Strathmore University Business School Management Mathematics I CAT II Date Due: Friday, 18/12/2020 at 5:00 Pm INSTRUCTIONS: 1. Answer all the questions 2. The CAT is to be done in groups (the syndicate groups used for CAT 1) 3. Indicate clearly the registration numbers of all the group members 4. Scan and submit your work as a single pdf document to [email protected] DO NOT submit jpg images. 1. A processing industry sought to maximize profits, P, using two resources as shown below: Maximize: P = 3𝑥1 + 5𝑥2 Subject to: 2𝑥1 + 3𝑥2 ≤ 90 [Resource 1] 6𝑥1 + 4𝑥2 ≤ 125 [Resource 2] where 𝑥1 , 𝑥2 ≥ 0 Using simplex analysis, the following optimal tableau was obtained from the industrial process (units are in ‘000 dollars). Basic variable P 𝑥2 𝑥1 𝑥1 P 1 0 0 0 0 1.0 𝑥2 0 1 0 𝑠1 1.2 0.6 -0.4 𝑠2 0.1 -0.2 0.30 solution 120.5 29 1.50 (a) Explain the terms “dual value” as used in sensitivity analysis. (b) State and explain the status of resource 1 and 2 (c) What happens to the optimal value of P if 𝑠2 is increased from its current value? 2. Consider the following transportation problem involving three sources and three destinations. Sources Destination Supply (Industries) 1 4 6 8 400 2 3 5 2 200 3 3 9 6 600 Demand 400 450 350 1200 (a) Develop a linear program model for this problem (b) Using the North-West corner method, obtain an initial basic feasible solution for the problem 3. What is the initial cost of transportation incurred by the company? Given the linear programming problem as Minimize: z = 15𝑦1 + 12𝑦2 + 3𝑦3 Subject to: 𝑦1 + 2𝑦2 + 𝑦3 ≥ 15 −2𝑦2 + 𝑦3 ≥ 3 3𝑦1 + 𝑦2 − 𝑦3 ≥ 2 where 𝑦1 , 𝑦2 , 𝑦3 ≥ 0 (a) Find the dual of the problem given (b) Solve the dual problem obtained using simplex algorithm 4. A manufacturing company is considering the construction of a new factory building. The following list shows the project activities, precedence relationships and the time estimate in days. Activity Immediate Predecessors Duration (Days) A = Definition of the 3 problem B = Preliminary study of A 3 costs and constraints C = Problem analysis in A 3 existing building D = Incorporation of C 5 requirements E = Detailed drawings of B, C 6 new building F = Building prototype D, E 9 G = Cost analysis E 5 H = Feasibility review G 3 I = Building begins G, F 5 J = Building inspection I, H 6 K = Final Plant layout J 4 (a) Construct the network diagram for this project and (b) Hence identify the critical path and the duration the project takes to be completed.