Theoretical questions and problems to practice Advanced Mathematics and Statistics (MSc course) Faculty of Business Administration M.Sc. English, 2015/16 first semester Topics (1) Complex numbers. Complex numbers and their binom, trigonometric and exponential form. Geometric representation of complex numbers on the complex plane. Operations with complex numbers and their geometrical meanings.calculation of powers and roots. (2) Difference equations. Existence and uniqueness theorem for first and second order difference equations. Solution of linear homogeneous difference equations with constant coefficients. Method of undetermined coefficients. (3) Linear programming. Problems leading to linear programming (LP) problems. Mathematical formulation. Graphical solution of two variable problems. Simplex method. The WinQSB software its usage and applications for solution of various problems: LP-ILP, transportation, work assignment, transshipment, traveling salesman etc. Important definitions, theorems (1) Complex numbers. The absolute value, argument of a complex number. Conjugate of a complex number. (2) Difference equations: general form of first and second order difference equations, Casorati determinant, characteristic equation of second order linear homogeneous equations,existence and uniqueness theorem for first and second order difference equations. (3) Linear programming: general formulation of standard normal maximum problems (see p.19/46 in Linear programming transparencies), explanation of the data (reduced cost, shadow price etc) in combined report of an LP solution in WinQSB (see p. 22-25/46) (1) Let z 1 = 2 8i, z 2 = 2 2i, z 3 = 5+3i, z 4 = 8i, z 5 = 1+i, z 6 = 3 8i. Calculate the complex numbers z 1 + z 3, z 2 3iz 2, z 3 z 4, z 2 /z 6, z 3, z 6. Draw the complex numbers z 2, z 4, z 5, z 6 on the complex plane! Write the numbers z 2, z 4, z 5 in trigonometric and exponential form! Calculate z5 20 -t! Calculate and represent on the plane all values of the roots 3 z 2, 4 z 4, 3 z 5! (2) Find the general solutions of the difference equations (a) y(n + 2) 6y(n + 1) + 8y(n) = 0, (b) y(n + 2) + 2y(n + 1) + 3y(n) = 0, (c) y(n + 2) 8y(n + 1) + 16y(n) = 0, (d) 3y(n + 2) + 2y(n) = 4, (e) y(n + 2) + 2y(n + 1) + y(n) = 9 2 n, (f) y(n + 1) + 2y(n) = 5.
2 (3) Solve with WinQSB the LP problem: z = 3x 1 + 2x 2 + 5x 3 max subject to the constraints x 1 + 2x 2 x 3 15 4x 1 x 2 + 2x 3 5 x 2 + x 3 17 x 1, x 2, x 3 0 Explain the meaning of shadow price and reduced cost! (4) An LP problem was solved by WinQSB. The data entry and solution table is given below (a) Write down the LP problem in the usual (standard) form (denote the variables by x 1, x 2, x 3, x 4 nad by z the objective function)! (b) Explain the meaning of reduced cost, slack or surplus, shadow price in the solution table. (5) (a) Find the graphical solution of the LP problem : z = x 1 + x 2 maximum, subject to 3x 1 + 2x 2 6, 9x 1 + 2x 2 6, 2x 1 2, 3x 2 3, x 1, x 2 0. (b) Find the solution by WinQSB! (c) Find the integer solutions by WinQSB!
(6) (Transportation problem) Goods should be transported from 3 factories (T 1, T 2, T 3 ) to four supermarkets (S 1, S 2, S 3, S 4 ). The good can be shipped from any factory to any location. The productions (supplies) of the factories, the demands of the locations and the transportation costs/ unit of goods are given in the table below. Find the quantities of goods shipped from T i to S j such that the shipping costs be minimal and from any factory one cannot ship more than the production (supply) and, if possible the demand of the locations should be satisfied. 3 Supply S 1 S 2 S 3 S 4 T 1 400 4 6 8 10 T 2 500 7 5 3 3 T 3 300 11 5 6 4 Demand 300 420 180 200 Also give the LP model of the problem. (7) (Optimal roster of workers) A supermarket is open on all days of the week. Each worker of the supermarket works for five consecutive days then gets two free days. The roster changes in every month. Corresponding to this there are seven possible work schedules for each worker. We assume that all workers get the same wage, for weekdays 50$ per day, for Saturdays 70$ and for Sundays 100$. The number of workers needed for each day is known and given in the next table. It also gives the present roster: there are 32 workers, out of them working 4,4,6,6,4,4,4 with schedules 1.,2.,3.,4.,5.,6.,7. respectively. Schedule Free days working M Tu W Th F Sa S 1. Monday, Tuesday 4 0 0 1 1 1 1 1 2. Tuesday, Wednesday 4 1 0 0 1 1 1 1 3. Wednesday, Thursday 6 1 1 0 0 1 1 1 4. Thursday, Friday 6 1 1 1 0 0 1 1 5. Friday, Saturday 4 1 1 1 1 0 0 1 6. Saturday, Sunday 4 1 1 1 1 1 0 0 7. Sunday, Monday 4 0 1 1 1 1 1 0 Total no. of w.: 32 24 24 22 20 22 24 24 Requ. no. of w.: 17 13 14 15 18 24 22 Presently the weekly wage of the 32 workers is 32 250 + 24 20 + 24 50 = 9680$ as 24 workers work on Saturday and on Sunday Find the minimum numbers of workers and their roster such that on each day at least the required number of workers should work and the total weekly wage be minima. Give the LP model of the problem and find the solution by WinQSB! Hint: Denote by x 1, x 2, x 3, x 4, x 5, x 6, x 7 the number of workers working according to the schedule 1.,2.,3.,4.,5.,6.,7. respectively. The objective function is the total weekly wage, ILP (integer linear programming) module should be used the constraints come from the minimum number of workers required on each day. (8) (New data) A small airline flies to four different cities A,B,C,D from its Boston base. It owns one B707 jet, one propeller-driven Electra plane and one DC9 jet. Assuming constant flying conditions and passenger usage, the following data is available (trip cost return flight in $, trip revenue return flight in $, time in hours) Average flying data
4 City Trip cost Trip revenue time B707 A 6.000 5.000 1 B707 B 7.000 7.000 2 B707 C 8.000 10.000 5 B707 D 10.000 16.000 7 Electra A 1.000 3.000 2 Electra B 2.000 4.000 4 Electra C 4.000 6.000 8 Electra D 6.000 8.000 12 DC9 A 2.000 4.000 1 DC9 B 3.500 5.500 2 DC9 C 6.000 8.000 6 DC9 D 10.000 14.000 9 Formulate constraints to take into account the following: i) city D must be served twice daily; cities A, B, and C must be served three times daily; ii) limitation on number of planes available, assuming that each plane can fly at most 18 hours/day. Formulate objective functions for: i) cost minimization; ii) profit maximization; iii) fleet flying-time minimization. Indicate when a continuous linear-programming formulation is acceptable, and when an integer-programming formulation is required. Solve the problems i)-iii) by WinQSB. Solution: denote the number of daily (return) flights by B707 to cities A,B,C,D by x 1, x 2, x 3, x 4 respectively, by Electra to cities A,B,C,D by x 5, x 6, x 7, x 8 respectively,by DC9 to cities A,B,C,D by x 9, x 10, x 11, x 12 respectively: A B C D B707 x 1 x 2 x 3 x 4 Electra x 5 x 6 x 7 x 8 DC9 x 9 x 10 x 11 x 12 We have 4 constraints concerning the flight frequency and three constraints concerning the flight time of the planes: x 1 + x 5 + x 9 = 3 number of flights to A x 2 + x 6 + x 10 = 3 number of flights to B x 3 + x 7 + x 11 = 3 number of flights to C x 4 + x 8 + x 12 = 2 number of flights to D x 1 + 2x 2 + 5x 3 + 7x 4 18 flying hours of B707 2x 5 + 4x 6 + 8x 9 + 12x 10 18 flying hours of Electra x 9 + 2x 10 + 6x 11 + 9x 12 18 flying hours of DC9 The objective function for cost minimization (cost is measured in thousand dollar units) is z 1 = 6x 1 +7x 2 +8x 3 +10x 4 +x 5 +2x 6 +4x 7 +6x 8 +2x 9 +3.5x 10 +6x 11 +10x 12 minimum The data entry and solution tables for z 1 are:
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6 The objective function for profit maximization is z 2 = x 1 + 2x 3 + 6x 4 + 2x 5 + 2x 6 + 2x 7 + 2x 8 + 2x 9 + 2x 10 + 2x 11 + 4x 12 maximum The data entry and solution tables for z 2 are:
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8 The objective function for fleet flying-time minimization is z 3 = x 1 + 2x 2 + 5x 3 + 7x 4 + 2x 5 + 4x 6 + 8x 7 + 12x 8 + x 9 + 2x 10 + 6x 11 + 9x 12 minimum The data entry and solution tables for z 3 are:
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10 (9) A portfolio manager in charge of a bank portfolio has $10 million to invest. The securities available for purchase, as well as their respective quality ratings, maturities in years, and yields in %, are shown in the next table. Lower bank rating means better quality. Name Type Moody s Quality Maturity Yield After tax yield A1 Municipal Aa 2 9 4.3 4.3 A2 Agency Aa 2 15 5.4 2.7 A3 Government Aaa 1 4 5.0 2.5 A4 Government Aaa 1 3 4.4 2.2 A5 Municipal Ba 5 2 4.5 4.5 The bank places the following policy limitations on the portfolio manager s actions: (a) Government and agency bonds must total at least $4 million. (b) The average quality of the portfolio cannot exceed 1.4 on the bank s quality scale. (Note that a low number on this scale means a high-quality bond.) (c) The (weighted) average years to maturity of the portfolio must not exceed 5 years.
Assume that all securities are purchased at par (face value) and held to maturity and that the income on municipal bonds is tax-exempt. (a) The objective of the portfolio manager is to maximize after-tax earnings and that the tax rate is 50 percent, what bonds should he purchase? (b) If it became possible to borrow up to $1 million at 5.5 percent before taxes, how should his selection be changed? Formulate the LP model of the above problems and solve them by WinQSB! Leaving the question of borrowed funds aside for the moment, the decision variables for this problem are simply the dollar amount of each security to be purchased: x 1 =Amount to be invested in bond A; in millions of dollars. x 2 =Amount to be invested in bond B; in millions of dollars. x 3 =Amount to be invested in bond C; in millions of dollars. x 4 =Amount to be invested in bond D; in millions of dollars. x 5 =Amount to be invested in bond E; in millions of dollars. We must now determine the form of the objective function. Assuming that all securities are purchased at par (face value) and held to maturity and that the income on municipal bonds is tax-exempt, the after-tax earnings are given by: z = 0.043x 1 + 0.027x 2 + 0.025x 3 + 0.022x 4 + 0.045x 5. Now let us consider each of the restrictions of the problem. The portfolio manager has only a total of ten million dollars to invest, and therefore: x 1 + x 2 + x 3 + x 4 + x 5 10. Further, of this amount at least $4 million must be invested in government and agency bonds. Hence, x 2 + x 3 + x 4 4. The (weighted) average quality of the portfolio, which is given by the ratio of the total quality to the total value of the portfolio, must not exceed 1.4: 2x 1 + 2x 2 + x 3 + x 4 + 5x 5 x 1 + x 2 + x 3 + x 4 + x 5 1.4. Note that the inequality is less-than-or-equal-to, since a low number on the bank s quality scale means a high-quality bond. Multiplying by the denominator and re-arranging terms, we find that this inequality is clearly equivalent to the linear constraint: 0.6x 1 + 0.6x 2 0.4x 3 0.4x 4 + 3.6x 5 0. The constraint on the average maturity of the portfolio is a similar ratio. The average maturity must not exceed five years: which is equivalent to the linear constraint: 9x 1 + 15x 2 + 4x 3 + 3x 4 + 2x 5 x 1 + x 2 + x 3 + x 4 + x 5 5, 4x 1 + 10x 2 x 3 2x 4 3x 5 0. Note that the two ratio constraints are, in fact, nonlinear constraints, which would require sophisticated computational procedures if included in this form. However, simply multiplying both sides of each ratio constraint by its denominator (which must be nonnegative since it is the sum of nonnegative variables) transforms this nonlinear constraint 11
12 into a simple linear constraint. We can summarize our formulation in tableau form, as follows: z = 0.043x 1 + 0.027x 2 + 0.025x 3 + 0.022x 4 + 0.045x 5 maximum, subject to x 1 + x 2 + x 3 + x 4 + x 5 10 (cash limit) x 2 + x 3 + x 4 4 (government limit) 0.6x 1 + 0.6x 2 0.4x 3 0.4x 4 + 3.6x 5 0 (quality limit) 4x 1 + 10x 2 x 3 2x 4 3x 5 0 (maturity limit) Solution by WinQSB: data entry and solution table
13 The optimal solution is x 1 = 2.1818 x 2 = 0 x 3 = 7.3636 x 4 = 0 x 5 = 0.4545 and z max = 0.2984. Now consider the additional possibility of being able to borrow up to $1 million at an after-tax rate of 2.75. Essentially, we can increase our cash supply above ten million by borrowing at the rate of 2.75 percent. We can define a new decision variable as follows: x 6 = amount borrowed in millions of dollars. There is an upper bound on the amount of funds that can be borrowed, and hence x 6 1. The cash constraint is then modified to reflect that the total amount purchased must be less than or equal to the cash that can be made available including borrowing: x 1 + x 2 + x 3 + x 4 + x 5 10 + x 6. Now, since the borrowed money costs 2.75 percent after taxes, the new after-tax earnings are: z = 0.043x 1 + 0.027x 2 + 0.025x 3 + 0.022x 4 + 0.045x 5 0.0275y. We summarize the formulation when borrowing is allowed as follows: z = 0.043x 1 + 0.027x 2 + 0.025x 3 + 0.022x 4 + 0.045x 5 0.0275x 6 maximum, subject to x 1 + x 2 + x 3 + x 4 + x 5 10 + x 6 (cash limit) x 2 + x 3 + x 4 4 (government limit) 0.6x 1 + 0.6x 2 0.4x 3 0.4x 4 + 3.6x 5 0 (quality limit) 4x 1 + 10x 2 x 3 2x 4 3x 5 0 (maturity limit) x 6 1 (borrowing limit) The data entry and solution tables are:
14 In this case the optimal solution is x 1 = 2.4000 x 2 = 0 x 3 = 8.1000 x 4 = 0 x 5 = 0.5000 x 6 = 1.0000 and z max = 0.3007. The maximum of one million was borrowed and the interest increased only by 2300$