MATH 445/545 Homework 1: Due February 11th, 2016

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1 MATH 445/545 Homework 1: Due February 11th, 2016 Answer the following questions Please type your solutions and include the questions and all graphics if needed with the solution 1 A business executive has the option to invest money in two plans: Plan A guarantees that each dollar invested will earn $070 a year later, and plane B guarantees that each dollar invested will earn $2 after 2 years In plan A, investments can be made annually, and in plan B investments are allowed for periods that are multiples of 2 years only How should the executive invest $100,000 to maximize the earnings at the end of 3 years? Use Lindo to solve your linear program Let x i,j be the amount of money invested in year i for a term of j years Then The linear program becomes: max z = 17x 2,1 + 3x 1,2 st x 0,1 + x 0, x 1,1 + x 1,2 17x 0,1 x 2,1 17x 1,1 + 3x 0,2 with x i,j 0 for all i and j combinations Solving the problem with lindo gives: max 17x21 + 3x12 st x01 + x02 <= x11 + x12-17x01 <= 0-17x11-3x02 + x21 <= 0 ==== Output ==== Global optimal solution found Objective value: Infeasibilities: Total solver iterations: 0 Elapsed runtime seconds: 485 Model Class: LP Total variables: 5 Nonlinear variables: 0

2 Integer variables: 0 Total constraints: 4 Nonlinear constraints: 0 Total nonzeros: 10 Nonlinear nonzeros: 0 Variable Value Reduced Cost X X X X X Row Slack or Surplus Dual Price The optimal value of the investment plans is $510,000 investing all money in plan A for a year and then investing the money in the two year plan B 2 Model each of the following decision making situations as a linear program (LP) For each LP clearly define your decision variables, objective function and constraints (a) Suppose that the latest scientific studies indicate that cattle need certain amounts b 1,, b m of nutrients N 1,, N m respectively More over, these nutrients are currently found in n commercial feed materials F 1,, F n as indicated by coefficients a ij, i = 1,, m, j = 1,, n, that denote the number of units of nutrientn i per pound of feed material F j Each pound of F j costs the rancher c j dollars How can a rancher supply these minimal nutrient requirements to his prize bull while minimizing his feed bill? Linear Program: The goal of the problem is to minimize the cost of the bulls food while maintaining the bulls healthy diet

3 Let x j be the number of pounds that the farmer should purchase of commercial feed F j where, j = 1,, n Thus the farmer wishes to minimize the following cost equation (where z denotes the farmers total cost): Minimize: z = c 1 x 1 + c 2 x c n x n = n c j x j j=1 In order to meet the nutrient requirements the farmer needs to impose the following constrictions on his cost equation: a 11 x 1 + a 12 x 2 + a 13 x a 1n x n b 1 a 21 x 1 + a 22 x 2 + a 23 x a 2n x n b 2 a 31 x 1 + a 32 x 2 + a 33 x a 3n x n b 3 a m1 x 1 + a m2 x 2 + a m3 x a mn x n b m where x j 0, j = 1,, n Let x = [x 1, x 2,, x n ] T, and c = [c 1, c 2,, c n ] T We can also denote [b 1, b 2,, b m ] T by b, and coefficient matrix a 11 a 12 a 13 a 1n a 21 a 22 a 23 a 2n a 31 a 32 a 33 a 3n by A a m1 a m2 a m3 a mn Thus the farmer is minimizing z = c T x subject to, Ax b where x j 0, j = 1,, n (b) Suppose that a small chemical company produces one chemical product which it stores in m storage facilities and sells in n retail outlets The storage facilities are located in different communities The shipping costs from storage facility F i to retail outlet O j is c ij dollars per unit of the chemical product On a given day, each outlet O j requires exactly d j units of the product Moreover, each facility F i has s i units of the product available for shipment The problem is to determine how many units of the product should be shipped from each storage facility to each outlet in order to minimize the total daily shipping costs Page 3

4 Linear Program: The goal of the problem is to minimize the cost of product shipping while maintaining product inventory and not overdrawing from supply Let x ij be the number of units of chemical shipped from facility F i to retail outlet O j, where i {1,, m}, and j {1,, n} Let z denote the total cost of shipping Thus, the chemical company wishes to optimize the following objective function, m n Minimize: z = c ij x ij In minimizing the above equation the company must meet the following constraints: x 11 + x x m1 = m i=1 x i1 = d 1 x 12 + x x m2 = m i=1 x i2 = d 2 x 1n + x 2n + + x mn = m i=1 x in = d n That is each outletj requires d j of chemical where j {1,, n} Each Facility i also only has a limited supply of the chemical it can ship s i where i {1,, m} Thus, the problem has the constrictions: i=1 j=1 x 11 + x x 1n = n j=1 x 1j s 1 x 21 + x x 2n = n j=1 x 2j s 2 x m1 + x m2 + + x mn = n j=1 x mj s m where x ij 0, i {1,, m} and j {1,, n} The notation of the problem can be simplified if matrix notation is used: x 11 x 12 x 13 x 1n x 21 x 22 x 23 x 2n = X x m1 x m2 x m3 x mn d 1 s 1 d 2 s 2 d n = d, and s m = s Denote a column of 1 s that is k rows long by 1 k and he linear program can now take the following form: m n Minimize: z = c ij x ij i=1 j=1 Page 4

5 Subject to: X1 n s X T 1 m = d where all elements of X are greator than or equal to zero (c) An investor has decided to invest a total of $50,000 among three investment opportunities: savings certificates, municipal bonds, and stocks The annual return in each investment is estimated to be 7%, 9%, and 14%, respectively The investor does not intend to invest her annual interest returns (that is, she plans to use the interest to finance her desire to travel) She would like to invest a minimum of $10,000 in bonds Also, the investment in stocks should not exceed the combined total investment in bonds and savings certificates And, finally, she should invest between $5,000 and $15,000 in savings certificated The problem is to determine the proper allocation of the investment capital among the three investment opportunities in order to maximize her yearly return Linear Program: The goal of the problem is to maximize the return on investment subject to several constraints Let x c be the amount to be invested in savings certificates Let x b be the amount to be invested in bonds, and Let x s be the amount to be invested in stocks The goal of the problem is to now maximize the following return on investment objective function given by z: Maximize: z = 07x c + 09x b + 14x s The problem also has the following constraints, or is Subject to: x c + x b + x s = 50, 000 x b 10, 000 x s x b + x c 5, 000 x c 15, 000 with x s 0 This can be solved using LINDO and the following values are obtained (See Attached LINDO print out): Solution x c = $5,000 x b = $20,000 x s = $25,000 Page 5

6 (d) The owner of a small chicken farm must determine a laying and hatching program for 100 hens There are currently 100 eggs in the hen house, and the hens can be used either to hatch existing eggs or to lay new ones In each 10-day period, a hen can either hatch 4 eggs or lay 12 new eggs Chicks that are hatched can be sold for 60 cents each, and every 30 days an egg dealer gives 10 cents each for the eggs accumulated to date Eggs not being hatched in one period can be kept in a special incubator room for hatching in a later period The problem is to determine how many hens should be hatching and how many should be laying in each of the next three periods so that total revenue is maximized Linear Program: The goal of the problem is to maximize the total revenue over the next 30 day period subject to several constraints Let H hi be the number of hens hatching eggs in period i, i {1, 2, 3} Let H li be the number of hens laying eggs in period i, i {1, 2, 3} Let E s0 be the number of eggs initially in storage (100 eggs) Let E si be the number of eggs in storage at the end of period i, i {1, 2, 3} The goal of the problem is to now maximize the profit made on the eggs and chicks produced by the chickens defined by the objective function z That is: Maximize: z = 6 4 (H h1 + H h2 + H h3 ) + 1E s3 Subject to: H hi + H li = 100 4H hi E s(i 1) E s(i 1) 4H hi + 12H li = E si H hi, H li, E si 0, where i {1, 2, 3} When the above program is optimized using LINDO(see the attached print out) After solving the following values are obtained: H h1 = 25 H h2 = 100 H h3 = 100 H l1 = 75 H l2 = 0 H 13 = 0 E s0 = 100 E s1 = 900 E s2 = 500 E s3 = 100 This is sensible, because the return from a hatched chicken is 60 cents, and that of an egg is only 10 cents Thus, a Hen laying 12 eggs a period can generate $120 and a Hen hatching 4 eggs can generate $240 Page 6

7 Additional Homework for MATH Show that for any two matrices A and B, (AB) T = B T A T Assume that A is an m n matrix and B is an n q matrix so that there product is defined Thus, AB is an m q matrix, and its transpose is a q m matrix We can also note that B T A T is the product q n matrix and a n m matrix resulting in again a q m So each side of the equality is of the same size We now only need to show that the arbitrary element in row i and column j of (AB) T is the same as the row i column j element in B T A T Denoting the entry in row i column j in a matrix M as M i,j we can consider the following algebra: which concludes the proof (AB) T i,j = (AB) j,i n = A j,k B k,i = = k=1 n k=1 n k=1 A T k,jb T i,k B T i,ka T k,j = (B T A T ) i,j 2 The Gotham City Police Department employs 30 police officers Each officer works 5 days per week The crime rate fluctuates with the day of the week, so the number of police officers required each day depends on which day of the it is: Sunday, 18; Monday, 18; Tuesday, 24; Wednesday, 25; Thursday, 16; Friday, 21; Saturday, 28 The police department wants to schedule police officers to minimize the number whose days off are not consecutive Formulate an LP that will accomplish this goal Type your LP into LINDO 1 and compute the optimal number of officers that do not get consecutive days off 1 Note LINDO is available in the class lab, as well as free to download for academic use Page 7

8 For i < j let x ij be the number of workers who get day i and j off each week Here i = 1 is Sunday, and i = 7 is Saturday Thus x 12 is the number of workers getting Sunday and Monday off, and x 17 are the number of workers who get Saturday and Sunday off The objective function is to maximize the following pairs: max z = x 12 + x 17 + x 23 + x 34 + x 45 + x 56 + x 67 Subject to each day of the week we need the proper number of officers to have the day off x 12 + x 13 + x 14 + x 15 + x 16 + x 17 = 12 Sunday x 12 + x 23 + x 24 + x 25 + x 26 + x 27 = 12 Monday x 13 + x 23 + x 34 + x 35 + x 36 + x 37 = 6 Tuesday x 14 + x 24 + x 34 + x 45 + x 46 + x 47 = 5 Wednesday x 15 + x 25 + x 35 + x 45 + x 56 + x 57 = 14 Thursday x 16 + x 26 + x 36 + x 46 + x 56 + x 67 = 9 Friday x 17 + x 27 + x 37 + x 47 + x 57 + x 67 = 2 Saturday All variables are non-negative That is x ij 0 i, j 1, 7such thati < j Page 8

MATH 445/545 Test 1 Spring 2016

MATH 445/545 Test 1 Spring 2016 MATH 445/545 Test Spring 06 Note the problems are separated into two sections a set for all students and an additional set for those taking the course at the 545 level. Please read and follow all of these