Simple Bounds in the Dual

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1 Simple Bounds in the Dual Many poblems have simple bounds on pimal viables How do these show up in the dual? Also, what if we have simple bounds on the dual viables? Conside the following elastic LP: max z = c subject to In this LP, evey constaint is eally a goal 1 1 Ax l s x, s + x 2 + Objective function has ewds and penalties fo deviations The auxiliy viables e slacks (s 1 ) and supluses (s 2 ) s 1 c 2 u 0 s 1 s c 2 = 3 s b 2 Lesson 6-1, p. 1

2 The Dual of the Elastic LP The pimal bounds end up in the dual objective, and the pimal ewds/penalties become dual bounds min y = w1b + w2u w3l subject to w1 A + w2 w3 c2 w1 c3 w, w c 1 This is a useful model when: It is uncle what the RHS should be It is uncle if the RHS can even be achieved (FOOTSTOMP) You can estimate the feasible ange of the shadow pices Lesson 6-1, p. 2

3 Adding Constaints to an LP Suppose I have the following intege pogam: min subject to I employ the paye method (solve as an LP and hope the answe s integal) and get: x 1 = 4/5, x 2 = 8/5 Now what? z = 3x 3x x x , x + 4x + 2 x 2 + 2x , 4, o 3x o x and intege, s x x 2, s 2 s 1 s 2 = 0 = 4 4 Lesson 6-1, p. 3

4 Adding a Cut I will now do something vey stange; I add the following constaint to the model: s1 + s2, o - s1 s2 + s3 =, s This is called a Gomoy dual factional cut What exactly is getting cut? We will touch on this moe in the IP pt of the couse Now, do we want to solve the poblem all ove again? Seems like we could do some sot of estt Howeve, adding this constaint will make the poblem infeasible Lesson 6-1, p. 4

5 Gaphical Depiction of the Cut 5 feasible points 4 Feasible Region cut 3 x2 factional LP soln x1 Lesson 6-1, p. 5

6 Adding the Constaint to the Tableau Hee s the LP tableau at optimality (via LINDO): Row z x1 x2 s1 s2 RHS BV /5-9/5 44/5 z /5 1/5 4/5 x /5-3/5 8/5 x2 note since this is a min poblem Hee s the new tableau with the constaint, slack s 3 : Row z x1 x2 s1 s2 s3 RHS BV /5-9/5 0 44/5 z /5 1/5 0 4/5 x /5-3/5 0 8/5 x /5-2/5 1-3/5 s3 Is this pimal feasible? Dual feasible? Lesson 6-1, p. 6

7 Intoduction to Dual Simplex The tableau is dual feasible Adding a ow to the dual is the same as adding a column to the pimal Can you make the pimal infeasible by adding moe viables? Leads to an altenative scheme, called dual simplex Discoveed by C. E. Lemke in 1954 (Lemke was Geoge Dantzig s fist doctoal student) Iteates among dual feasible solutions in a pimal tableau Impovements in dual simplex e esponsible in damatic impovements in LP solve times in the 1990 s Moe impotantly, a key method fo adding constaints in intege pogamming Lesson 6-1, p. 7

8 Pivoting in Dual Simplex This method is a tanspose of pimal simplex The pivot ow is the most negative RHS We only pivot on columns with negative coefficients The atio test is computed using the objective function ow; take the atio with the smallest absolute value Example: Row z x1 x2 s1 s2 s3 RHS BV /5-9/5 0 44/5 z /5 1/5 0 4/5 x /5-3/5 0 8/5 x /5-2/5 1-3/5 s3 atio = 2 atio = 9/2 Lesson 6-1, p. 8

9 Dual Simplex Temination Dual simplex finishes when the tableau is pimal feasible Recall that we stted, and stay, dual feasible If both pimal and dual e feasible, then whee e we? Row opeations e exactly the same in dual simplex Once you pick a pivot element, you get a 1 thee, and 0 s in the est of the column Hee s the tableau afte the pivot: Row z x1 x2 s1 s2 s3 RHS BV z x x s1 It s optimal, and intege Lesson 6-1, p. 9

10 Dual Simplex as a Solution Method Conside the stting tableau fo the same poblem: Row z x1 x2 s1 s2 RHS BV z x x2 We can t do pimal simplex; no BFS, need Phase I Row z x1 x2 s1 s2 RHS BV z x x2 This equivalent tableau, howeve, is dual feasible; we can do dual simplex immediately Lesson 6-1, p. 10

11 The Pivots Row z x1 x2 s1 s2 RHS BV z s s2 Row z x1 x2 s1 s2 RHS BV z 1 1 1/3-1/3 0 4/3 x /3-1/3 1-8/3 s2 Row z x1 x2 s1 s2 RHS BV /5-9/5 44/5 z /5 1/5 4/5 x /5-3/5 8/5 x2 Lesson 6-1, p. 11

12 Compehensive Example This is a small poblem Intended to show the entie pocess Initial poblem statement Fist fomulations Fist solutions Refomulations and modifications Subsequent solutions Sensitivity analysis Typical stumbling blocks Lesson 6-2, p. 1

13 The Situation A goup of investos wants to stt a small passenge ailine opeation The ea they e tgeting is cuently only seved by inconvenient hub-and-spoke outes They believe they can compete and not get cushed in a pice w; specialize in chtes They have a oute stuctue and can lease vious aicaft The need to schedule thei outes They call you in to assist Afte some convesation, you believe you can model the poblem You e sent off to gathe elevant data Lesson 6-2, p. 2

14 Initial Infomation You meet with othes involved in the new company Most e iitated an outside has been bought in Coopeation is gudging; management has to theaten one goup (the mket foecastes) to get them to talk to you Hee s the initial infomation The ailine wants to cove 5 outes They have a foecast fo demand on each oute They have leased 4 diffeent aicaft types Tentative opeating costs ($/ac/oute) e available fo each aicaft type The pax capacity of each aicaft is known Lesson 6-2, p. 3

15 What s the Objective and the Constaints? Minimize oveall cost? Only costs we have e opeating (mginal) costs Company claims to have fixed costs in hand, so you don t have to woy about them Othe questions you might ask Does it matte whethe we have multiple aicaft types? (no, all lease, with contact maintenance) Can we get diffeent aicaft configuations? (No) Ae thee limits on the numbe of aicaft available (No, they don t think so) Does all demand have to be met? (Yes) Is thee a maximum opeating cost? (No but they hadn t consideed this yet) Lesson 6-2, p. 4

16 You Initial Fomulation Detemine the aicaft mix that: Minimizes total opeating cost, and Coves all demand Management agees Indicies: a: aicaft types : outes Data DEMAND = passenges flying oute pe month (100 s) COST = $1K/month to opeate aicaft type a on oute CAP a = maximum monthly capacity of aicaft type a (100 s) Lesson 6-2, p. 5

17 Fomulation, cont d Viables aca = # of aicaft a assigned to oute pe day Objective and Constaints min z = subject to a aca CAP a, COST * aca 0 fo all a, aca DEMAND fo all Lesson 6-2, p. 6

18 MPL Code INDEX a := (ac1, ac2, ac3, ac4); { aicaft types } := (1, 2, 3, 4, 5); { outes } DATA COST[a,] := (18,21,18,16,10, { cost of aicaft a on oute, $1k/month } 0,15,16,14,9, 0,10,0,9,6, 17,16,17,15,10); { NOTE: 0 cost means can't fly that oute! } CAP[a,] := (16,15,28,23,81, { capacity of aicaft a on oute, 100 s/month } 0,10,14,15,57, 0,5,0,7,29, 9,11,22,17,55); { NOTE: 0 capacity means can't fly that oute! } DEMAND[] := (253,120,180,80,600); { demand pe month (100 s) on oute } DECISION VARIABLES aca[a,]; { numbe of aicaft a flying on oute } MODEL MIN totexpcost = SUM(a,: COST[a,]*aca[a,]); SUBJECT TO demeq[]: { demand constaints } SUM(a: CAP[a,]*aca[a,]) > DEMAND[] ; END Lesson 6-2, p. 7

19 Initial Solution Initial solution: use nothing but aicaft type 1 Optimal cost: $698K/month Assignment data: VARIABLE aca[a,] : a Activity Reduced Cost ac ac ac ac ac What do you think the optimal intege solution is? Why? Change MPL code as follows to see: INTEGER VARIABLES aca[a,]; { numbe of aicaft a flying on oute } Lesson 6-2, p. 8

20 Intege Solution and Fist Revisions The best intege solution is NOT to use all AC 1: Optimal cost: $720K/month Aicaft assignments: a Activity Reduced Cost ac ac ac ac ac ac ac ac You pesent this to management They say we fogot; we can t get that many of AC 1 It tuns out thee s limits on availability of all the aicaft types Lesson 6-2, p. 9

21 Model Adjustments Data AVAIL a = numbe of aicaft a available New Model min z = subject to a aca CAP aca a, COST * aca AVAIL a 0 fo all a, aca DEMAND fo all a fo all Lesson 6-2, p. 10

22 Fist Model Death Hee s the MPL changes: AVAIL[a] := (10,19,25,15); { aicaft availability } acavail[a]: { aicaft availability } SUM(: aca[a,]) < AVAIL[a]; You un the model, and MPL says intege infeasible What happened? Change it back to an LP, see if it solves; it s still infeasible Now what? Solve a diffeent poblem Minimize the unmet demand, given the aicaft availability See if you can figue out what combinations e causing touble Lesson 6-2, p. 11

23 The Next Model - Whee Ae We Shot? Hee s the new fomulation; minimize unmet demand min z = subject to a aca ( CAP * aca ) aca unmet unmet AVAIL 0 fo all a, 0 fo all + unmet a fo all a DEMAND fo all Is this ight? Why? Lesson 6-2, p. 12

24 The Answes LP esults - close, but can t satisfy Route 1 VARIABLE unmet[] : Activity Reduced Cost CONSTRAINT acavail[a] : a Slack Shadow Pice ac ac ac ac Which aicaft type do we pobably want moe of? Note that the intege answe is somewhat wose: VARIABLE unmet[] : Activity Reduced Cost Lesson 6-2, p. 13

25 Negotiations with the Custome Mketing goup is upset; claims answe is wong They show the following table: Aicaft Capacity Route AC 1 AC 2 AC 3 AC 4 Max AC Avail How would you gue you way out of this? But, suppose you win Management says, get with mketing and figue this out Lesson 6-2, p. 14

26 Adding a Bumping Cost Mketing says, we can bump people at a pice Data: BPCOST = $K lost pe 100 passenges bumped on oute Viable: bumped = passenges bumped on oute (100 s) New model: min z = subject to a aca CAP aca 0 a, COST * aca AVAIL + a aca bumped fo all a + fo all a, ; bumped BPCOST DEMAND 0 fo all * bumped fo all Lesson 6-2, p. 15

27 The New Solution LP solution: z = $999K/month VARIABLE bumped[] : Activity Reduced Cost CONSTRAINT acavail[a] : a Slack Shadow Pice ac ac ac ac Intege solution: z = $1012K/month VARIABLE bumped[] : Activity Reduced Cost CONSTRAINT acavail[a] : a Slack Shadow Pice ac ac ac ac Lesson 6-2, p. 16

28 The Management Responds Leadeship doesn t like the answe Almost all the bumping occus in oute 5 Wants the isk of bumping spead out moe evenly acoss outes Now what? Fist, check fo multiple optima in the solution May be an altenative that is cost optimal, but speads out bumps But, thee e none in the LP solution This means that speading out bumping will cost moe Note, howeve, that this is based on expected demand Mketing says foecasts pobably good to within 5% Implies that total costs have about 5% accuacy as well This is how we will ty to spead out bumping Lesson 6-2, p. 17

29 New Model to Spead Out Bumping Pevious objective function now becomes a constaint We add a new viable, maxbump Hee s the new model: NOTE min z = maxbump subject to a, maxbump a aca COST CAP aca 0 aca * aca bumped AVAIL fo all + bumped a + fo all a BPCOST fo all a, ; bumped * bumped DEMAND 0 fo all 1.05* fo all z oiginal Lesson 6-2, p. 18

30 And, What Happens? This solution does indeed spead out bumping: VARIABLE bumped[] : Activity Reduced Cost This does not eally make things equitable Demand diffes on each oute Management wants an equal chance of bumping on each oute Need to ecast maxbump as a popotion of oute demand Lesson 6-2, p. 19

31 Bumping As a Popotion This solution has the optimal maxbump at 2.01% New bumping esults: VARIABLE bumped[] : Hee s how it vies by total cost: Activity Reduced Cost Max Bumping % $999K $1009K $1019K $1029K $1039K $1049K Expected Total Cost Lesson 6-2, p. 20

32 Demand Scutiny Howeve, this whole execise causes scutiny of demand foecast Management to mketing: Whee the did this come fom? Mketing digs though the files, comes up with the following speadsheet data Lesson 6-2, p. 21

33 The Oiginal Demand Data Hee s whee the expected demand was deived fom: DEMAND LIKELIHOOD Demand State Route Demand State Route Looks like it s time fo a ecouse model Lesson 6-2, p. 22

34 Extacting Scenios Note that this data is by oute The joint distibution of demand is uncle Seems easonable, though, that if demand is high on one oute, it is pobably also high on anothe We decide to use 6 scenios: scenio s1 s2 s3 s4 s5 s6 pobability oute oute oute oute oute Lesson 6-2, p. 23

35 The New Scenio Model Go back to minimizing cost, but add: Index s = scenio Data SPROB s = pobability of scenio s Add s index to demand data and bumping viables New model min z = subject to a aca CAP aca 0 a, COST * aca AVAIL aca + bumped a s fo all a + fo all a, ; bumped, s s SPROB DEMAND 0 fo all, s s * BPCOST s fo all, s * bumped s Lesson 6-2, p. 24

36 As You Would Expect... This answe is nowhee ne as osy Total expected cost: $1562K Opeating cost: $887K Bumping cost: $678K One oute/scenio combo has 26,000 pax/month unmet demand Convesation ensues Fist question: what if the oute data is all independent? Second question: If the 6-scenio model is valid, what s the minimum numbe of aicaft needed to ensue a less than 10% chance of bumping on any oute? Lesson 6-2, p. 25

37 Assume the Route Demands e Independent Model mods: Index d = demand state (1-5) Data DPROB d = pobability of demand state d on oute Data DDEM d = demand on oute in demand state d Viable bumped d = numbe bumped fom oute in demand state d New Model min aca z subject to a CAP aca = 0 a, COST * aca AVAIL aca + bumped a fo all a + d fo all a, ; bumped, d d DPROB DDEM 0 d d fo all, d * BPCOST fo all, d * bumped d Lesson 6-2, p. 26

38 Answe to the Independent Demand Case Total expected cost: $1566K $883K opeating cost $683K bumping cost Bumping statistics simil to scenio case Intege answe: $1580K (vey simil) Inteesting esult: total expected cost is slightly highe than the scenio case Lesson 6-2, p. 27

39 Homewok Answe the second question Fomulate and solve in MPL the case that minimizes the numbe of aicaft equied to get less than 10% bumping fo any oute and scenio Tun in sepate fomulation (witten out, NOT MPL code) Povide MPL code fo new model Also, investigate sensitivity of the solution fo the ange 5-15% Changes in total costs Changes in optimal fleet mixes I have povided MPL code fo the fist question; wok fom thee Also: p. 335: 2a Lesson 6-2, p. 28

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