Algorithms to prioritize equipment renewals

Question

I am looking for algorithms to prioritize equipment renewals.

Input: (years since last renewal, cost of renewal, importance of renewal).

Output: An ordering of the equipment according to which it will be renewed.

I do not know if there are any algorithms for this particular problem. If you have any idea how to fit this problem into a more general context, that would be useful too.

A way to rephrase the problem:

You have $n$ pieces of equipment $E_1,\ldots,E_n$. For each piece $E_i$ you have a triple $(\text{age}_i,\text{cost}_i,\text{importance}_i)$. At the beginning of the year you have $X$ amount of money. You want to spend these money in order to minimize the function $\sum_i \text{age}_i\cdot \text{importance}_i$ at the end of the year. So, during the year you have to select a subset $S$ of $\{1,\ldots,n\}$ such that: $$\sum_{i\in S} \text{cost}_i\le X \text{ (cost constraint)}$$ and the sum $$\sum_{i\in S} \text{age}_i\cdot \text{importance}_i$$ is maximal among all subsets of $\{1,\ldots,n\}$ that satisfy the cost constraint.

Any help?

Answer 1

Given the clarification to the question, it is a 0-1 knapsack problem (your knapsack is the money available, the value is the objective function), and look for ways to solve that one, for example following the Wikipedia lead.

Edit: A simple approximate algorithm would be just sort equipment on increasing cost per unit (age * importance), and in that order include each if it fits (doesn't run over maximal cost). In the case of continuous knapsack (i.e., can include a fraction of a item) that strategy is optimal. Should work fine if there are many items, and the data isn't too spread out.

Answer 2

It all depends on how you define how you decide a an equipment is to be renewed. But at the end, you may sort the lists lexicographically. That is, define a comparison function that takes as input two sets $X = (x _1, ..., x_k)$ and $Y = (y _1, ..., y _k)$, it will return true if $X \succ Y$ and false otherwise, where $X \succ Y$ if:

• $x _1 \succ y _1$ or:
• if there is a $j$ s.t. $1 < j \leq k$, $x _j \succ y _j$ and $x _i = y _i$ for each $1 \leq i < j$.

Use this function with any sorting algorithm you know. (but instead of sorting numbers, you sort lists according to the set comparison function you define).