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I'm going through some past papers for an exam which has the production planning problem and formulating linear programming problems in it but I've come across a question that is an odd mix of both.

It gives a limit to weekly production, the demands for the next four weeks, the costs of backlogging and inventory storage (plus their limits) and the cost of production. The objective function you're seeking is for the optimal production plan (so I assume maximising profit) and it specifically states that you should not solve the problem.

It gives all the information you'd need to solve this as a production planning problem but I've never come across a combination of the two. How do I formulate this?

So far all I've worked out is that the decision variables are the four amounts produced which must be above zero and below the weekly production limit.

Any help greatly appreciated.

EDIT: I have typed out the whole question as best I can below

A factory can make at most $10$ units a week and there is demand forecast to be $8, 14, 11, 7$ for the next four weeks. $7$ units can be held in inventory at a cost of £10 a unit/week, orders for up to $3$ units can be backlogged at a cost of £15 a unit/week. There is currently no inventory and no orders backlogged and we want none at the end of the four weeks.

The cost of production is $0$ if $n=0$

$100$ if $0<n\leq6$

$130$ if $6<n\leq8$

$150$ if $9<n\leq10$

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  • $\begingroup$ Additional information about the problem would be helpful. For example, how many different products are there? What are the prices they are sold at, how much are the costs for storage, etc.? $\endgroup$ – YukiJ Apr 23 '18 at 14:45
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In addition to production volumes, the amount of inventory carried and the amount of demand backlogged in each week are decisions, and you will need constraints (sometimes referred to as "flow balance" constraints) to make sure the amounts match up correctly.

You also need to be concerned whether the production costs are per-unit or for a batch, and if per-unit whether they are incremental or cumulative. Looking at what you wrote and postulating a production run of 9.2 units, I can picture three possibilities for production cost:

  • 9.2 * 150;
  • 150 (for a batch of anywhere between 9 and 10); or
  • 100 * 6 + 130 * 2 + 150 * 1.2 (incremental).

I believe that two of those cases will require integer variables.

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