R be a non-commutative prime ring of characteristic different from two with involuation of the second kind. If for any α∈R, [αr,αr∗]−[r,r∗]=0 for every r∈R, then α∈C and α2=1.
Proof. For any r∈R, we have
[αr,αr∗]−[r,r∗]=0. | (3.1) |
This can also be written as
α2[r,r∗]+α[r,α]r∗+α[α,r∗]r−[r,r∗]=0 | (3.2) |
for every r∈R. Replace r by r+s′ in above equation, where s′∈K(R)∩Z(R), we get
α2[r,r∗]+α[r,α]r∗−α[r,α]s′+α[α,r∗]r+α[α,r∗]s′−[r,r∗]=0 | (3.3) |
for every r∈R and s′∈K(R)∩Z(R). In view of (3.2), we have
α[α,r]s′+α[α,r∗]s′=α[α,r+r∗]s′=0 | (3.4) |
for every r∈R and s′∈K(R)∩Z(R). Since the involution is of the second kind, so we have
α[α,r+r∗]=0 | (3.5) |
for every r∈R. For r=w+s, where w∈W(R) and s∈K(R), observe that
α[α,w]=0 | (3.6) |
for every w∈W(R). Substitute ss′ for w in above expression, we get
α[α,s]=0 | (3.7) |
for every s∈K(R), since the involution is of the second kind. Observe from Fact 2.2 that 2α[α,r]=α[α,2r]=α[α,w+s]=α[α,w]+α[α,s]=0. Thus we have α∈C and, by our hypothesis, (α2−1)[r,r∗]=0, for every r∈R. By the primeness of R, it follows that either α2=1 or [r,r∗]=0, for any r∈R. Thus we are led to the required conclusion by Fact 2.1.
Theorem 3.1. Let R be a non-commutative prime ring of characteristic different from two. If H is a non-zero b−generalized derivation on R associated with a derivation δ on R and ψ is a non-zero map on R satisfying [ψ(r),H(v)]=[r,v] for every r,v∈R. Then there exists 0≠λ∈C and an additive map μ:R→C such that H(r)=λr, ψ(r)=λ−1r+μ(r), for any r∈R.
Proof. Notice that if either δ=0 or b=0, the map H reduces to a centralizer, that is H(v)=αv, for any v∈R. Then the conclusion follows as a reduced case of [25,Theorem 1.1]. Hence, in the rest of the proof we assume both b≠0 and δ≠0. By Fact 2.3, there exists α∈H such that H(r)=αr+bδ(r) for every r∈R. By the hypothesis
[ψ(r),αv+bδ(v)]=[r,v] | (3.8) |
for every r,v∈R. Taking of vz for v in above expression gives
(αv+bδ(v))[ψ(r),z]+[ψ(r),bvδ(z)]=v[r,z] | (3.9) |
for every r,v,z∈R.
Suppose firstly that δ is not an inner derivation of R. In view of (3.9) and Lemma 2.2, we observe that
(αv+bv′)[ψ(r),z]+[ψ(r),bvz′]=v[r,z] | (3.10) |
for every r,v,z,v′,z′∈R. In particular, for v=0 we have
bv′[ψ(r),z]=0 | (3.11) |
for every r,z,v′∈R. By the primeness of R and since b≠0, it follows that ψ(r)∈Z(R), for any r∈R. On the other hand, by ψ(r)∈Z(R) and (3.8) one has that [r,v]=0 for any r,v∈R, which is a contradiction since R is not commutative.
Therefore, we have to consider the only case when there is q∈H such that δ(r)=[r,q], for every r∈R. Thus we rewrite (3.10) as follows
((α−bq)v+bvq)[ψ(r),z]+[ψ(r),bvzq−bvqz]=v[r,z] | (3.12) |
that is
{(α−bq)vψ(r)+bvqψ(r)−ψ(r)bvq−vr}z+{bqv−αv}zψ(r)+ψ(r)bvzq−bvzqψ(r)+vzr=0 | (3.13) |
for every r,v,z∈R.
Suppose there exists v∈R such that {bv,v} are linearly C-independent. From relation (3.13) and Lemma 2.1, it follows that, for any r∈R, both r and qψ(r) are C-linear combinations of {1,q,ψ(r)}. In other words, there exist α1,α2,α3,β1,β2,β3∈C, depending by the choice of r∈R, such that
r=α1+α2ψ(r)+α3q | (3.14) |
and
qψ(r)=β1+β2ψ(r)+β3q. | (3.15) |
Notice that, for α2=0, relation (3.14) implies that q commutes with element r∈R. On the other hand, in case α2≠0, by (3.14) we have that
ψ(r)=α−12(r−α1−α3q). | (3.16) |
Then, by using (3.16) in (3.15), it follows that
β1+β3q=α−12(q−β2)(r−α1−α3q). | (3.17) |
So, by commuting (3.17) with q, we get α−12(q−β2)[r,q]=0, implying that [r,q]=0 in any case.
Therefore q commutes with any element of R and this contradicts the fact that δ≠0. Therefore, for any v∈R, {v,bv} must be linearly C-dependent. In this case a standard argument shows that b∈C, which implies that H(r)=(α−bq)r+r(bq), for any r∈R. Hence H is a generalized derivation of R and once again the result follows from [25,Theorem 1.1].
The following theorem is a generalization of [8,Theorem 2.3].
Theorem 3.2. Let R be a non-commutative prime ring with involution of the second kind of characteristic different from two. If H is a b−generalized derivation on R associated with a derivation δ on R such that [H(r),H(r∗)]=[r,r∗] for every r∈R, then there exists λ∈C such that λ2=1 and H(r)=λr for every r∈R.
Proof. By the given hypothesis, we have
[H(r),H(r∗)]−[r,r∗]=0 for all r∈R. | (3.18) |
Taking r as r+v in (3.18) to get
[H(r),H(v∗)]+[H(v),H(r∗)]−[r,v∗]−[v,r∗]=0 | (3.19) |
for every r,v∈R. Substitute vs′ for v, where s′∈K(R)∩Z(R), in above relation, we obtain
0=−[H(r),H(v∗)]s′−[H(r),bv∗]δ(s′)+[H(v),H(r∗)]s′+[bv,H(r∗)]δ(s′)+[r,v∗]s′−[v,r∗]s′ | (3.20) |
for every r,v∈R and s′∈K(R)∩Z(R). Multiply (3.19) with s′ and combine with (3.20) to get
2[H(v),H(r∗)]s′−2[v,r∗]s′−[H(r),bv∗]δ(s′)+[bv,H(r∗)]δ(s′)=0 | (3.21) |
for every r,v∈R and s′∈K(R)∩Z(R). Again substitute v as vs′ in (3.21), we get
0=2[H(v),H(r∗)]s′2+2[bv,H(r∗)]δ(s′)s′−2[v,r∗]s′2+[H(r),bv∗]δ(s′)s′+[bv,H(r∗)]δ(s′)s′ | (3.22) |
for every r,v∈R and s′∈K(R)∩Z(R). In view of (3.21), we have
2[bv,H(r∗)]δ(s′)s′+2[H(r),bv∗]δ(s′)s′=0 | (3.23) |
for every r,v∈R and s′∈K(R)∩Z(R). Since char(R)≠2 and the involution is of the second kind, so [bv,H(r∗)]+[H(r),bv∗]=[H(r),bv∗]−[H(r∗),bv]=0 for every r,v∈R or δ(s′)s′=0 for every s′∈K(R)∩Z(R). Observe that s′=0 is also implies δ(s′)=0 for every s′∈K(R)∩Z(R). Assume that δ(s′)≠0, therefore we have
[H(r),bv∗]−[H(r∗),bv]=0 | (3.24) |
for every r,v∈R. Taking r=v=w+s in above expression, we obtain
[H(s),bw]−[H(w),bs]=0 | (3.25) |
for every w∈W(R) and s∈K(R). Replace s by s′ in (3.25), we get
[H(s′),bw]−[H(w),b]s′=0 | (3.26) |
for every w∈W(R) and s′∈K(R)∩Z(R). Substitute ss′ for w in last expression, we get
[H(s′),bs]s′−[H(s),b]s′2+b[b,s]δ(s′)s′=0 | (3.27) |
for every s∈K(R) and s′∈K(R)∩Z(R). One can see from (3.24) that [H(s′),bs]=0 and [H(s),b]s′=0 for every s∈K(R) and s′∈K(R)∩Z(R). This reduces (3.27) into
b[b,s]δ(s′)s′=0 | (3.28) |
for every s∈K(R) and s′∈K(R)∩Z(R). This implies either b[b,s]=0 for every s∈K(R) or δ(s′)=0 for every s′∈K(R)∩Z(R). Suppose b[b,s]=0 for every s∈K(R). Take s=w0s′ and use the fact that the involution is of the second kind, we get b[b,w0]=0 for every w0∈W(R). An application of Fact 2.2 yields b∈C. One can see from (3.27) that b[H(s′),s]=0 for every s∈K(R) and s′∈K(R)∩Z(R). If b=0 and in light of Fact 2.3, H has the following form: H(r)=αr, for some fixed element α∈H. Thus, by Lemma 3.1 and since R is not commutative, we get the required conclusion α∈C and α2=1.
So we assume b≠0 and [H(s′),s]=0 for every s∈K(R) and s′∈K(R)∩Z(R). Again taking s as w0s′ and making use of Fact 2.2 in last relation gives H(s′)∈Z(R) for every s′∈K(R)∩Z(R). Next, take r=w and v=s in (3.19), we get
[H(s),H(w)]+[w,s]=0 | (3.29) |
for every s∈K(R) and w∈W(R). Substitute w0s′ for s in above relation, we get
[H(w0s′),H(w)]+[w,w0]s′=0 | (3.30) |
for every s′∈K(R)∩Z(R) and w,w0∈W(R). It follows from the hypothesis that
[H(w0),H(w)]s′+[bw0,H(w)]δ(s′)+[w,w0]s′=0 | (3.31) |
for every s′∈K(R)∩Z(R) and w,w0∈W(R). For w0=w, we have
b[H(w),w]δ(s′)=0 | (3.32) |
for every s′∈K(R)∩Z(R) and w∈W(R). Since δ(s′)≠0 and b≠0, so [H(w),w]=0 for every w∈W(R). Since b∈C, so we observe from (3.25) that
[H(s),w]−[H(w),s]=0 | (3.33) |
for every w∈W(R) and s∈K(R). Repalcement of s by s′h in above expression and making use of H(s′)∈Z(R) for every s′∈K(R)∩Z(R) and b≠0 yields [δ(w),w]=0 for every w∈W(R). In light of Fact 2.4, we have δ(s′)=0 for every s′∈K(R)∩Z(R). Finally, we suppose δ(s′)=0 and substitute v by vs′ in (3.19), we obtain
−[H(r),H(v∗)]s′+[H(v),H(r∗)]s′+[r,v∗]s′−[v,r∗]s′=0 | (3.34) |
for every r,v∈R and s′∈K(R)∩Z(R). Combination of (3.19) and (3.34) gives
([H(v),H(r∗)]−[v,r∗])s′=0 | (3.35) |
for every r,v∈R and s′∈K(R)∩Z(R). This implies that
[H(v),H(r∗)]−[v,r∗]=0 | (3.36) |
for every r,v∈R. In particular
[H(r),H(v)]−[r,v]=0 | (3.37) |
for every r,v∈R. As a special case of Theorem 3.1, H is of the form H(r)=λr, where λ∈C and λ2=1.
The following theorem is a generalization of [8,Theorem 2.4].
Theorem 3.3. Let R be a non-commutative prime ring with involution of the second kind of characteristic different from two. If H is a b−generalized derivation on R associated with a derivation δ on R such that [H(r),δ(r∗)]=[r,r∗] for every r∈R, then there exists λ∈C such that H(r)=λr for every r∈R.
Proof. By the hypothesis, we have
[H(r),δ(r∗)]−[r,r∗]=0 | (3.38) |
for every r∈R. The derivation δ satisfies δ(R)⊈Z(R), otherwise the hypothesis [H(r),δ(r∗)]=[r,r∗] for every r∈R, would reduce to [r,r∗]=0 for all r∈R, and, therefore R would be commutative, by Fact 2.1. A linearization of (3.38) yields
[H(r),δ(v∗)]+[H(v),δ(r∗)]−[r,v∗]−[v,r∗]=0 | (3.39) |
for every r,v∈R. Replace r by z∈Z(R) in (3.39), we obtain [H(z),δ(v)]=0 for every v∈R and z∈Z(R). Observe from [18,Theorem 2] that H(z)∈Z(R) for every z∈Z(R). Now take r as rs′ in (3.38) and suppose δ(s′)≠0, we get
0=−[H(r),δ(r∗)]s′2−[H(r),r∗]δ(s′)s′−b[r,δ(r∗)]δ(s′)s′−[b,δ(r∗)]rδ(s′)s′−b[r,r∗]δ(s′)2−[b,r∗]rδ(s′)2+[r,r∗]s′2 | (3.40) |
for every r∈R and s′∈K(R)∩Z(R). In view of (3.38), we have
0=−[H(r),r∗]δ(s′)s′−b[r,δ(r∗)]δ(s′)s′−[b,δ(r∗)]rδ(s′)s′−b[r,r∗]δ(s′)2−[b,r∗]rδ(s′)2 | (3.41) |
for every r∈R and s′∈K(R)∩Z(R). Replace r by r+s′ in last expression and use the fact that H(z)∈Z(R) for every z∈Z(R), we get
0=−[H(r),r∗]δ(s′)s′−b[r,δ(r∗)]δ(s′)s′−[b,δ(r∗)]rδ(s′)s′−b[r,r∗]δ(s′)2−[b,r∗]rδ(s′)2−[b,δ(r∗)]δ(s′)s′2−[b,r∗]δ(s′)2s′ | (3.42) |
for every r∈R and s′∈K(R)∩Z(R). Observe from (3.41)
[b,δ(r∗)]δ(s′)s′2+[b,r∗]δ(s′)2s′=0 | (3.43) |
for every r∈R and s′∈K(R)∩Z(R). Replace r by rs′ and use (3.43), we get [b,r∗]δ(s′)2s′=0 This forces that [b,r∗]=0 for every r∈R since δ(s′)≠0. One can easily obtain from last relation that b∈C. On the other hand
0=−H(s′)[r,δ(r∗)]s′−b[δ(r),δ(r∗)]s′2−[b,δ(r∗)]δ(r)s′2−H(s′)[r,r∗]δ(s′)−b[δ(r),r∗]δ(s′)s′−[b,r∗]δ(r)δ(s′)s′+[r,r∗]s′2 | (3.44) |
for every r∈R and s′∈K(R)∩Z(R). Since b∈C, so we have
0=−H(s′)[r,δ(r∗)]s′−b[δ(r),δ(r∗)]s′2−H(s′)[r,r∗]δ(s′)−b[δ(r),r∗]δ(s′)s′+[r,r∗]s′2 | (3.45) |
for every r∈R and s′∈K(R)∩Z(R). Now substitute w and s for r and v in (3.39), respectively. This yields
[H(s),δ(w)]−[H(w),δ(s)]+2[s,w]=0 | (3.46) |
for every w∈W(R) and s∈K(R). Take s as s′w in last relation and use b∈C, we see that
H(s′)[w,δ(w)]−[H(w),w]δ(s′)=0 | (3.47) |
for every w∈W(R) and s′∈K(R)∩Z(R). On the other hand
b[h,δ(w)]δ(s′)−[H(w),w]δ(s′)=0 | (3.48) |
for every w∈W(R) and s′∈K(R)∩Z(R). In view of (3.47) and (3.48), we get
(H(s′)+bδ(s′))[w,δ(w)]=0 | (3.49) |
for every w∈W(R) and s′∈K(R)∩Z(R). Since H(s′)∈Z(R), δ(s′)∈Z(R) and b∈Z(R), so either H(s′)+bδ(s′)=0 for every s′∈K(R)∩Z(R) or [w,δ(w)]=0 for every w∈W(R). If [w,δ(w)]=0 for every w∈W(R), then δ(s′)=0 for every s′∈K(R)∩Z(R) from Fact 2.4. Therefore consider H(s′)=−bδ(s′) for every s′∈K(R)∩Z(R) and use it in (3.45), we obtain
0=b[r,δ(r∗)]δ(s′)s′−b[δ(r),δ(r∗)]s′2+b[r,r∗]δ(s′)2−b[δ(r),r∗]δ(s′)s′+[r,r∗]s′2 | (3.50) |
for every r∈R and s′∈K(R)∩Z(R). For r=w, we have
2b[h,δ(h)]δ(s′)s′=0 | (3.51) |
for every w∈W(R) and s′∈K(R)∩Z(R). Thus b=0 or [w,δ(w)]=0 for every w∈W(R) or δ(s′)=0 for every s′∈K(R)∩Z(R). If b=0 and since s′ is not a zero-divisor, the relation (3.50) reduces to [r,r∗]=0, for every r∈R. Thus the commutativity of R follows from Fact 2.1, a contradiction. The rest of two relations yields δ(s′)=0 for every s′∈K(R)∩Z(R). Finally consider δ(s′)=0 and replace r by rs′ in (3.38) and use the facts that the involution is of the second kind, we see that
[H(r),δ(v∗)]−[H(v),δ(r∗)]−[r,v∗]+[v,r∗]=0 | (3.52) |
for every r,v∈R. Combination of (3.39) and (3.52) yields
[H(r),δ(v∗)]−[r,v∗]=0 | (3.53) |
for every r,v∈R. In particular,
[H(r),δ(v)]−[r,v]=0 | (3.54) |
for every r,v∈R. In view of Theorem 3.1, there exist 0≠λ∈C and an additive map μ:R→C such that H(r)=λr and δ′(r)=λ−1r+μ(r) for every r∈R, where δ′=−δ. Commute the latter case with r, we get [δ′(r),r]=0 for every r∈R. Since δ′=−δ≠0, so R is commutative from [29,Lemma 3], this leads to again a contradiction. This completes the proof of the theorem.
The following example shows that the condition of the second kind involution is essential in Theorems 3.2 and 3.3. This example collected from [2,Example 1].
Example 4.1. Let
R={(β1β2β3β4)|β1,β2,β3,β4∈Z}, |
which is of course a prime ring with ususal addition and multiplication of matrices, where Z is the set of integers. Define mappings H,δ,∗:R→R such that
H(β1β2β3β4)=(0−β2β30), |
δ(β1β2β3β4)=(0−β2β30), |
and a fixed element
b=(1001), |
(β1β2β3β4)∗=(β4−β2−β3β1). |
Obviously,
Z(R)={(β100β1)|β1∈Z}. |
Then r∗=r for every r∈Z(R), and hence Z(R)⊆W(R), which shows that the involution ∗ is of the first kind. Moreover, H, δ are nonzero b−generalized derivation and associated derivation with fixed element b defined as above, such that the hypotheses in Theorems 3.2 and 3.3 are satisfied but H is not in the form H(r)=λr for every r∈R. Thus, the hypothesis of the second kind involution is crucial in our results.
We conclude the manuscript with the following example which reveals that Theorems 3.2 and 3.3 cannot be extended to semiprime rings.
Example 4.2. Let (R,∗) be a ring with involution as defined above, which admits a b−generalized derivation H, where δ is an associated nonzero derivation same as above and R1=C with the usual conjugation involution ⋄. Next, let S=R×R1 and define a b−generalized derivation G on S by G(r,v)=(H(r),0) associated with a derivation D defined by D(r,v)=(δ(r),0). Obviously, (S,τ) is a semiprime ring with involution of the second kind such that τ(r,v)=(r∗,v⋄). Then the b−generalized derivation G satisfies the requirements of Theorems 3.2 and 3.3, but G is not in the form G(r)=λr for every r∈R, and R is not commutative. Hence, the primeness hypotheses in our results is not superfluous.
We recall that "a generalized skew derivation is an additive mapping G:R→R satisfying the rule G(rv)=G(r)v+ζ(r)∂(v) for every r,v∈R, where ∂ is an associated skew derivation of R and ζ is an automorphism of R". Following [9], De Filippis proposed the new concept for further research and he defined the following: "Let R be an associative algebra, b∈Q, ∂ be a linear mapping from R to itself, and ζ be an automorphism of R. A linear mapping G:R→R is called an X−generalized skew derivation of R, with associated term (b,ζ,∂) if G(rv)=G(r)v+bζ(r)∂(v) for every r,v∈R". It is clear from both definitions, the notions of X−generalized skew derivation, generalize both generalized skew derivations and skew derivations. Hence, every X−generalized skew derivation is a generalized skew derivation as well as a skew derivation, but the converse statement is not true in general.
Actuated by the concept specified by De Filippis [9] and having regard to our main theorems, the following are natural problems.
Question 5.1. Let R be a (semi)-prime ring and L be a Lie ideal of R. Next, let F and G be two X−generalized skew derivation with an associated skew derivation ∂ of R such that
[F(r),G(v)]=[r,v],for everyr,v∈L. |
Then what we can say about the behaviour of F and G or about the structure of R?
Question 5.2. Let R be a prime ring possessing second kind involution with suitable torsion restrictions and L be a Lie ideal of R. Next, let F and G be two X−generalized skew derivation with an associated skew derivation ∂ of R such that
[F(r),G(r∗)]=[r,r∗],for everyr∈L. |
Then what we can say about the behaviour of F and G or about the structure of R?
The characterization of strong commutative preserving (SCP) b−generalized derivations has been discussed in non-commutative prime rings. In addition, the behavior of b−generalized derivations with ∗−differential/functional identities on prime rings with involution was investigated. Besides, we present some problems for X−generalized skew derivations on rings with involution.
We are very grateful to the referee for his/her appropriate and constructive suggestions which improved the quality of the paper. This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. (G: 156-662-1439). The authors, therefore, gratefully acknowledge the DSR technical and financial support.
The authors declare that they have no competing interests.
[1] |
L. Cooper, Location-allocation problems, Oper. Res., 11 (1963), 331–343. https://doi.org/10.1287/opre.11.3.331 doi: 10.1287/opre.11.3.331
![]() |
[2] |
L. F. Gelders, L. M. Pintelon, L. N. V. Wassenhove, A location-allocation problem in a large Belgian brewery, Eur. J. Oper. Res., 28 (1987), 196–206. https://doi.org/10.1016/0377-2217(87)90218-9 doi: 10.1016/0377-2217(87)90218-9
![]() |
[3] |
L. Nick, A. V. Felipe, Points of distribution location and inventory management model for post-disaster humanitarian logistics, Transport. Res. Part E: Logist. Transport. Rev., 116 (2018), 1–24. https://doi.org/10.1016/j.tre.2018.05.003 doi: 10.1016/j.tre.2018.05.003
![]() |
[4] |
A. Moreno, D. Alem, D. Ferreira, A. Clark, An effective two-stage stochastic multi-trip location-transportation model with social concerns in relief supply chains, Eur. J. Oper. Res., 269 (2018), 1050–1071. https://doi.org/10.1016/j.ejor.2018.02.022 doi: 10.1016/j.ejor.2018.02.022
![]() |
[5] |
C. A. Irawan, D. Jones, Formulation and solution of a two-stage capacitated facility location problem with multilevel capacities, Ann. Oper. Res., 272 (2019), 41–67. https://doi.org/10.1007/s10479-017-2741-7 doi: 10.1007/s10479-017-2741-7
![]() |
[6] |
Z. M. Liu, R. P. Huang, S. T. Shao, Data-driven two-stage fuzzy random mixed integer optimization model for facility location problems under uncertain environment, AIMS Math., 7 (2022), 13292–13312. https://doi.org/10.3934/math.2022734 doi: 10.3934/math.2022734
![]() |
[7] |
N. Noyan, Risk-averse two-stage stochastic programming with an application to disaster management, Comput. Oper. Res., 39 (2012), 541–559. https://doi.org/10.1016/j.cor.2011.03.017 doi: 10.1016/j.cor.2011.03.017
![]() |
[8] |
N. Ricciardi, R. Tadei, A. Grosso, Optimal facility location with random throughput costs, Comput. Oper. Res., 29 (2002), 593–607. https://doi.org/10.1016/S0305-0548(99)00090-8 doi: 10.1016/S0305-0548(99)00090-8
![]() |
[9] |
S. Baptista, M. I. Gomes, A. P. Barbosa-Povoa, A two-stage stochastic model for the design and planning of a multi-product closed loop supply chain, Comput. Aided Chem. Eng., 30 (2012), 412–416. https://doi.org/10.1016/B978-0-444-59519-5.50083-6 doi: 10.1016/B978-0-444-59519-5.50083-6
![]() |
[10] |
J. Qin, H. Xiang, Y. Ye, L. L. Ni, A simulated annealing methodology to multiproduct capacitated facility location with stochastic demand, Sci. World J., 2015, 1–9. https://doi.org/10.1155/2015/826363 doi: 10.1155/2015/826363
![]() |
[11] | I. Litvinchev, M. Mata, L. Ozuna, Lagrangian heuristic for the two-stage capacitated facility location problem, Appl. Comput. Math., 11 (2012), 137–146. |
[12] |
Z. M. Liu, S. J. Qu, Z. Wu, D. Q. Qu, J. H. Du, Two-stage mean-risk stochastic mixed integer optimization model for location-allocation problems under uncertain environment, J. Ind. Manag. Optim., 17 (2021), 2783–2804. https://doi.org/10.3934/jimo.2020094 doi: 10.3934/jimo.2020094
![]() |
[13] | A. L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Oper. Res., 21 (1973), 1154–1157. |
[14] | A. Ben-Tal, L. E. Ghaoui, A. Nemirovski, Robust optimization, Princeton: Princeton University Press, 2009. |
[15] |
D. Bertsimas, D. B. Brown, C. Caramanis, Theory and applications of robust optimization, SIAM Rev., 53 (2011), 464–501. https://doi.org/10.1137/080734510 doi: 10.1137/080734510
![]() |
[16] |
Z. Liu, Z. Wu, Y. Ji, S. J. Qu, H. Raza, Two-stage distributionally robust mixed-integer optimization model for three-level location-allocation problems under uncertain environment, Phys. A: Stat. Mech. Appl., 572 (2021), 125872. https://doi.org/10.1016/j.physa.2021.125872 doi: 10.1016/j.physa.2021.125872
![]() |
[17] |
X. J. Chen, A. Shapiro, H. L. Sun, Convergence analysis of sample average approximation of two-stage stochastic generalized equation, SIAM J. Optim., 29 (2019), 135–161. https://doi.org/10.1137/17M1162822 doi: 10.1137/17M1162822
![]() |
[18] |
X. J. Chen, H. L. Sun, H. F. Xu, Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems, Math. Program., 177 (2019), 255–289. https://doi.org/10.1007/s10107-018-1266-4 doi: 10.1007/s10107-018-1266-4
![]() |
[19] |
R. P. Huang, S. J. Qu, X. G. Yang, Z. M. Liu, Multi-stage distributionally robust optimization with risk aversion, J. Ind. Manag. Optim., 17 (2021), 233–259. https://doi.org/10.3934/jimo.2019109 doi: 10.3934/jimo.2019109
![]() |
[20] |
A. Klose, An LP-based heuristic for two-stage capacitated facility location problems, J. Oper. Res. Soc., 50 (1999), 157–166. https://doi.org/10.1057/palgrave.jors.2600675 doi: 10.1057/palgrave.jors.2600675
![]() |
[21] |
B. Li, Q. Xun, J. Sun, K. L. Teo, C. J. Yu, A model of distributionally robust two-stage stochastic convex programming with linear recourse, Appl. Math. Model., 58 (2018), 86–97. https://doi.org/10.1016/j.apm.2017.11.039 doi: 10.1016/j.apm.2017.11.039
![]() |
[22] |
B. Li, J. Sun, H. L. Xu, M. Zhang, A class of two-stage distributionally robust games, J. Ind. Manag. Optim., 15 (2019), 387–400. https://doi.org/10.3934/jimo.2018048 doi: 10.3934/jimo.2018048
![]() |
[23] |
Z. M. Liu, S. J. Qu, M. Goh, Z. Wu, R. P. Huang, G. Ma, Two-stage mean-risk stochastic optimization model for port cold storage capacity under pelagic fishery yield uncertainty, Phys. A: Stat. Mech. Appl., 541 (2020), 123338. https://doi.org/10.1016/j.physa.2019.123338 doi: 10.1016/j.physa.2019.123338
![]() |
[24] |
V. Rico-Ramirez, G. A. Iglesias-Silva, F. Gomez-De la Cruz, S. Hernandez-Castro, Two-stage stochastic approach to the optimal location of booster disinfection stations, Ind. Eng. Chem. Res., 46 (2007), 6284–6292. https://doi.org/10.1021/ie070141a doi: 10.1021/ie070141a
![]() |
[25] |
J. Sun, L. Z. Liao, B. Rodrigues, Quadratic two-stage stochastic optimization with coherent measures of risk, Math. Program., 168 (2018), 599–613. https://doi.org/10.1007/s10107-017-1131-x doi: 10.1007/s10107-017-1131-x
![]() |
[26] |
M. Dillon, F. Oliveira, B. Abbasi, A two-stage stochastic programming model for inventory management in the blood supply chain, Int. J. Prod. Econ., 187 (2017), 27–41. https://doi.org/10.1016/j.ijpe.2017.02.006 doi: 10.1016/j.ijpe.2017.02.006
![]() |
[27] |
K. L. Liu, Q. F. Li, Z. H. Zhang, Distributionally robust optimization of an emergency medical service station location and sizing problem with joint chance constraints, Transport. Res. Part B-Meth., 119 (2019), 79–101. https://doi.org/10.1016/j.trb.2018.11.012 doi: 10.1016/j.trb.2018.11.012
![]() |
[28] |
F. Maggioni, F. A. Potra, M. Bertocchi, A scenario-based framework for supply planning under uncertainty: stochastic programming versus robust optimization approaches, Comput. Manag. Sci., 14 (2017), 5–44. https://doi.org/10.1007/s10287-016-0272-3 doi: 10.1007/s10287-016-0272-3
![]() |
[29] |
R. Venkitasubramony, G. K. Adil, Designing a block stacked warehouse for dynamic and stochastic product flow: a scenario-based robust approach, Int. J. Prod. Res., 57 (2019), 1345–1365. https://doi.org/10.1080/00207543.2018.1472402 doi: 10.1080/00207543.2018.1472402
![]() |
[30] |
C. L. Hu, X. Liu, J. Lu, A bi-objective two-stage robust location model for waste-to-energy facilities under uncertainty, Decis. Support Syst., 99 (2017), 37–50. https://doi.org/10.1016/j.dss.2017.05.009 doi: 10.1016/j.dss.2017.05.009
![]() |
[31] |
S. Mišković, Z. Stanimirović, I. Grujičić, Solving the robust two-stage capacitated facility location problem with uncertain transportation costs, Optim. Lett., 11 (2017), 1169–1184. https://doi.org/10.1007/s11590-016-1036-2 doi: 10.1007/s11590-016-1036-2
![]() |
[32] |
J. Portilla, Image restoration through l0 analysis-based sparse optimization in tight frames, 2009 16th IEEE International Conference on Image Processing (ICIP), 2009, 3909–3912. https://doi.org/10.1109/ICIP.2009.5413975 doi: 10.1109/ICIP.2009.5413975
![]() |
[33] |
M. Zibulevsky, M. Elad, L1-L2 optimization in signal and image processing, IEEE Signal Proc. Mag., 27 (2010), 76–88. https://doi.org/10.1109/MSP.2010.936023 doi: 10.1109/MSP.2010.936023
![]() |
[34] | L. Xu, S. C. Zheng, J. Y. Jia, Unnatural l0 sparse representation for natural image deblurring, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2013, 1107–1114. |
[35] | B. Y. Liu, L. Yang, J. Z. Huang, P. Meer, L. G. Gong, C. Kulikowski, Robust and fast collaborative tracking with two stage sparse optimization, In: K. Daniilidis, P. Maragos, N. Paragios, Computer vision–ECCV 2010, Lecture Notes in Computer Science, Vol. 6314, Springer, Berlin, Heidelberg, 2010. https://doi.org/10.1007/978-3-642-15561-1_45 |
[36] |
X. F. Zhu, X. L. Li, S. C. Zhang, Block-row sparse multiview multilabel learning for image classification, IEEE Trans. Cybern., 46 (2016), 450–461. https://doi.org/10.1109/TCYB.2015.2403356 doi: 10.1109/TCYB.2015.2403356
![]() |
[37] |
H. Zhang, V. M. Patel, Convolutional sparse and low-rank coding-based image decomposition, IEEE Trans. Image Proc., 27 (2018), 2121–2133. https://doi.org/10.1109/TIP.2017.2786469 doi: 10.1109/TIP.2017.2786469
![]() |
[38] |
D. Bertsimas, R. Cory-Wright, A scalable algorithm for sparse portfolio selection, INFORMS J. Comput., 34 (2022), 1489–1511. https://doi.org/10.1287/ijoc.2021.1127 doi: 10.1287/ijoc.2021.1127
![]() |
[39] |
M. Dyer, L. Stougie, Computational complexity of stochastic programming problems, Math. Program., 106 (2006), 423–432. https://doi.org/10.1007/s10107-005-0597-0 doi: 10.1007/s10107-005-0597-0
![]() |
[40] |
M. Bansal, K. L. Huang, S. Mehrotra, Decomposition algorithms for two-stage distributionally robust mixed binary programs, SIAM J. Optim., 28 (2018), 2360–2383. https://doi.org/10.1137/17M1115046 doi: 10.1137/17M1115046
![]() |
[41] |
M. Bansal, S. Mehrotra, On solving two-stage distributionally robust disjunctive programs with a general ambiguity set, Eur. J. Oper. Res., 279 (2019), 296–307. https://doi.org/10.1016/j.ejor.2019.05.033 doi: 10.1016/j.ejor.2019.05.033
![]() |
[42] | L. R. Medsker, Hybrid intelligent systems, Boston: Kluwer Academic Publishers, 1995. |
[43] |
M. Rahman, N. S. Chen, M. M. Islam, A. Dewand, H. R. Pourghasemie, R. M. A. Washakh, et al., Location-allocation modeling for emergency evacuation planning with GIS and remote sensing: a case study of Northeast Bangladesh, Geosci. Front., 12 (2021), 101095. https://doi.org/10.1016/j.gsf.2020.09.022 doi: 10.1016/j.gsf.2020.09.022
![]() |
[44] |
M. Y. Qi, R. W. Jiang, S. Q. Shen, Sequential competitive facility location: exact and approximate algorithms, Oper. Res., 2022. https://doi.org/10.1287/opre.2022.2339 doi: 10.1287/opre.2022.2339
![]() |
[45] | M. Rahbari, S. H. R. Hajiagha, H. A. Mahdiraji, F. R. Dorcheh, J. A. Garza-Reyes, A novel location-inventory-routing problem in a two-stage red meat supply chain with logistic decisions: evidence from an emerging economy, Int. J. Syst. Cybern., 4 (2022), 1498–1531. |
[46] |
Y. J. Yang, Y. Q. Yin, D. J. Wang, J. Ignatius, T. C. E. Cheng, L. Dhamotharan, Distributionally robust multi-period location-allocation with multiple resources and capacity levels in humanitarian logistics, Eur. J. Oper. Res., 2022. https://doi.org/10.1016/j.ejor.2022.06.047 doi: 10.1016/j.ejor.2022.06.047
![]() |
[47] |
T. Q. Liu, F. Saldanha-da-Gama, S. M. Wang, Y. C. Mao, Robust stochastic facility location: sensitivity analysis and exact solution, INFORMS J. Comput., 34 (2022), 2776–2803. https://doi.org/10.1287/ijoc.2022.1206 doi: 10.1287/ijoc.2022.1206
![]() |
[48] |
K. S. Shehadeh, Distributionally robust optimization approaches for a stochastic mobile facility fleet sizing, routing, and scheduling problem, Transport. Sci., 2022. https://doi.org/10.1287/trsc.2022.1153 doi: 10.1287/trsc.2022.1153
![]() |
[49] |
H. Soleimani, P. Chhetri, A. M. Fathollahi-Fard, S. M. J. Mirzapour Al-e-Hashem, S. Shahparvari, Sustainable closed-loop supply chain with energy efficiency: lagrangian relaxation, reformulations, and heuristics, Ann. Oper. Res., 318 (2022), 531–556. https://doi.org/10.1007/s10479-022-04661-z doi: 10.1007/s10479-022-04661-z
![]() |
[50] |
S. Kim, S. Weber, Simulation methods for robust risk assessment and the distorted mix approach, Eur. J. Oper. Res., 298 (2022), 380–398. https://doi.org/10.1016/j.ejor.2021.07.005 doi: 10.1016/j.ejor.2021.07.005
![]() |
[51] |
P. Embrechts, A. Schied, R. D. Wang, Robustness in the optimization of risk measures, Oper. Res., 70 (2021), 95–110. https://doi.org/10.1287/opre.2021.2147 doi: 10.1287/opre.2021.2147
![]() |
[52] |
W. Liu, L. Yang, B. Yu, Distributionally robust optimization based on kernel density estimation and mean-Entropic value-at-risk, INFORMS J. Optim., 2022. https://doi.org/10.1287/ijoo.2022.0076 doi: 10.1287/ijoo.2022.0076
![]() |
[53] |
A. M. Fathollahi-Fard, M. A. Dulebenets, M. Hajiaghaei-Keshteli, R. Tavakkoli-Moghaddam, M. Safaeian, H. Mirzahosseinian, Two hybrid meta-heuristic algorithms for a dual-channel closed-loop supply chain network design problem in the tire industry under uncertainty, Adv. Eng. Inform., 50 (2021), 101418. https://doi.org/10.1016/j.aei.2021.101418 doi: 10.1016/j.aei.2021.101418
![]() |
[54] |
N. Noyan, Risk-Averse stochastic modeling and optimization, INFORMS TutORials Oper. Res., 2018,221–254. https://doi.org/10.1287/educ.2018.0183 doi: 10.1287/educ.2018.0183
![]() |
[55] |
P. Artzner, F. Delbaen, J. M. Eber, D. Heath, Coherent measures of risk, Math. Finance, 9 (1999), 203–228. https://doi.org/10.1111/1467-9965.00068 doi: 10.1111/1467-9965.00068
![]() |
[56] |
W. Ogryczak, A. Ruszczynski, Dual stochastic dominance and related mean-risk models, SIAM J. Optim., 13 (2002), 60–78. https://doi.org/10.1137/S1052623400375075 doi: 10.1137/S1052623400375075
![]() |
[57] | R. T. Rockafellar, S. Uryasev, Optimization of conditional value-at-risk, J. Risk, 2 (2000), 21–41. |
[58] |
P. M. Esfahani, D. Kuhn, Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations, Math. Program., 171 (2018), 115–166. https://doi.org/10.1007/s10107-017-1172-1 doi: 10.1007/s10107-017-1172-1
![]() |
[59] |
M. Clerc, The swarm and queen: towards a deterministic and adaptive particle swarm optimization, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99, 3 (1999), 1951–1957. https://doi.org/10.1109/CEC.1999.785513 doi: 10.1109/CEC.1999.785513
![]() |
[60] |
M. Clerc, J. Kennedy, The particle swarm-explosion, stability, and convergence in a multidimensional complex space, IEEE T. Evolut. Comput., 6 (2002), 58–73. https://doi.org/10.1109/4235.985692 doi: 10.1109/4235.985692
![]() |
[61] |
J. Kennedy, R. C. Eberhart, A discrete binary version of the particle swarm algorithm, 1997 IEEE International Conference on Systems, Man, and Cybernetics. Computational Cybernetics and Simulation, 5 (1997), 4104–4108. https://doi.org/10.1109/ICSMC.1997.637339 doi: 10.1109/ICSMC.1997.637339
![]() |
[62] | Z. Ji, H. L. Liao, Q. H. Wu, Particle swarm optimization algorithm and its application, Science Press, 2009. |
[63] | S. Gao, K. Tang, X. Jiang, J. Yang, Convergence analysis of particle swarm optimization algorithm, Sci. Technol. Eng., 6 (2006), 1625–1627. |
[64] | J. Nocedal, S. Wright, Numerical optimization, Springer, 2006. |
[65] |
J. Moosavi, A. M. Fathollahi-Fard, M. A. Dulebenets, Supply chain disruption during the COVID-19 pandemic: recognizing potential disruption management strategies, Int. J. Disast. Risk Re., 75 (2022), 102983. https://doi.org/10.1016/j.ijdrr.2022.102983 doi: 10.1016/j.ijdrr.2022.102983
![]() |
1. | Shuai Huang, Youwu Lin, Jing Zhang, Pei Wang, Chance-constrained approach for decentralized supply chain network under uncertain cost, 2023, 8, 2473-6988, 12217, 10.3934/math.2023616 | |
2. | Tareq Oshan, Nuha Hamada, Mahmoud Sharkawi, 2024, Optimizing Facility Location Under Disruptions: A Big Data-Driven Approach, 979-8-3503-5475-1, 225, 10.1109/IDSTA62194.2024.10746991 |