MASSACHUSETTS INSTITUTE OF TECHNOLOGY 15.053 Optimizatio Methods i Maagemet Sciece (Sprig 2007) Problem Set 3 Due March 1 st, 2007 at 4:30 pm You will eed 116 poits out of 137 to receive a grade of 5. Problem 1: Britey s New Life (30 Poits) This problem allows you to practice the simplex algorithm ad to review some cocepts from geometry. Sice splittig with Kevi, Britey is slowly learig the harsh realties of havig to maage o her ow. She o loger has Kevi to help her with takig care of the kids, cleaig the house, maagig the garde, doig laudry, ad most importatly solvig her liear programs. Havig ot take a liear programmig class, she turs to you for help. Britey s goal is to maximize her utility poits. Curretly Britey eeds to divide her hours each day betwee activities icludig: partyig with Paris ad Lidsey, cuttig her hair, ad takig care of her childre. Accordig to her mother, each hour of partyig is worth 5 utility poits; each hour she speds o her hair is worth 3 utility poits; ad each hour takig care of the kids is worth 1 utility poit. Due to the fact she has ot toured or sold a CD i more the 5 years, Britey caot sped more the $6 i a give day. Partyig costs Britey $1 a hour. (Fortuately, Paris pays for all her driks. Hair stylig costs $1 a hour, ad carig for the kids costs $3 a hour. (She actually watches while she pays a local teeager to care for them.) I a give day, Britey has at most 15 uits of eergy to sped. Partyig for oe hour takes up 6 eergy uits; hair stylig for a hour takes up 3 eergy uits; ad carig for the kids for oe hour takes up 6 eergy uits. We assume that the total time spet o these activities ca be at most 24 hours. Ay amout of time ot sped o these activities, she speds sleepig. Usig Britey s cosideratios above formulate a liear program that will optimally allocate the umber of hours to partyig, hair stylig, ad carig for the kids. (Hit: Your LP should have exactly three variables) Page 1 of 9
Covert the LP i Part A to stadard form by addig variables. For each variable you added write a setece about what that variable represets. Is the costrait that models the total amout of time spet i a day o the three activities must be 24 hours or less redudat. If it is explai why ad remove it, if ot explai why ad leave it i the formulatio. Idetify a feasible solutio where all basic variables are slack variables. Part E: Usig your solutio i part D fill out the first Simplex Tableau. Part F: Solve the problem usig the Simplex Method. For each iteratio write dow the startig Tableau ad idicate the pivot elemet. Part G: Does Britey have multiple optimal ways to divide her time? Please explai your aswer. Part H: I the form of a lie segmet express all of the optimal solutios to Britey s problem. Problem 2: Turkey Tim ad the Simplex Tableau (25 Poits) This problem is meat to help defie some of the coditios that arise whe ruig the simplex method o a stadard form problem. Turkey Tim was tryig to fid the optimal solutio to a maximizatio problem i decisio variables x j 0 (for j = 1, 2,,7). After performig several pivots, he came up with a tableau similar to the oe below. However, Turkey Tim was watchig the West crush the East i the NBA all star game while workig o the problem, ad he smudged some of the elemets i the simplex tableau with buttered popcor. Ollie replaced these questioable elemets i the tableau with the letters (variables) A,B,C,D,E, ad F hopig that you (a bright studet takig 15.053) will be able to explai to Tim iformatio about the values that these letters ca represet. Cosider this liear programmig problem i caoical form (depedig o F), described i terms of the followig iitial tableau: Page 2 of 9
z x 1 x 2 x 3 x 4 x 5 x 6 x 7 RHS 1 0 0 0 A 3 B C 1 0 0 1 0 D 1 0 3 F 0 0 0 1-2 2 E -1 2 0 1 0 0 0-1 2 1 3 What is the curret objective value? Which of the variables are curretly i the basis? What is the curret basic feasible solutio? You may express your aswer i terms of the letters A to F if eeded. Parts D-G: for each statemet below, give sufficiet coditios o all six ukows A, B, C, D, E, F such that the statemet is true. If there is othig that ca be doe to make the statemet true please explai why. If there are several ways of accomplishig this, please state oly oe. The correspodig basic solutio is feasible but ot optimal Part E: The correspodig basic solutio is feasible, ad there is a choice of the eterig variable so that the first simplex iteratio idicates that the optimal cost is ifiity (ubouded from above). Part F: The correspodig basic solutio is feasible, x 6 is a cadidate for eterig the basis. Moreover, if x 6 is the eterig variable, the x 3 leaves the basis. Part G: The correspodig basic solutio is feasible, x 7 is a cadidate for eterig the basis. Moreover, if x 7 eters the basis, the the solutio ad the objective value remai uchaged after the pivot. This is called a degeerate BFS ad a degeerate pivot, ad we will lear more about these types of BFS s ext week Page 3 of 9
Part H: Suppose A=4, B=0, C=-2 ad F=3. Perform a pivot usig the simplex algorithm. Idicate the variable that eters the basis, the variable that leaves the basis, ad the total chage i profit. Also, write the resultig tableau. (The aswer for some of the coefficiets of the tableau will be i terms of D ad E, which were ot specified.) Why is the Simplex Method ot so simple at first? Relax, Tim. Everythig is difficult at first. After a bit of practice the Simplex Method will be as easy as carvig a Turkey. Bur!!!! Problem 3: Simplex Paths (25 Poits) This problem is meat to help you develop a uderstadig of how the simplex method moves form corer to corer. It should give some isights ito pivotig. Our four frieds: Ollie the Owl, Tim the Turkey, Cleaver the Beaver, ad Nooz the Fox are all performig the simplex method with the feasible regio show below, all usig the same objective fuctio. Labels of the corer poits A: (0,0,0) F: (0,1,1) B: (1,0,0) G: (1,0,1) C: (0,1,0) H: ( 1,1/2/,1) D: (0,0,1) I: (1,1,1/2) E: (1,1,0) J: (1/2,1,1) Poit (0, 0, 0) is ot visible here. (0, 0, 1) (1, 0, 1) (1,.5, 1) (.5, 1, 1) (0, 1, 1) (1, 0, 0) (1, 1,.5 ) (1, 1, 0) (0, 1, 0) Page 4 of 9
Tim suggests the followig the followig pairs of corer poits could result from successive iteratios. (A, B) (B, D) (E, H) (A, I) For each pair determie if Tim is correct ad explai your reasoig. Suppose each character starts the simplex method at Poit A. listed below are the paths each foud by ruig the algorithm o the objective fuctio, resultig i the optimal solutio H. Ollie s Path: A B G H Cleaver s Path: A E I H Nooz s Path: A C E B A D G H For each of the above, determie from the iformatio give if the path could have resulted whe ruig the simplex method o the problem. If ot explai why ot. Now suppose each character decides to use a differet objective fuctio. They will start ruig the simplex method at poit A. Their objectives are listed below: Tim s Objective: Mi z = x 1 + 2x 2 3x 3 Ollie s Objective Mi z = 5x 1 2x 2 4x 3 Nooz s Objective Mi z = 2x 1 7x 2 2x 3 For each of the followig objective fuctios determie which variable eters the basis as the first iteratio, ad what the ext corer poit will be. What is the improvemet i the objective fuctio. (If there are differet choices of a eterig variable, choose ay of the oes that are possible.) Does the feasible regio i the picture represet a problem i stadard form, or does it represet a problem with iequality costraits. How may iequalities are there, other tha oegativity costraits. Please explai your aswer. Problem 4: Variables that come ad Go (21 Poits) Page 5 of 9
This problem is meat to build isight ito how the simplex method works ad to coect the mathematics behid the simplex method with the geometry of the simplex method. Suppose we are solvig a miimizatio problem ad the variable x 3 is about to leave the basis. What ca you say about the z-row coefficiet of x 3 prior to the pivot? After the pivot is carried out, ca the z-row coefficiet of x 3 ca be less the zero. Explai your aswer. Is it true that a variable that has just left the basis ca ot reeter o the very ext iteratio? Briefly explai. Commet: a variable that has left the basis ca reeter o ay subsequet iteratio after the first iteratio. For a liear program with variables, how may times ca a variable (say x 1 ) eter ad leave the basis? (Give oe aswer, ad you do ot eed to justify it.) a. at most 1 time. b. at most 2 times. c. possibly more tha 2-1 times. Part E: What is the maximum umber of corer poits (bfs s) for a liear program with 3 liearly idepedet equality costraits ad 5 variables as well as o-egativity costraits? Problem 5: Plaig for Expressjet Airlies (15 Poits) The idea behid the problem is to cotiue to build your skills at formulatig liear programs ad to practice abstractio. Page 6 of 9
Expressjet Airlies will o loger operate flights exclusively for Cotietal Airlies but also for their ow idepedet ew airlies Express Jet. Maagemet believes that they will eed the followig umber of pilots over the ext five years. Year 1: 60 Pilots Year 2: 70 Pilots Year 3: 50 Pilots Year 4: 65 Pilots Year 5: 75 Pilots At the begiig of each year, the compay must decide how may pilots should be fired or hired. It costs Expressjet $4000 to hire a pilot ad $2000 to fire a pilot. A pilot s salary is $10,000 per year, as they are all flyig o UROP wages. At the begiig of year 1, Expressjet has 50 pilots. A pilot hired at the begiig of a year may be used to meet the curret year s requiremets ad is paid full salary for the curret year. It is feasible (but expesive) to have too may pilots Formulate a LP to miimize Expressjets s labor costs over the ext five years. Defie a set of variables ad data poits ad formulate the abstracted versio of the problem. Be sure to clearly idicate the costats ad variables you defie. (For example, let h j be the cost of hirig a pilot i year j; let f j be the cost of firig a pilot i year j; ad make up otatio for ay other terms you eed.) Problem 6: The Kapsack Problem (20 Poits) This problem is meat to help you practice abstract formulatio ad to build isight ito the feasible regios of stadard form problems with a sigle costrait. New best frieds, Lace Armstrog ad Matthew McCoaughey have decided to go o a campig trip to Napa Valley to sped some quality time aloe together. (Nothig is implied by this statemet.). They ca pick betwee two types of food: Lea Cuisie ad Hugry Ma diers to put i their back packs. Lea Cuisies weigh a 1 pouds ad have a utility of c 1. Hugry ma diers weight a 2 pouds ad have a utility of c 2. The kapsack they will carry ca hold at most b pouds. Page 7 of 9
Assumig fractioal items are allowed, formulate a liear program that will maximize the utility of Lace ad Matt s backpack. Is it true that if: c 2 c 1 a 2 a 1 b The the pair ca maximize utility by fillig the kapsack with Hugry Ma diers. a 2 Briefly, explai your aswer. (Please ote that a formal argumet or proof is ot ecessary.) Suppose ow that there are items. Each item i has weight a i ad utility c i. Formulate the abstracted versio of part A. What is the optimal solutio to part C i terms of the coefficiet vectors a ad b? (HINT: geeralize what you leared i part b.) Part E: Give a set of coditios that eed to be preset so that multiple optimal solutios exist. (HINT: your aswer should follow from your aswer to part D.) Challege Problem C: (10 Poits) Cosider the followig LP: Page 8 of 9
Maximize subject to j i =1 cx j aij x j = b i for i = 1 to m i =1 x j 0 for j = 1 to Suppose that x, x 1 2,..., x is a optimal solutio. Suppose that c1 is decreased ad all of the * * * other c s are kept the same, ad that x1, x2,..., x is the ew optimal solutio. Show that * x 1 x 1. Part B (5 poits): Suppose that 0 is a corer poit for the followig feasible regio. ij i =1 ax 0 for i = 1 to m j Show that it is the oly corer poit. A formal proof is ot eeded Page 9 of 9