Let X(k) be the number of products at the workstation at time k = 0, 1, 2, … , and assume for convenience that X(0) = 0.

3.   Draw a state-transition diagram for the Markov chain representing the workstation under study, using circles for states and arrows for transitions. Label each arrow with the corresponding transition probability. Finally, write down the   transition probability matrix for the same chain. [30 marks]

4.   What is the probability that in period 4 there is only a single product at the workstation? [10 marks]

5.   Does this Markov chain admit a unique equilibrium distribution? Justify your answer and, if a unique equilibrium distribution indeed exists, compute it. [30 marks]

QUESTION 2

1.   Next Monday you are being interviewed by Lothian Buses for a business analyst position. Fortunately for you, the company’s Head of Commercial has just given  a related guest talk on your Decision Analytics course. In preparation for the interview, and with direct reference to what the guest speaker said as well as    your own notes from the guest talk, say how you would put your learnings from the Decision Analytics course to the service of the Analytics Team at Lothian Buses. Provide one or two examples of decision making/support at Lothian Buses – as you imagine it, based on the partial information you have from the guest talk– and explain how you would employ techniques from the course to help Lothian Buses enhance their decision-making processes. [25 marks]

HGC is a renowned garden centre in the beautiful market town of Haddington, East   Lothian. During weekends, you work at HGC, helping to run their busy café , and you  come across a conversation where the HGC owners seem to agree that there might   be something wrong with the way they manage their stock in relation to some of their products, with regards to one model of chainsaw. Having just passed your Decision Analytics exam with a first-class mark, you offer the owners of HGC your help, saying you are happy to look into their problem.

Of course, you develop a Markov decision process model of the inventory of the chosen single type of chainsaw. The state of the process is the number of chainsaws in stock at the beginning of a month. The set of possible states is S = {0, 1, 2, 3}, as   the owners believe that within the limitations of their current footprint, and the extremely wide range of products kept on stock at any one time , there will hardly ever be space for 4 chainsaws or more. At the beginning of each month HGC decides how many chainsaws to order for delivery at the end of the month. They believe that no more than 2 chainsaws can be ordered in any month. The owners are quite confident that each month there is demand for 0, 1 or 2 chainsaws. Thanks to the available records, you estimate that the probabilities of each demand level are 0.5, 0.4 and 0.1 respectively.

1.   With reference to our discussion of the paper “Model validation in operations research” (Landry etal., 1983), which you were also required to read and debate in your coursework group ahead of starting your joint work on the TurnCo case study, discuss the various types of validation that are involved in quantitative modelling and analysis of decision problems. [20 marks]

A small manufacturer is considering investing in the production of innovative components for electric cars. With the help of an expensive external consultant, four alternative options have been singled out for a new production process. For convenience, these options are labelled as A, B, C and D. The external consultant has also estimated a set of three alternative future scenarios for the product demand that could be used as a reference for the evaluation of the best investment option. The scenarios represent respectively ‘ High’ , ‘ Medium’ and ‘ Low’ demand volumes. Armed with an extensive report (of 120 pages in length!) put together by the same consultant, the senior management team now wishes to decide which of the four alternative options should be pursued. The core of the report is the following table, which shows the expected monetary gains (profits, in £/year) over the next ten years, for each alternative option/state of nature combination:

2.   Based on the maximin payoff criterion, which option should the senior managers choose? [15 marks]

3.   Based on the minimax regret criterion, which option should the senior managers choose? [15 marks]

4.   The same extensive report also contains a three-page section which offers partial information (not necessarily easily interpretable) on the likelihoods of the   three scenarios. On studying this section in detail, you understand that the ‘High’ demand scenario should be about four times as likely as the ‘Low’ demand scenario, and that the ‘High’ and ‘Low’ scenarios together should be roughly as likely as the ‘Medium’ demand scenario. Compute the best estimates of the likelihoods of the three scenarios, to help the senior managers further appreciate what the future might hold for their company. [5 marks]

5.   Draw a decision tree for the problem and use it to find the option that maximises the expected monetary value criterion. [25 marks]

6.   Before converging towards a final decision, the senior managers wish to evaluate the opportunity losses pertaining to the various options under study. This could be done by examining the probabilities estimated at point 4. Help the  senior management team with this exploration. What alternative is suggested as best course of action under this criterion? [15 marks]

7.   Using the findings from the above analysis (points 2 to 6), what option would you recommend the senior management team to choose, and why? [5 marks]

QUESTION 4

1.   With reference to our discussion of the paper “Model validation in operations research” (Landry etal., 1983), which you were also required to read and debate in your coursework group ahead of starting your joint work on the TurnCo case study, discuss the various types of validation that are involved in quantitative modelling and analysis of decision problems. [20 marks]

2.   Briefly describe the batch sampling and sequential sampling approaches to repetitive testing problems. [10 marks]

Murano Doors specialises in the manufacture of luxury glass door units for retail premises. Each unit costs £1,000 to produce and sells for £3,000. If a door unit is not strong enough, the glass will shatter during installation and Murano Doors then incurs a penalty cost of £10,000 (and so makes a loss of £8,000 on the unit). Past experience has shown that the manufacturing process produces units of the required strength 90% of the time. The company can perform a test to estimate the strength of a door unit. This test costs £200 and does not give perfect information. Past experience suggests that if the unit is of the required strength, there is an 80% chance that the test will give a positive result, but if the unit is not of the required strength, there is only a 5% chance that the test will give a positive result. If the company suspects that a unit is not of the required strength, it can scrap the unit.   The value of the steel and glass that can be salvaged from a scrapped door unit is £250.