趙 傳 立

(沈陽師范大學(xué) 數(shù)學(xué)與系統(tǒng)科學(xué)學(xué)院, 沈陽 110034)

具有學(xué)習(xí)效應(yīng)的排序問題的某些新進(jìn)展

趙 傳 立

(沈陽師范大學(xué) 數(shù)學(xué)與系統(tǒng)科學(xué)學(xué)院, 沈陽 110034)

Biskup 首先將學(xué)習(xí)效應(yīng)的的概念引入到排序問題中,并且在2008年給出具有學(xué)習(xí)效應(yīng)的排序問題的全面綜述。從此以后,具有學(xué)習(xí)效應(yīng)的排序問題持續(xù)引起研究者的興趣。除了Biskup綜述所提到的模型外,文獻(xiàn)中還有其他學(xué)習(xí)效應(yīng)模型,對(duì)近年來文獻(xiàn)中出現(xiàn)的具有學(xué)習(xí)效應(yīng)的排序模型做一簡要介紹。主要考慮3類模型:工件的實(shí)際加工時(shí)間依賴于具體的位置函數(shù)的模型,工件的實(shí)際加工時(shí)間依賴于抽象的位置函數(shù)的模型,工件的實(shí)際加工時(shí)間依賴于截?cái)嗍綄W(xué)習(xí)函數(shù)的模型。

排序; 學(xué)習(xí)效應(yīng); 單機(jī); 平行機(jī); 流水作業(yè)

作為新型排序的一個(gè)組成部分,具有學(xué)習(xí)效應(yīng)的排序問題持續(xù)受到研究者的關(guān)注。在Biskup的綜述之后,近年又出現(xiàn)了許多新模型。這些新模型一部分是利用已有模型加以組合而成,另一部分則是對(duì)已有模型做某些推廣。本文僅介紹后一類模型,關(guān)注2009年以后的某些研究進(jìn)展。主要介紹各個(gè)模型中常見目標(biāo)函數(shù)在一般情況下的結(jié)論,關(guān)注多項(xiàng)式最優(yōu)算法,不涉及啟發(fā)式算法、分支定界法等。對(duì)于特殊情況和特殊目標(biāo)函數(shù)的有關(guān)結(jié)論不做介紹。

1 主要模型

主要介紹近年來產(chǎn)生的3類新模型:工件的實(shí)際加工時(shí)間是具體函數(shù)的模型,工件的實(shí)際加工時(shí)間是抽象函數(shù)的模型,工件的實(shí)際加工時(shí)間是截?cái)嗍綄W(xué)習(xí)函數(shù)的模型。

1.1工件的實(shí)際加工時(shí)間是具體函數(shù)的模型

表1 加工時(shí)間是具體函數(shù)的模型

續(xù)表1

加 工 時(shí) 間 問 題 計(jì)算復(fù)雜性或算法參考文獻(xiàn)pAj[r]=pj(αa∑r-1k=1p[k]+β)1pAj[r]，SpsdfSPT[16]α≥0，β≥0，00pAj[r]=pj(αa∑r-1k=1lnp[k]+β)1pAj[r]fSPT[21]α≥0，β≥0，0

1.2工件的實(shí)際加工時(shí)間是抽象函數(shù)的模型

表2 加工時(shí)間是抽象函數(shù)的模型

續(xù)表2

加 工 時(shí) 間 問題計(jì)算復(fù)雜性或算法 參考文獻(xiàn)pAj[r]=pjf(∑r-1k=1p[k])g(r)1pAj[r]f，f∈{Cmax，∑Ckj}SPT[36?38]f不增，f′不減，g不增1pAj[r]，Spsdf，f∈{Cmax，∑Cj}SPT1pAj[r]，qpsdf，f∈{Cmax，∑Ckj}SPTpAj[r]=f(j，r)Fmprop，pAj[r]CmaxO(n5)[40?42]PmpAj[r]∑mi=1CiO(nm+2)[41]PmpAj[r]，rej∑ACj+∑RejO(nm+3)[43]PmpAj[r]，rej∑mi=1Cmaxi+∑RejO(nm+3)[43]pAj[r]=pjf(∑r-1k=1pA[k]∑nk=1pk)g(r)1pAj[r]，Spsdf，f∈{Cmax，∑Cj}SPT[44]f不增，f′不減，g不增pAj[r]=pjf(∑r-1k=1βkp[k])g(r)1pAj[r]f，f∈{Cmax，∑Ckj}SPT[45]f不增，f′不減，g不增，0<βj不減pAj[r]=f(pj，r)1pAj[r]，PRMαCmax+β∑Ej+λ∑TjO(n4)[46]pAj[r]=pjf(∑r-1k=1βkp[k]，r)，βj不減1pAj[r]，Spsdf，f∈{Cmax，∑Cj}SPT[47]

1.3工件的實(shí)際加工時(shí)間是截?cái)嗍降哪Ｐ?/p>

表3 截?cái)嗍綄W(xué)習(xí)效應(yīng)模型

續(xù)表3

加 工 時(shí) 間 問題計(jì)算復(fù)雜性或算法 參考文獻(xiàn)pAj[r]=pjmax{raj，β}，0<β<11pAj[r]，CON∑(αEj+βTj+γd)O(n3)[57]1pAj[r]，SLK∑(αEj+βTj+γdj)1pAj[r]ff∈{Cmax，∑Cj，∑∑Ci-CjpAj[r]=pjmax{(1-∑r-1k=1p[k]∑nk=1p[k])a，β}1pAj[r]f，f∈{Cmax，∑Cj}SPT[58]a>1，0<β<1pAj[r]=pjmax{(aα-∑r-1k=1p[k]∑nk=1pk+b)，β}1pAj[r]f，f∈{Cmax，∑Ckj}SPT[60]a≥0，b≥0，a+b=1，a≥1，0<β<1

2 結(jié) 論

關(guān)于具有學(xué)習(xí)效應(yīng)的排序問題還有許多結(jié)論,限于本文涉及內(nèi)容和資料的局限,不可能完全提及。即使對(duì)于提到的模型的結(jié)論也難免掛一漏萬,因此文中內(nèi)容僅供感興趣的讀者參考。從上節(jié)所提各個(gè)模型的研究成果來看,對(duì)于工件的實(shí)際加工時(shí)間是具體函數(shù)的模型,雖模型較多,但研究結(jié)論主要集中在單機(jī)問題。具有一般性結(jié)論的目標(biāo)函數(shù)主要是最大完工時(shí)間和完工時(shí)間和,且所用方法是常見的SPT規(guī)則。對(duì)于工件的實(shí)際加工時(shí)間是抽象函數(shù)的模型,某些模型的研究結(jié)果還相對(duì)較少,有些主要集中在某一具體機(jī)器環(huán)境的特定目標(biāo)函數(shù)。對(duì)于截?cái)嗍綄W(xué)習(xí)效應(yīng)的排序模型,由于提出的時(shí)間相對(duì)較短,目前研究成果相對(duì)較少。在未來的工作中,對(duì)于各個(gè)模型可以考慮如下研究方向:對(duì)于實(shí)際加工時(shí)間是具體函數(shù)的模型,針對(duì)具體模型的機(jī)器環(huán)境是平行機(jī)還是車間作業(yè),研究目標(biāo)函數(shù)范圍更廣泛的問題;對(duì)于工件的實(shí)際加工時(shí)間是抽象函數(shù)的模型,可研究包括多個(gè)具體函數(shù)的抽象函數(shù)模型,使其結(jié)論具有一般性;對(duì)于截?cái)嗍綄W(xué)習(xí)效應(yīng)的排序模型,根據(jù)不同的加工時(shí)間函數(shù),考慮其他形式的階段學(xué)習(xí)效應(yīng)模型。

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Recentadvanceinschedulingwithlearningeffects

ZHAOChuanli

(School of Mathematics and Systems Science, Shenyang Normal University, Shenyang 110034, China)

The concept of a learning effect on scheduling problems was first introduced by Biskup. The comprehensive survey paper on scheduling problems with learning effects was conducted by Biskup in 2008. Since then there has been an increasing interest in scheduling problems with learning effects. Therefore, besides the scheduling models mentioned in the survey of Biskup, there are several other new learning effect scheduling models in the literature. The goal of this paper was to provide a briefly review of the literature on the scheduling problems with learning effects in recent years. We mainly consider three models: 1) the actual processing time of jobs depend on specific position function; 2) the actual processing time of jobs depend on abstract position function; 3) the actual processing time of jobs depend on a truncated learning functions.

scheduling; learning effects; single machine; parallel machine; flow shop

2014-09-01。

遼寧省教育廳科學(xué)技術(shù)研究項(xiàng)目(L2014433)。

趙傳立(1958-),男,黑龍江泰來人,沈陽師范大學(xué)教授,博士,碩士研究生導(dǎo)師。

1673-5862(2014)04-0453-08

O223

: A

10.3969/ j.issn.1673-5862.2014.04.001