Practice: Constrained Optimization

A bakery decides how many loaves and how many cakes to bake tomorrow. Ovens allow at most 9 hours of baking; a loaf takes 15 minutes, a cake 40. Loaves sell for 3, cakes for 7.

Which of the following is a decision variable?

A workshop builds $c$ chairs and $t$ tables. A chair takes $3$ hours of labour and $2$ metres of timber; a table takes $5$ hours and $4$ metres. There are $120$ labour hours and $80$ metres of timber available, and the workshop cannot build a negative number of either product.

Select every constraint of the corresponding linear program.

A workshop builds two products. Each chair yields \$40 of profit and each table \$65. The workshop wants the most profitable production plan it can manage within its labour and timber limits.

Select every statement that correctly states the objective of the corresponding linear program.

For each situation, state the decision variables, the objective and its direction, and the constraints.

(a) A haulier has three lorries and must deliver to five depots today. Each lorry costs a different amount per kilometre, each depot must be visited once, and no lorry may drive more than 400 km. The haulier wants the cheapest schedule.

(b) A student has 20 hours before three exams. Each hour spent on a subject raises the expected mark in it, with diminishing returns, and each exam has a pass threshold that must be met. The student wants the highest total mark.

For one of the two, name a quantity mentioned in the description that is not a decision variable, and say why.

A student writes:

A courier must deliver 12 parcels using vans that each hold at most 5 parcels and cost 30 per van sent out. Let $C$ be the total cost. Minimise $C$ subject to all 12 parcels being delivered.

This formulation cannot be solved as written. Say precisely what has gone wrong, and rewrite it correctly.