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A linear programming problem from a mechanical engineering exam at cork institute of technology. The problem involves a manufacturing company, dme4 ltd., which produces three products x, y, and z, and must decide how many units of each product to produce in-house and buy from outside sources to meet demand while maximizing profit. The problem is formulated as a linear programming problem and includes machine time requirements and profit per unit for in-house and bought-in products.
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Instructions: Answer FOUR questions. All questions carry equal marks. Statistical tables are available.
Examiners: Mr. D. O’Hare Mr. P. Clarke Prof. M. Gilchrist
Minimise 2 9 4 subject to 2 3 4 12 2 4 4 8 3 , , 0.
z x x x x x x x x x x x x x x x
(13 marks)
(b) A personnel manager has recently hired 25 workers, each of whom will work in either assembly, machining, packing or inspection. 15 of the workers are classified as apprentices while the remaining 10 are classified as experienced. The company has developed productivity measures for the workers as follows:
Productivity per worker by position Assembly Machining Packing Inspection Apprentice 58 62 76 29 Experienced 73 94 81 56 Seven workers each are required in assembly, machining, and packing, while four are needed in inspection. Solve the problem of allocating workers to departments so as to maximise the overall sum of productivity measures. (14 marks)
(c) The following is an ANOVA table for a randomised block design:
Source Sum of squares df Mean Square F Factor * 4 13307.75 * Blocks * * 38739.25 * Error * 12 * Total 171119.75 19
(i) Fill in the values denoted by * in the ANOVA table. (ii) How many factor levels and how many blocks are involved? (iii) What conclusion do you draw from the table and why? (8 marks)
The following data were gathered:
Box Type Compressive Strength 1 565.5 788.3 734.3 721. 2 789.2 772.5 786.9 686. 3 535.1 628.7 542.8 559. Produce a one-way ANOVA table based on these data and state your conclusions clearly. (10 marks) (b) The data below resulted from a 2^3 experiment designed to study the effects of concentration of detergent (A), concentration of sodium carbonate (B), and concentration of sodium carboxymethyl cellulose (C) on cleaning ability of a solution in washing tests. A larger number indicates better cleaning ability than a smaller number. Two observations were taken at each combination of factor levels.
Test Observations (1) 106, 93 a 198, 200 b 197, 202 ab 329, 331 c 149, 169 ac 243, 237 bc 255, 230 abc 383, 360 (i) Calculate estimates of the main effects and the two-factor interactions. (ii) Test the significance of the various effects. (iii) What would your recommendation be based on the evidence of these data? (15 marks) Note that (^) S (^) e^2 = 112
