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# Introduction to Finite Mathematics with Applications MA 114

NCS

GPA 3.93

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This 56 page Class Notes was uploaded by Braeden Lind on Thursday October 15, 2015. The Class Notes belongs to MA 114 at North Carolina State University taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/223719/ma-114-north-carolina-state-university in Mathematics (M) at North Carolina State University.

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Date Created: 10/15/15

Vegetarian Diet Vern decides to adopt a vegetarian diet consisting of fruits grains and vegetables His minimum daily requirements are 14 units of protein 16 units of carbohydrates and 12 units of fiber Suppose a serving of fruits can supply him with 1 unit of protein 2 units of carbohydrates and 1 unit of fiber while a serving of grains provides 3 units of protein 2 units of carbohydrates and 3 units of fiber A serving of vegetables provides 4 units of protein 3 units of carbohydrates and 2 units of fiber If fruit costs 30 per serving grains cost 60 per serving and vegetables cost 70 per serving how many servings of each type of food should he eat per day in order to satisfy his daily food requirements at minimum cost r r Nlt servings fruit servings grain servings veggies servings fruit servings grain servings veggies Constraints X 3y 42 2 14 protein 2X 2y 32 2 16 carbos X 3y 22 2 12 fiber X 2 0 y 2 0 2 2 O X Y 2 X servings fruit y servings grain 2 servings veggies Constraints X 3y 42 2 14 protein 2X 2y 32 2 16 carbos X 3y 22 2 12 fiber X 2 0 y 2 0 2 2 O minimize C 30X 60y 702 X 3y 42 2 14 protein 2X 2y 32 2 16 carbos X 3y 22 2 12 fiber X 2 0 y 2 0 2 2 O minimize C 30X 60y 702 X3y4ZS14 2X 2y 32 S 16 X3y22312 maximize D 30X 60y 702 X3y4zS 14 2X 2y 32 S 16 X3yZZS 12 maximize D 30x 60y 702 X3y4zu14 2X2y3zv16 X3y22w12 30X60y7OZDO X3y4zu14 2X2y3zv16 X3y22w12 30X60y7OZDO x y 1 3 2 2 1 3 60 OOOHC ooolt gowoog H o o o D 14 16 12 O 14 16 6 l Legal choices for lst pivot element 100 30 2 0 1 2001042 306070000 1 41000 14 3 1 2 30100 16 l 12 0 1 3 2001 1 0 306070000 We ll use this one uva 1 Z 4 Y 3 14 12 0 0 1 1 0 0 0 2 0 0 1 30 60 70 0 0 0 l y z u v w D 2 1 0 6 1 0 0 0 8 2 0 1 0 4 30 25 0 15 0 1 240 Next pivot element y z u v w D 2 1 0 6 1 0 0 0 8 0 1 0 4 30 25 0 15 0 1 240 Possible pivot element y z u v w D 2 1 0 1 0 0 0 2 0 1 0 30 25 0 15 0 1 We ll use this one 240 X y z u v w D 0 0 2 1 0 1 0 2 1 3 2 0 0 1 0 12 0 4 1 0 1 2 0 8 0 30 10 0 0 30 1 360 Here s where we are after the next complete step The 5th column has been put into unit form We re at x12 y0 z0 but the 2 in the far right column tells us that we are not yet in the solution region X 0 1 0 0 yzuva 0 2100 2 3200 1012 4101 20 8 30 10 0 0 301 360 Choose next pivot element We ll use the 1 in column 6 l kox O O y z O 2 3 4 4 5 3O 50 3O l OOlt O ooobkg E O 2 O 14 O 12 1 420 VVhen cohunn 6is putintOlu tfornL Unsis the tableau We re at the point x14 y0 z0 There are no negative number above the 420 Hithelastcohunnand HHStEHSLE that we finally are in the solution region X y z u v w D 0 0 1 0 1 o 2 1 3 4 1 0 0 0 14 0 4 5 2 1 0 0 12 0 3O 50 3O 0 0 1 420 The problem now becomes like a standard problem We choose our next pivot element by taking the pivot column to be the one with the most negative number in the bottom row excluding the bottom right number 420 X y z u v w D O O 1 L1 0 39139 O 1 2 2 1 3 O 1 O 2 O 10 1 5 O 4 O E 1 3 O 7 O 30 O 5 O 25 1 370 After this step x10 y0 z1 We re in the solution region but we re not at the optimal solution w 12 10 370 30050251 0 Here s our next pivot element 5 7 A5 X11 16 MD 5 h 51 m Ely yzuvw 00 001 Finished z1 x475 y175 D3175 Minimum cost is C 3175 obtained by eating 475 servings of fruit 175 servings of grains and 1 serving of vegetables Final Example maximize P 2x y subject to constraints X y S 1 X y S 1 X 2 O y 2 O Final Example maximize P 2x y subject to constraints X y S 1 X y S 1 X 2 O y 2 0 Add slackvariables X yu1 xyv1 2X yPO Add slack variables X yu xyv1 2X yPO X y uv P 1 110 01 1 1 0101 2 1 O 010 lt5 1 2 O 1 100 2 3201 0 X y u v P 1 1 1 0 0 1 0 0 1 1 0 2 0 3 2 0 1 2 What s the problem here We re not done but there s no possible pivot element This is the way that the simplex method signals that the problem has no solution Example A company has stores in Boston and Chicago and sells items to men women and children Sometimes items that do not sell well are transferred from one store to another Each week at the Boston store 10 of the items are sold to men 10 are sold to women 20 are sold to children 20 are shipped to Chicago and the remaining 40 are kept in Boston The Chicago store sells 10 to men 20 to women 30 to children ships 10 to Boston and keeps the remaining 30 MWCBOCH 0000 1 1000 0 C001 BO CH 0 0 2 4 2 1 1 1 3 2 3 MWCBOCH M10000 W01000 c00100 BO11242 CH12313 What is the probability that an item in the Boston store will be sold within 3 weeks BO CH O O O O O O m 192 226 418 086 078 CH 158 299 457 039 047 Transition matrix raised to third power shows the 3step transition probabilities BO CH O O O O O O m 192 226 418 086 078 CH 158 299 457 039 047 Transition matrix raised to third power shows the 3step transition probabilities Probabability that item in Boston store is sold within 3 weeks 192 226 418 836 MWCBOCH 0000 1 M W 000 0 0 0 C001 3 4 2 1 MWCBOCH 0000 1 000 0 W 0 0 C001 3 4 2 1 175 5 Na Qr1 M VV C BO CH O O O O 1 M VV 0 O O O O O C O O 1 225 275 5 175 325 5 5 1 75 25 15 a er N M w c BO 225 275 5 CH 175 325 5 What is the probability that an item in the Boston store will eventually be sold to a woman EGPYREGHT 2mm My AVON n PAGE M W C BO 225 275 5 CH 175 325 5 What is the probability that an item in the Boston store will eventually be sold to a woman Answer 275 M W C BO 225 275 5 CH 175 325 5 On average how long will an item in the Chicago store be held in one of the stores before being sold M w c BO 225 275 5 CH 175 325 5 BO c BO 175 5 CH 25 15 On average how long will an item in the Chicago store be held in one of the stores before being sold M W C BO 225 275 5 CH 175 325 5 BO c BO 175 5 CH 25 15 On average how long will an item in the Chicago store be held in one of the stores before being sold Answer 25 15 175 If the company starts outwith half their items in Boston and half in Chicago how will these original items be distributed 3 weeks later COPYRIGHT 2006 by LAVON B PAGE If the company starts outwith half their items in Boston and half in Chicago how will these original items be distributed 3 weeks later M W C BO CH M 1 O O O O W O 1 O O O c 0 0 1 0 0 BO 192 226 418 086 078 CH 158 299 457 039 047 1 0 0 0 0 0 1 0 0 0 0 c 0 5 5 o o 1 o o 192 226 418 086 078 158 299 457 039 047 M W C BO CH p0T3 175 438 0625 0625 Fraction of original items that have been bought by women at the end of 3 weeks Notice Where there is an absorbing state a transition matrix cannot possibly be regular So we can t use this example to illustrate the steadystate probability distributions of Section 63 MWCBOCH M10000 W01000 c00100 BO11242 CH12313 Example Bob Alice and Carol are playing Frisbee Bob always throws to Alice and Alice always throws to Carol Carol throws to Bob 23 of the time and to Alice 13 of the time In the long run what percentage of the time do each of the players have the F sbee OWZD HOgt OOUJ OOI Ln COPYRIGHT 2006 by LAVON B PAGE saw u o pBB O 2 A DE 0 A pBB p2 O BA 0 98 0 pBB O p2 BA 0 DE 0 DE 0 A pBB O PEA 0 DE 0 DE 0 B C In 5 throws the frisbee can 0 from anyone to anyone T4B kDILwINKOII l 9 KDII lwll lwlN gt INmolwwINCcn w i uoh nwn s IanINwli I O KOIh 39 If Bob has the frisbee to start with it s impossible for him to have it again after it has been thrown 4 times T5 0 03 gt EIGKOIHKOILgt lwtolwtolhuj KOIhwlNKOII ln No matter who has the frisbee to start with it s possible for anyone to have it after it has been thrown 5 times This is a regular transition matrix ESPYREQM mm may LAVEEM Ei PAEE Math WebAssign Student Help Sheet 1 Log In You can do these assignments on any computer provided that you have Internet Access with Netscape Navigator 40 or higher or Internet Explorer 50 or higher or Mozilla or Safari If you are doing this from home go to step 3 If you have logged into the campus system before then your login ID and password is the same as before If this is your first time logging in then your login ID is generally the first letter of your first name lst letter of your middle name and the first 6 letters of your last name unless you have 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