Objectives This assignment addresses the following objectives for the course as outlined in the course specification: • use a variety of operational research techniques to analyse complex management problems and to synthesise and evaluate possible solutions to these problems. • analyse production and supply in terms of various inputs, types of costs, break even analysis, and the effect of time on the value of money. QUESTION 1 Mechatronics, Inc., is a relatively new firm making small calculators. Mechatronics entered the business with the production of an inexpensive handheld calculator, the Z-1000, which sells for $15. It has recently added a more powerful version of the Z-1000 called the Z-2000. The Z-2000 sells for $25. The variable costs of producing a Z-1000 and a Z-2000 are given in Table 1. Table 1 Z-1000 Z-2000 Labour $3.00 $5.00 Material $6.00 $12.00 Factory overhead $2.00 $2.00 Total $11.00 $19.00 Mechatronics produces its own circuit boards and purchases all other materials from other firms. Manufacturing of circuit boards is a complex operation, which requires precision equipment. Mechatronics has the capacity to produce, at most, 61,000 basic circuit boards per month. One of these circuit boards is used in each Z-1000 calculator. To manufacture the advanced circuit board for the Z-2000 calculator takes three times as long on this precision equipment as the Z-1000. Therefore, if Mechatronics made no basic circuits at all, it could produce no more than 20,333 (i.e. 61,000/3) of the Z-2000 boards. Mechatronics can manufacture any combination of Z-1000 and Z-2000 circuit boards, as long as the combined production time does nor exceed the available capacity. Assembly time for the two calculators is:  Z-1000: 0.2 hour  Z-2000: 0.25 hour. If the company maintains its current two-shift operation, it has available 8000 hours of assembly time per month. The marketing manager has undertaken a detailed study of the calculator market and foresees a monthly demand of 40,000 units for Z-1000 calculator and 18,000 units for Z-2000. If Mechatronics would like to maximize its profit, how many Z-1000 and Z-2000 calculators it should produce? (List out the objective function as well as all limitations and solve the problem mathematically and graphically). QUESTION 2 The performance of two machines that dispense set weight of a product into containers is being assessed with a view to purchasing one of the machines. Sixteen (16) samples of three (3) containers from each machine are taken at regular intervals, and their contents weighed. The result of this sampling are detailed below. Sample number Machine A Machine B Sample average (g) Sample range (g) Sample average (g) Sample range (g) 1 501 2.0 504 6.0 2 500 3.5 500 4.4 3 502 8.5 498 2.8 4 499 7.0 497 1.8 5 502 4.0 492 3.0 6 498 5.5 496 5.5 7 501 6.0 501 7.2 8 498 1.0 498 4.8 9 502 2.8 503 2.4 10 499 7.6 502 3.6 11 502 7.5 505 6.4 12 499 2.0 508 3.0 13 500 5.6 503 2.8 14 498 4.5 499 3.6 15 500 3.5 497 1.5 16 499 1.0 497 2.0 = 8000 = 72.0 = 8000 = 60.8 The potential purchaser requires that the contents of the containers should be 500 ± 10.00 grams. (a) Calculate range and sample average control limits and plot charts for both machines. (36 marks) (b) Find the relative precision class of each machine and construct modified control limits if this is warranted. (6 marks) (c) Compare the characteristics of the machines in detail, and recommend which machine should be purchased. In particular, indicate clearly whether or not the process is in control and if scrap is being produced. (8 marks) QUESTION 3 A factory has three departments A, B and C that have an excess of pallets over and above their needs and four departments, D, E, F and G that require pallets. The excesses and requirements are shown in the following table: Department A B C D E F G Excess pallets Pallets required 75 90 60 45 45 60 45 The times involved in moving the pallets from the departments with excesses to those needing pallets are shown in the following table: Time, minutes From To D E F G A B C 12 9 23 25 23 15 13 14 12 21 18 16 What is the optimum distribution plan to minimize total delivery time? What is the total minimum time involved in distributing the excess pallets? Assume that the pallets are moved one at a time and use the North-West programming method to answer these questions.