Solve with only correct answers

INSTRUCTIONS: ANSWER ALL QUESTIONS QUESTION A1 [25 marks] The Delux Nut Company produces a deluxe mix composed of almonds, cashews, peanuts and walnuts. The deluxe mix must contain at least 10% of each kind of nut. At least half of the mix must be composed of almonds and cashews and it can contain at most 20% peanuts. The company also produces a companion mix composed of cashews, walnuts and raisins. This mix must contain at least 10% cashews, at least 30% walnuts, and between 20% and 40% raisins. Raisins are available in unlimited supply at a purchasing cost of R3,60 per kilogram. The quantities of nuts available and the purchasing costs in rand per kilogram are as follows: Nut Purchasing costs in rand Kilogram available Almonds Cashews Peanuts Walnuts 12,00 10,60 2,80 8,40 400 200 600 300 Both mixes are sold in 100 g packages. The company must produce at least 4 000 packages of the deluxe mix and at least 5 000 packages of the companion mix. (a) Formulate an LP model for this problem. (5) (b) Solved the LP using LINGO or SOLVER (5) (c) Write down the optimal purchasing plan and the associated purchasing costs. (3) (d) The suppliers inform the company that the availability of almonds has been reduced to 200 kg and that of peanuts to 100 kg. How will this influence the optimal plan and purchasing costs? Give reasons for your answer. (3) (e) An extra 50 kg of walnuts can be bought at R11,00 per kg. Would you make use of this offer? Give reasons for your answer. (3) (f) Assume that the optimal plan is obtained from the original available quantities of nuts. The company now decides that it must produce at least 4 500 packages of the deluxe mix. Will the total purchasing costs be influenced? Give reasons for your answer. (3) (e) The supplier of peanuts announces a 40% increase in price. What will the effect of this be? Give reasons for your answer. (3)

4 PBA4804 OCTOBER/NOVEMBER 2017 PORTFOLIO EXAMINATION [TURN OVER] SECTION B (75 MARKS) INSTRUCTIONS: ANSWER ANY FIVE QUESTIONS: QUESTION B1 [15 marks] The Bloom & Blossom Company wants to expand its activities. In the next few years a number of large capital amounts will become untied, and they wish to plough these back into the company. The amounts that will become available at the beginning of each year are given in the following table: Year Amount 1 R150 000 2 R135 000 3 R210 000 4 R120 000 5 R 90 000 Four projects are being considered. The capital needed per year for each project (in R1 000) and the present value of the net return (in R1 000) on each project are given in the table below. Capital needed Net Project 1 2 3 4 5 return 1. Expand plant in Alberton 30 60 81 60 30 240 2. Build new plant in Port Elizabeth 60 30 120 60 60 300 3. Enlarge small machine capacity in Rosslyn 30 15 60 30 0 120 4. Enlarge large machine capacity in Rosslyn 90 60 30 30 30 210 If a project is selected, it has to be completed as a whole. Money not used can be invested at 12% per annum for a year. Bloom & Blossom wants to select one or more of the projects in such a way that the net return on the chosen projects will be a maximum. Formulate the problem as linear integer programming model. Do not solve the model!

5 PBA4804 OCTOBER/NOVEMBER 2017 PORTFOLIO EXAMINATION [TURN OVER] QUESTION B2 [15 marks] (a) Give a full definition of the concept of simulation. (1) (b) Set out the general simulation methodology in ten steps using a flow diagram. (5) At a bakery, the daily demand for bread (in loaves) has the following probability distribution: Demand (loaves) Probability 1 000 0,10 2 000 0,30 3 000 0,45 4 000 0,15 (c) What is the expected daily demand for bread at the bakery? (2) (d) In a simulation run, the daily demand for bread is generated from the probability distribution by sequentially using the uniformly distributed random numbers U1, U2, : : :, as given in Appendix A. (i) What is the demand for bread on the first day of the simulation run? (1) (ii) What is the average demand for bread over the first four days of the simulation run? (3) (iii) Draw a flow diagram of the logic to calculate the average daily demand for bread over the duration of a year (365 days). (Define any variables that you use clearly) (3)

6 PBA4804 OCTOBER/NOVEMBER 2017 PORTFOLIO EXAMINATION [TURN OVER] QUESTION B3 [15 Marks] A decision maker faced with four decision alternatives and four sates of nature develops the following profit payoff table. States of nature Decision alternative S_1 S_2 S_3 S_4 d_1 14 9 10 5 d_2 11 10 8 7 d_3 9 10 10 11 d_4 8 10 11 13 (a) State and use the average payoff strategy to choose the best decision. (3) (b) State and use the aggressive strategy to choose the best decision. (3) (c) State and use the conservative strategy to choose the best decision. (3) (d) State and use the opportunity loss strategy to make the best decision. (3) (e) Suppose the decision maker obtains information that enables the following probabilities assessments: P(s_1) = 0.5; P(s_2) = 0.2; P(s_3) = 0.2; and P(s_4) = 0.1. Use the expected value approach to determine the optimal strategy. (3)

7 PBA4804 OCTOBER/NOVEMBER 2017 PORTFOLIO EXAMINATION [TURN OVER] QUESTION B4 [15 Marks] (a) What do you understand by a time series forecasting approach? Describe each of the four factors in this approach, and indicate how they are measured or determined. (5) (b) The historical demand for a product is as follows: Month Demand January 12 February 11 March 15 April 12 May 16 June 15 Stating any assumptions that you make answer the following questions. (i) Using a weighted moving average with weights of 0.60 for June; 0.30 for May and 0.10 for April, find the July forecast. (3) (ii) Using a simple three-month moving average, find the July forecast. (3) (iii) Using simple exponential smoothing with =0, 2 and a June forecast of 13 find the July forecast. (4) QUESTION B5 [15 marks] Phalaborwa Development Corporation, PDC, purchased land that will be a site of a new luxury double storey complex. The location provides a spectacular view of the surrounding countryside, including mountains and rivers. PDC plans to price the individual units between R300 000 and R1 400 000. PDC commissioned preliminary architectural drawings for three different projects; one with 30 units, one with 60 units and one with 90 units. The financial success of the project depends upon the size of the complex and the chance event concerning the demand of the units. The statement of the decision problem is to select the size of the new complex that will lead to the largest profit given the uncertainty concerning the demand of for the units. The information for the PDC case (in terms of action and states of nature), including the corresponding payoffs can be summarised as follows:

8 PBA4804 OCTOBER/NOVEMBER 2017 PORTFOLIO EXAMINATION [TURN OVER] States of Nature Probability Decision Alternative Strong Demand (s_1) Weak Demand (s_2) 0.8 0.2 Small Complex (d_1) Medium Complex (d_2) Large Complex (d_3) 8 7 14 5 20 -9 The management of PDC is considering a six month market research study designed to learn more about the potential market's acceptance of the PDC project. Management anticipates that the market research study will provide the following two results: (1) Favourable report: A significant number of the individuals contacted express interest in purchasing a PDC unit (2) Unfavourable report: very few individuals contacted express interest in purchasing a PDC unit. If the market study is undertaken P(Favourable report) = 0.77 P(Unfavourable report) = 0.23 If the market report is favourable P(strong demand given favourable report) = 0.94 P(weak demand given favourable report) = 0.06 If the market report is unfavourable P(strong demand given unfavourable report) = 0.35 P(weak demand given an unfavourable report) = 0.65. (a) Given the above information, use the decision tree approach, by only carrying out the forward calculations, to advise PDC on the best strategy. (12) (b) Based on your analysis, provide a corresponding risk profile for the optimal decision strategy. (3)

9 PBA4804 OCTOBER/NOVEMBER 2017 PORTFOLIO EXAMINATION [TURN OVER] QUESTION B6 [15 marks] (a) 15 samples of n = 8 have been taken from a cleaning operation. The average sample range for the 20 samples was 0.016 minute, and the average mean was 3 minutes. Determine the three- sigma control limits for this process. (3) (b) 15 samples of n = 10 observations have been taken from a milling process. The average sample range is 0.01 centimetres. Determine upper and lower control limits for sample ranges. (3) (c) Determine which of these three processes are capable: (9) Process Mean Standard Deviation Lower Specification Upper Specification 1 6.0 0.14 5.5 6.7 2 7.5 0.10 7.0 8.0 3 4.6 0.12 4.3 4.9 10 PBA4804 OCTOBER/NOVEMBER 2017 PORTFOLIO EXAMINATION [TURN OVER] ANNEXURE-A: Table 1: Control Chart parameters Number of Observations in Subgroup n _ Factor for X Chart A2 FACTORS FOR R CHARTS Lower Control Limit D3 FACTORS FOR R CHARTS Upper Control Limit D4 2 1.88 0 3.27 3 1.02 0 2.57 4 0.73 0 2.28 5 0.58 0 2.11 6 0.48 0 2.00 7 0.42 0.08 1.92 8 0.37 0.14 1.86 9 0.34 0.18 1.82 10 0.31 0.22 1.78 11 0.29 0.26 1.74 12 0.27 0.28 1.72 13 0.25 0.31 1.69 14 0.24 0.33 1.67 15 0.22 0.35 1.65 16 0.21 0.36 1.64 17 0.20 0.38 1.62 18 0.19 0.39 1.61 19 0.19 0.40 1.60 20 0.18 0.41 1.59 Table 2: Random numbers The following sequence of 20 random numbers were generated sequentially from a U[0; 1) distribution. U1 = 0; 48 U5 = 0; 08 U9 = 0; 71 U13 = 0; 69 U17 = 0; 78 U2 = 0; 03 U6 = 0; 23 U10 = 0; 63 U14 = 0; 98 U18 = 0; 92 U3 = 0; 52 U7 = 0; 40 U11 = 0; 57 U15 = 0; 60 U19 = 0; 19 U4 = 0; 83 U8 = 0; 24 U12 = 0; 41 U16 = 0; 20 U20 = 0; 56