MAT1005 Mathematics for Business and Economics (Semester 1, 2020-21) ASSIGNMENT 2 Deadline: November 6, 2020 (Friday) noon Topics 2 Linear Equation, Linear Inequalities & Linear Programming (1) Given the following Linear Programming problem: Maximize 45𝑥$ + 30𝑥( + 24𝑥* Subject to: 2𝑥$ + 3𝑥( + 4𝑥* ≤ 64 (Constraint 1) 2𝑥$ + 𝑥( + 𝑥* ≤ 32 (Constraint 2) 𝑥$ + 𝑥( + 𝑥* ≤ 20 (Constraint 3) 𝑥$ , 𝑥( , 𝑥* ≥ 0 (a) Prepare the initial Simplex tableau as discussed in the class. (b) Solve the problem by using the Simplex method as discussed in the class. (You are requested to show your full working to gain full credit) (c) Find the shadow price for constraint 2 and 3 (2) Alicia Liquide (Alicia), one of the top chemical manufacturer in the local region, is planning to setup another production plant in Bougainvillea. The new plant will have 24/7 operation. Each day of every week is divided into two 12-hour shift periods, 8am-8pm and 8pm-8am, denoted by morning shift and night shift. The minimum number of workers required for each of these shifts over any week is listed below Sun. Mon. Tue. Wed. Thu. Fri. Sat. Morning Shift 8 14 16 18 15 13 6 Night Shift 6 16 18 22 12 10 4 The code of practice of Alicia has the following requirements governing acceptable shifts for workers (1) Each worker works 4 consecutive days during any seven-day period (2) Each worker is assigned to work either a morning shift or a night shift. Workers assigned to a shift must remain on the same shift every day that they work (3) Number of workers in the night shift is limited to a maximum of 40% of the total workers in this plant Workers are paid daily. The daily salaries are listed below: Weekday (Mon. – Fri.) Weekend (Sat. – Sun.) Morning Shift ¥500 /day ¥600 /day Night Shift ¥650 /day ¥800 /day You, the production manager of Alicia, are attempting to develop the working schedule to minimize the weekly salary expenses. Formulate this problem as a linear programming problem. (You are NOT required to solve it) (3) Given the following Linear Programming problem: Maximize 2𝑦$ + 5𝑦( + 9𝑦* Subject to: 3𝑦$ + 3𝑦( + 𝑦* ≤ 90 5𝑦$ − 𝑦( + 𝑦* ≥ 120 4𝑦$ + 2𝑦( + 𝑦* ≤ 150 (a) (b) (c) (Constraint 1) (Constraint 2) (Constraint 3) 𝑦$ , 𝑦( , 𝑦* ≥ 0 Prepare the initial Simplex tableau as discussed in the class. Determine the entering and leaving variables in the first iteration. Find out the Simplex tableau after the first iteration Topics 3 Matrix & System of Linear Equations (4) Let 𝐴 = 𝑥 7 −1 and 𝐵 = 𝑦 2 4 1 , where 𝑥, 𝑦 and 𝑧 are non-zero constant 𝑧 Given that (1) 𝑥 6𝑥 =𝐴 𝑦 6𝑦 (2) 6 + 𝑥 5𝑧 = 1 𝑧 𝐴8 (a) Find 𝑥, 𝑦 and 𝑧 (b) Determine if 𝐵 is a non-singular matrix. (c) Evaluate 𝐵9$ 𝐴𝐵 (d) By using (c), determine 𝐴$: (you may use 𝑚 0 0 𝑛 = = 𝑚= 0 0 without proof) 𝑛=