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I'm helping a wholesaler in perishable products that sales 1200 units a week of a product buyed $30€$ by unit. The order placement cost is $40€$ and the possession cost is $25\%$ a year. We will take, for computations, a stable demand distributed on a year of $50$ weeks of $6$ days. The corporation could buy a preservative that would allow to extend the shelf life of the product that is $6$ days for the moment. $200€$ a day would allow to extend it to $12$ days. (In this case, the preservative has to be used everyday).

What is the best replenishment quantity and the replenishment frequency. May the wholesaler use the preservative ? How can I compute the annual total cost corresponding to this replenishment quantity policy ?

My attempt

With the following variables :

  • $c_t$ cost of buying product at the $t$ period = $30$.
  • $X_t$ amount of product bought
  • $hc$ host cost $200$ a day
  • $x_t$ inventory quantity at period $t$

Therefore the objective function would be :

\begin{cases} \min &\sum_t 30X_t+(200*6)*s_t\\ &X_t*s_t\le 1200\\ X_t,s_t\ge 0 \end{cases}

But I can't have $t$, the frequency nor $X_t$, the amount to buy.

Revolucion for Monica
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  • Wait, wait, wait: You buy and sell 1200 units a week, each time buying costs 40 Euro. So in total, you would have to pay 4050 = 2000 Euro each year for order placement. You want to cut down on that by only ordering after two weeks, as you increase your lifespan. This would decrease the ordering cost by 1000 Euro (while the product price stays the same), but you would have to pay 200 a day for it. Thus for saving 1000 Euro, you pay 20050*6 = 60000 a year? oO So I either got you totally wrong or some of your numbers are mixed up... – Dirk May 05 '17 at 09:13
  • @Bemte I think the preservative cost of $200€$ is for the whole supply. I would therefore pay $200*6=1200$ max if i keep them $6$ days but I can keep them less time ! – Revolucion for Monica May 05 '17 at 09:31
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    So if you want to increase the lifespan by one day, you pay 200 a year, for two days you pay 400 a year, etc. up to 1200 a year for +6 days? In this case I would simply compute all cases (there are only seven...) and then check which one is the best. – Dirk May 05 '17 at 09:33
  • @Bemte Thank you for this insight. Nevertheless, isn't it "for $x$ days you pay $x\times 200$ a day"? – Revolucion for Monica May 05 '17 at 09:48
  • That would be even worse? The 40 Euro order placement cost are almost nothing, compared to the 1200*30 = 36000 euro product cost payed per order. As I mentioned before, you can save at most 1000 a year by doubling the lifespan and thus halving the placement cost. As a year in your calculation has 300 days, such a product that doubles the lifespan should cost at most 3 Euro a day, so 200 - or even $200 \times x$ a day should be not even discussed. Again, I might get some numbers wrong, because I still don't see why such a thing would even be considered. – Dirk May 05 '17 at 09:59

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