Here is the word problem.

You are the manager at Nanuq Australia’s Hill Valley factory, which produces edge trimmers, leaf blowers and mowers.

This morning, your boss, the regional manager, met with you and reluctantly explained that the company is struggling, and some contraction will be necessary.

In particular, you have been informed that, unless your company can make a profit of at least $250 000 in the next 29 working days, it will be closed down.

Resource requirements and profits per unit of production are as follows.

| Edge trimmer | Leaf blower | Mower |

Labour (minutes) | 4 | 6 | 9 |

Energy (kWh) | 1.5 | 2 | 5 |

Plastic (kg) | 1.5 | 2 | 0.5 |

Copper (kg) | 0.4 | 0.8 | 2.4 |

Steel (kg) | 1.2 | 0.3 | 30 |

Profit per unit | $18 | $24 | $101 |

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Table 1: Resource requirements and profits per unit of production.

Over the planning period (that is, the next 29 working days),

there will be 550 hours of labour, 15MWh of energy, 6.24 tonnes of copper and 63.06 tonnes of steel available at the Riverton factory.

The amount of plastic available for the planning period is not known exactly.

Your factory buys plastic in an unfinished form and uses a production line to prepare the plastic components needed in production of the edge trimmers, leaf blowers and mowers.

Unfortunately, the equipment in the plastic production line is old and tends to break down, reducing the total amount of finished plastic available.

In particular, each day, the plastic production line is either functioning or broken down.

If it is broken down on one day, then it will be fixed and functioning again the following day, and the required repairs will cost the factory $200 per breakdown.

On the other hand, if it is functioning one day, then on the following day it will either continue functioning or break down, with probabilities that depend on how recently it was fixed:

Probability that today the production line will be: | | |

Days since last breakdown: | Functioning: | Broken down: |

2–5: | 0.875 | 0.125 |

6 or more: | 0.75 | 0.25 |

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Table 2: Probabilities of the plastic production line breaking down today if it was functioning

yesterday.

You can assume that the plastic production line was fixed the day before the planning period starts, so that it is definitely functioning on the first day of the planning period.

It produces 200 kg of finished plastic per functioning day, so that the maximum amount available for production of edge trimmers,

leaf blowers and mowers is 5.8 tonnes (if no breakdowns occur).

Work through the following problems to estimate the probability that the factory will be shut down.

For Problem 2, the numerical answer is given (to allow you to approach Problem 3 with confidence), but you must show how to derive that numerical answer.