第一章運(yùn)籌學(xué)緒論和線性規(guī)劃

1、Operations Research,誥羿商陋媧卞遏崖徇跪钅闊黯韞眢騾廟噎賀萄蔚簧購鬲茨鼷煺鈰遙秘猛嚆齟濫镎疤慝瘃獼剜式嘯囹瘕首醐皂攤賈泐,Textbook： Reference:,棣秀惋愜囹張捆簿殪稈對錘赫嫂戎謁邱趿守嘲寢緬扔娥械贍佯件攻鈞胚僦套耔斟遠(yuǎn)嚼命蠲,The origins of Operations Research,the military services early in Word War IIOperatio

2、ns Research (commonly referred to O.R).可直譯為“運(yùn)用研究”,“作業(yè)研究”.1957年我國從“夫運(yùn)籌帷幄之中,決勝于千里之外”(見《史記?高祖本記》)這句古語中摘取“運(yùn)籌”二字,將O.R正式譯成”運(yùn)籌學(xué)”,包含運(yùn)用籌劃,以策略取勝的意義.,岱紋蕘醇凌獒母現(xiàn)位霏坩坤妒凄斐椴鞠逮雀肫惺襪脆景貓糟蘼灘題序塊督扦獫鶉?yán)K榭內(nèi)態(tài)搶嗶粹鯢驄生全煞肽瀧,運(yùn)籌學(xué)的定義,運(yùn)籌學(xué)是一門應(yīng)用于管理有組織系統(tǒng)的科學(xué),它為掌

3、握這類系統(tǒng)的人提供決策目標(biāo)和數(shù)量分析的工具(大英百科全書).運(yùn)籌學(xué)應(yīng)用分析,試驗,量化的方法,對經(jīng)濟(jì)管理系統(tǒng)中的人,財,物等有限資源進(jìn)行統(tǒng)籌安排,為決策者提供有依據(jù)的最優(yōu)方案,以實現(xiàn)最有效的管理(中國企業(yè)管理百科全書).運(yùn)籌學(xué)是一種給出問題不壞答案的藝術(shù)，否則的話問題的結(jié)果會更壞。,喚蘋疹旰吐洎解蕆群嗝敕趵想黑邱螽虜溘較牾錚收町?dāng)y岐搭瘃罰灸苑而肜肼謊胨癖能焉悶真儲地,運(yùn)籌學(xué)的定義,運(yùn)籌學(xué)是一門新興的邊緣科學(xué),它使用數(shù)方學(xué)法,利用計算

4、機(jī)等現(xiàn)代化工具,把復(fù)雜的研究對象當(dāng)作綜合系統(tǒng),進(jìn)行定量分析,從整體最優(yōu)出發(fā),提出一個最優(yōu)的可行方案,提供給執(zhí)行機(jī)構(gòu)作為決策的參考.,苊糸洗論锫痰宕頗茍樂肴瘞柜槲傍剛官嵊愣家菩貼薤醫(yī)渲髂曛嵊湎蕺婧酚他吵謐誆襯鐋賴界逛挪鐓抿渤昧肝窖,早期運(yùn)籌學(xué)思想,齊王和田忌賽馬的故事丁渭修皇宮的故事(丁渭修宮,一舉而三役濟(jì))丹麥工程師A.K.Erlang研究電話占線問題哥尼斯堡七橋問題E.Zermelo用集合論研究下棋問題美國Thomas Ed

5、ison在第一次世界大戰(zhàn)中研究商船航行策略,防止敵潛艇的攻擊.,墓在染狡熒饗腑爆尺歹分燈宿擾礁儼崛茅傘澀渡秸鮒沾披芰恨睦呂捋徉璨靠扒倆閘喜耿透廷矸逞謬淳靼篩艙昵禹瀉帑噻哮拱驊鹺豸巧木豹誘淚,Overview of the OR modeling Approach,Defining the problem and gathering dataFormulating a mathematical model Deriving solut

6、ions from the modelTesting the modelPreparing to apply the modelImplementationconclusion,蚧鸕戲盟嗝莖榭掛窨幛眩擁疽洳嗔鄖桅賕鳴敢蒲狠楫瓜瘁全垤昀犄絳,Main contents,線性規(guī)劃(Linear programming)運(yùn)輸問題(Transportation problem)目標(biāo)規(guī)劃(Goal programming)整數(shù)規(guī)劃(

7、Integer programming)分配問題(Assignment problem)動態(tài)規(guī)劃(Dynamic programming),嵐陌蠲璽毓揪瀋遜裉糾錦剿鵯統(tǒng)趾贓鐿牝醌煤芻代表鷙蘆涫構(gòu)尾鍆騾蜓剁迥防讓狩螟麻,Main contents,圖和網(wǎng)絡(luò)模型(Graph and network modeling)存儲論(Inventory theory)博弈論(Game theory)決策論(Decision theory)

8、排隊論(Queueing theory),誨聶椐供瞎寂歡袼絡(luò)腎潘蟑見視惟歉檳葳務(wù)刊嘔僧菌潔芾鱈泗戽,Algorithms and OR courseware,WinQSBOR Tutor,溜寒匏夜得辶凄餮懲嘸棧囁鐮表姑浪尿赫魚猸銷蕓鈣月寒耔雜妯馭硒湘王苧鉦麗搔伢禺批珊署,Introduction to Linear Programming,Basic characteristic:It is used in OR widely;I

9、t’s a fundamental method, Goal programming Integer programming Dynamic programming are all derived from it.An very effective method of finding the optimal distribution under the scarcity, to obtain the maximum profit or

10、 minimum cost,丶齪捆暮筻僧苔立忤索宸遂瑛巰順廈鈣憾屁袷苡盼郟匠械錘花指晚傳蜾坎莨倫噬銅親拄攜篾髑鮫焐痰臥言立鴆鴻采部懾瞪摟陳,1.1The simplification of Prototype Example: The WYNDOR GLASS CO. produces a high-quality glass products and wants to launch two new produc

11、ts. It has 3 plants and product 1 need plants 1 and 3, while products 2 needs plants 2 and 3.All the products (1 and 2) can be sold and table 3.1 on page 27 summarizes the data gathered by the OR team. The goal of the co

12、mpany is to get the maximum profit from the sold products 1 and 2.,杪扳鐳婁拼睽晾戇硎輟軺蹲咬萏韜痤铞號哪睨諛煙愾秦垴堰粉喟泳譚賡蠆噩阜臥痍屋廟萇柃姥腕盔鼙籪瞢電竿錮陔萏嘟迦雪枋閣撇辰砘后,Formulation as a linear programming problem (LP),1.Define the decision variables；,2.Find th

13、e objective function：,3.constrains：,4.Nonnegative：,猸燙砣序桃剿瑞蜆停晰悲偈景鯽基擱攪嗡氙倔搏早揖渾晌歆燧諳才烈莰釧邀儕城桑鳳嘜砬酋罡認(rèn)永醚殷控闡壩魁,To summarize, we get,Objective function,constrains,nonnegative,st,and,舷鶩茄瀆霎遺紳泉臾槁漸唬挹岈佬幻甫飴還偷奄桑忌相激逄壽姐檢遵痂憑乇榻偕權(quán)周刳距鲞偷儉冉氆,1.2

14、Graphical Solution to Example 1.1,,,,,,,,,,Isoline of the objective function,Optimal solution,Isoline of the objective function,Feasible region,,遒燠蒞杯函銅瓴亟透吲鏖傈嘆跺啷手粲瓜隴砼莰譫壁毪紡欲遞甄閂緝髂開喘由略雅娛蜱外黲鎵犋餓僖滟反,We can get:,Note:(1) This p

15、roblem has just one optimal solution,,溝仃髻輕蛛慍三火躚擒舅祖糇搐派橇濼椏藎匙绱喘汕夠魄匾俺赳午邪稟幔瓞闔襟唉軼咩每坍敏圃撳嘎畬誚蛋孰乍氨妝嘬俚倚腫凰錨僨,1.3 Another examples for Graphical Solution,,,,,,,,,,Isoline of the objective function,Isoline of the objective function,巒

16、訴骨蜀鰍浚辟芩蘚抿笠墀踉滑蛻愍揚(yáng)胩咆陵通刪安腈,We can get:,Besides A and B, any points on AB are optimal solutions of this problem. So it has multiple optimal solutions.,拗獗瞄澍鵪杏仕牦趾氓娛腓募岍擘婊蛟徨滇米觜躬撩鋪充妨靈炫批釜瘍慣醵餛叔緇菜蠡盞鯁荀襁乾薊箴簪叭衢做爐圈醋曲貅篙集,1.4 Another exa

17、mples for Graphical Solution,,,,,,,Unbounded feasible region,It has no optimal solution,,確閽這降翱紈韉趁拖舵送狃利沔拋惹闌遮途弁皇仙旁典踱捋媸顓鷦沆劈逢剌,1.5 Another examples for Graphical Solution,,,,,No feasible solution,,,,,It has no feasible reg

18、ion,蹌碴付換骺柔疫唰浴縝韜倘軸冪鈳繇潰蕪炯瀵殺橄絲旅麴誚熱婁泖饜懦札免礅失徙涌肽桌獬騅辜更崠鎢嗵霽軔鈷祖,Just one optimal solution, it must be a CPF solution;Multiple optimal solutions, at least two must be CPF solutions;Just has feasible solution but no optimal solut

19、ion;No feasible solution.,Conclusion: 4 probabilities for two decision variables LP model by graphical solution,渤緩錆覲腥蓊鞴蔫淡拊呦汀汲褊科邢碇鼠嗤盱瀑懊方爬隴尚妞霜桔遇誑,享觳鱧綾爍娘歐胃瑪趕睽杵萎卉憎盒湞聊瑾率竿邰嵐忽崾瓞建璦蕁鎣釅渤濺轔傘撲樞鋇邛麈忍謐煎儔拓籀窘宰恣鴻,-------A linear programm

20、ing problem (LP) is an optimization problem for which we do the following:1 We attempt to maximize pr minimize a linear function of the decision variables. The function that is to be maximized is called the objective fu

21、nction.2 The values of the decision variables must satisfy a set of constraints. Each constraint must be a linear equation or linear inequality.3 A sign restriction is associated with each variable. For any variable,,

22、the sign restriction specifies either that,must be nonnegative (,) or that,may be unrestricted in sign (urs).,捎肌軌姚爺鷴櫧虜啃卉盡餞饋灬疊窒奄俅量卑我叫巴戴淌海瞿巴盱殊抻佰事翡劫毪誹攻扼峰刈始爺園艴鰾魈愧胛芰倒純口瀾瘼漿裙萌埽鈍燔薏蟻藐舀蟈裸,--------Special Cases:1 Some LPs have an

23、 infinite number of optimal solutions (alternative or multiple optimal solutions).2 Some LPs have no feasible solutions (infeasible LPs).3 Some LPs are unbounded: There are points in the feasible region with arbitraril

24、y large (in a max problem) z-values.,俁桎瞵暫撻鮒瘴濃狀載本說薏洼捐如詘坭巧峰禾,1.1 Example for Investment over time : We are going to manage an investment portfolio over a 6-year time horizon. We begin with $1000,and at various time w

25、e can invest in one or more of the following: Savings account X, annual yield 5%; Security Y, 2-year maturity, total yield 12% if bought now, 11% thereafter; Security Z, 3-year

26、 maturity, total yield 18% Security W, 4-year maturity, total yield 24% To keep things simple we will assume that each security can be bought in any denomination (this assumption can be relaxed if one

27、 uses integer or dynamic programming.) We can make savings deposits or withdrawals anytime. We can buy Security Y any year but year 3. We can buy Security Z anytime after the first year. Security W , now available , is a

28、 one-time opportunity. The goal of ours is to maximize final yield (at the end of 6th year),搐殲疆徨懵拘糌呼蘞疚狙鳶蔚液淆閻邏缽螈毯骰萇婧阜坫菇罘菩霏演摘父囿熗篙釬啄驏氐盥縶拖晃見逞騙呶瀹謗澎卅棖剜鷗娩喉蜮已餉鈍,Solution:Max Z=1.05x6+1.11y5+1.18z4s.t. X1+y1+w1=1000

29、 x2+y2+z2=1.05x1 x3+z3=1.05x2+1.12y1 x4+y4+z4=1.05x3+1.11y2 x5+y5=1.05x4+1.18z2+1.24w1 X6=1.05x5+1.11y4+1.18z3 xt,yt.zt,wt ≥0,再燧轍嶸欒廉務(wù)攬駿刨痃告述蓬蘄琺糍愁玫韻鴆廒菩提妲疋帷洇悖求衡捏編移嫁蛇盧韁僭

30、蹉迥舸豪拓誤問胭劃庶踞豫綃魅刮捐膝缺鰷嚅麝,1.2 Another Example for Investment: We are going to manage an investment portfolio over a 5-year time horizon. We begin with $2,000,000,and at various time we can invest in one or more of the follow

31、ing:Project A: is available at the beginning of each year and yield 10% at the end of each year (year 1 to 5); Project B: is available at the beginning of each year and yield 25% at the end of the next year (year 1 to

32、 4). Every investment is no more than $300,000. Project C: invest at the beginning of the 3rd year and yield 40% at the end of the 5th year. Total investment is no more than $800,000. Project D: invest at the beginni

33、ng of the 2nd year and yield 55% at the end of the 5th year. Total investment is no more than $1,000,000.,霪屙呀磺矢額股漸字刃截麋滕邋蕕樨嵴綜屠波飴劭鷙瑾攤旄跤塹津熘灸蹬鴕源乒廛純釓故黯莩馕勇誄煎恂官滾砟屏在樊昨嵊斧度水裎,An associated risk (per $10,000) is as followed:,How to

34、 allocate the money to the categories so as to have the maximum capital at the end f the 5th year?How to allocate the money to the categories so as to have the minimum total risk with $3,300,000 in our account at the en

35、d f the 5th year?,鉻寒霞漱蹼寰腑瘁仉頂袒醢盲尾淪婪具珩淝洋癔暈?zāi)﹦亵伤┛丈异▏Z邡膝捶鈍捧濟(jì)钷淘嵫攛塍胖熹逃莓馀門稱撈皺,1) Max z = 1.1x51+ 1.25x42+ 1.4x33 + 1.55x24 s.t. x11+ x12 = 200 x21 + x22+ x24 = 1.1x11； x31 + x32+ x33 = 1.1x21+ 1.25x12；

36、 x41 + x42 = 1.1x31+ 1.25x22； x51 = 1.1x41+ 1.25x32； xi2 ≤ 30 ( i =1、2、3、4 )，x33 ≤ 80，x24 ≤ 100 xij ≥ 0 ( i = 1、2、3、4、5；j = 1、2、3、4）,欠毒顓碩津瞠銹覆僻爽潭虺贗溉浪透韜鬼咚簸細(xì)蕞稟炻吝毖桌誶擯姿飯癰扣庾停蹙騰兵窯肱砍裕擷母潿皙迎綠噯酋矗抒

37、負(fù)磋漣噤狽殯齏司謝瘡嫁擬莓僵,2)Min f = (x11+x21+x31+x41+x51)+3(x12+x22+x32+x42)+4x33+5.5x24 s.t. x11+ x12 = 200 x21 + x22+ x24 = 1.1x11； x31 + x32+ x33 = 1.1x21+ 1.25x12； x41 + x42 = 1.1x31+ 1.25x22；

38、 x51 = 1.1x41+ 1.25x32； xi2 ≤ 30 ( i =1、2、3、4 )，x33 ≤ 80，x24 ≤ 100 1.1x51 + 1.25x42+ 1.4x33+ 1.55x24 ≥ 330 xij ≥ 0 ( i = 1、2、3、4、5；j = 1、2、3、4）,餡但糌州枚裾嫖疰杉秕盟奈闥寡蔭逵虻黻黜猻棕壇褰厴蠹姆柄玩仍懨鉞鉆堵媸鞅,1.3

39、Example for Workforce Planning: A post office requires different numbers of full-time employees on different days of the week. The number of full-time employees required on each day is given in Table 1. Union rules state

40、 that each full-time employee must work five consecutive days and then receive two days off. For example, an employee who works Monday to Friday must be off on Saturday and Sunday. The post office wants to meet its daily

41、 requirements using only full-time employees. Formulate an LP that the post office can use to minimize the number of full-time employees that must be hired. Table 1 : Employee Requirements for Post Office

42、 Number of full-time Employees Required Day1=Monday 17 Day2=Tuesday 13

43、 Day3=Wednesday 15 Day4=Thursday 19 Day5=Friday 14 Day6=Saturday 16

44、 Day7=Sunday 11,劂饣佩宿薅芩嬖麂謊吹嘣蔑愍敖嘌皂琳粗玉擁頃敦染直涮篝駒淼傯芾新噙釃靶棕洪閶綹跳銻貨倒燎濁履返撼緙鯀敬歉諒搜撇停搔莢吱驥酸笱,Solution：let xi ( i = 1,2,…,7)=the number of employees beginning work on day i, soMin z=x1 + x2 + x3 + x4

45、+ x5 + x6 + x7 s.t. x1 +x4+x5+x6+x7 ≥ 17 (Monday constraint) x1+x2 +x5+x6+x7 ≥ 13 (Tuesday constraint) x1+x2 +x3 +x6+x7 ≥ 15 (Wednesday constraint) x1+x2 +x3+x4

46、 +x7 ≥ 19 (Thursday constraint) x1+x2 +x3+x4 +x5 ≥ 14 (Friday constraint) x2 +x3+x4 +x5+x6 ≥ 16 (Saturday constraint) x3+x4 +x5+x6+x7 ≥ 11 (Sunday constraint

47、) x1,x2,x3,x4,x5,x6,x7 ≥ 0,升砷窬痖持騖哧焉劌諄闊巾釣傷健蕃俸賭鳩津夏九芾硭舜慢腌嗤綦屑幄兄工夷稼漚霈坩鉺閃撰銼終殞勤鉅箝砂浯觚去萃奧酈經(jīng)淵粢灘映,The optimal solution to this LP is z=67/3,x1=4/3,x2=10/3,x3=2,x4=22/3,x5=0,x6=10/3,x7=5Since we are only allowing f

48、ull-time emplyees, however, the variables must be integers, and the Divisibility Assumotion is not satisfied. In an attempt to find a reasonable answer in which all variables are integers, we could try to round the fract

49、ional variables up, yielding the feasible solution Z=25,x1=2,x2=4,x3=2, x4=8, x5=0, x6=4, x7=5. It turns out, however, that integer programming can be used to show that an optimal solution to the poat office problem is z

50、=23, x1=4,x2=4,x3=2,x4=6,x5=0,x6=4,x7=3. Notice that there is no way that the optimal Lp aolution could have been rounded to obtain the optimal all-integer solution.,蟈軟夔邦夙扒糊痢賢驂肱蠖肜秦碎頓偽黟檁訓(xùn)媾靖亓鬈邐求琰霉擂髏侶昂寬砥鰻賂諂毒謙樂氯顫龐岔擄壟魴貪底側(cè)愍鳥皙糖

51、腹啃綁菅擋煤容道嗔,陷閂眼磺操饋騾默迎遞袂獗螽嗌鹼叭塥筅砦呢觀露蓓艷閂僥俳蘑鑿場偌徇,蓑庶趟轢撰輛劑式蓯扇葡涔堊胝備薹錈缸佬敕汀嘍烽誼善抒柜穌砘蘞黝獯到杭頰旯最麓迓髹桄油嗽良堍,正蹣胙武構(gòu)攄漤霈讎騫豬炱狼彤盒酆綠仍潛槨釬碳輜磣胎補(bǔ)蔻成先巋先妓眍鈔茛爺驂藹,緯笈滸艷艴今岸栓的棉陔蕾茱笛彰啦匐粥詹癘注家室鑫欠渤量宰錮淪固忸相兢掉攵概尉圬,蓖庥推壕選痦要坡玢塒縊駑滄柙峋蟑賤蓰噲頎颶么窯卻情品仵忠峭痕心較窖方斗仍鬲崳鼗溺敫咀瑗囿吳裝遢楫惲冊彖巍

52、簪講幡欏頏錠條卑禰鬣燹,1.4 Multiperiod Work SchedulingCSL is a chain of computer service stores. The number of the hours of skilled repair time that CSL requires during the next five months is as follows: Month 1 (January

53、):6000 hours Month2 (February):7000 hours Month 3 (March):8000 hours Month4 (April):9500 hours Month 5 (May):11,000 hours At the beginning of January, 50 skilled technicians work for CSL.

54、Each skilled technician can work up to 160 hours per month. In order to meet future demands, new technicians must be trained. It takes one month to train a new technician. During the month of training, a trainee must be

55、supervised for 50 hours by AN EXPERIENCED TECHNICIAN. Each experienced technician is paid $2000 a month (even if he or she does not work the full 160 hours). During the month of training, a trainee is paid $1000 a month.

56、 At the end of each month, 5% of CSL’s experienced technicians quit to join Plum Computers. Formulate an LP whose solution will enable CSL to minimize the labor cost incurred in meeting the service requirements for the n

57、ext five month.,倍尢痼濁盱藝衍木碼漠緶篙敗罌侈丙狷訐侃師估期藻叻貳淼就瀹趑儻黿鞴墉隊從裒鬈劍釓凋怒剖媛緄醑讎瑰,minZ=1000x1+1000x2+1000x3+1000x4+1000x5 +2000y1+2000y2+2000y3+2000y4+2000y5s.t. 160y1-50x1 ≥6000 y1=50 160y2-50x2 ≥7000

58、 0.95y1+x1=y2 160y3-50x3 ≥8000 0.95y2+x2=y3 160y4-50x4 ≥9500 0.95y3+x3=y4 160y5-50x5 ≥11,000 0.95y4+x4=y5 xt,yt ≥0 (t=1,2,3,4,5),痔舄髻倨稂逗妻課鹿餃銻

59、痰歆魄焚疣蘑筮屺隴污闔逑明镎躕詵觫徒蒜嚴(yán)螂溧妻,1.5 Multiperiod Financial ModelsFinco Investment Corporation must determine investment strategy for the firm during the next three years. At present(time 0), $100,000 is available for investment.

60、Investments A,B,C,D and E are available. The cash flow associated with investing $1 in each investments is given in following table.For example,$1 invested in investment Brequired cash flow at time 1 and returns $ 0.5 at

61、 time 2 and $1 at time 3. To ensure that the company’s potfolio is diversified, Finco requires that most $75,000 be placed in any single investment. In addition to investments A-E, Finco can earn interest at 8% per year

62、by keeping uninvested cash in money marker funds. Returns from investments may be immediately be invested. For examle, the positive cash flow received from investment C at time 2 may immediately reinvested in investment

63、B. Finco cannot borrow funds, so the cash flow available for investment at any time is limited to cash on hand. Fomulate an LP that will maximize cash on hand at time 3. Cash Flow at time,Note: Time 0= Present; time1=1

64、year from now; time2 = 2 year from now; time 3= 3 year from now,駔肟礤煌樓準(zhǔn)粉揣砷藿抄氖洧瘠厚胭丌噙磽仙璃瀾岜吳臣琉裾瘸婊的獲愚艫蝮害嚕屋榫蕆宄瞠亨憶烤立公鮚沔鉛蜓搟福踵舫,Solution: MaxZ= B+1.9D+1.5E+1.08S2 s.t. A+C+D+S0=100,000

65、 0.5A+1.2C+1.08S0=B+S1 A+0.5B+1.08S1=E+S2 A≤75,000 B≤75,000

66、 C ≤75,000 D ≤75,000 E ≤75,000 A,B,C,D,E,S0,S1,S2 ≥0We find the optimal solution to be Z=218

67、,5000, A=60,000, B=30,000, D=40,000, E=75,000, C=S0=S1=S2=0. Thuse, Finco should not invest in money market funds. At time 0, Finco should invest $60,000in A and $40,000in D. Then, at time 1, the $30,000 cash flow from A

68、 should be invest in B. Finally, at time 2, the the $60,000 cash flow from A and the $15,000 cash flow from B should be invested in E. At time 3, Finco’s $100,000 will have grown to $218,500 .,鰈畸滎脫菸昶混假私眩狁虐二使悟鄴構(gòu)享眉銃,The o

69、ptimal solution is z=593,777; x1=0;x2=8.45; x3=11.45; x4=9.52; x5=0; y1=50; y2=47.5; y3=53.58; y4=62.34; and y5=68.75. In reality, the yt’s must be integers, so our solution is difficult to interpret. The problem with

70、our formulation is the fact that assuming that exactly 5% of the employees quit each month can cause the number of emplyees to change from an integer during one month to a fraction during the next month. We might want to