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Assume the following scenario. 

I can 
    Sell P1 for a profit of  14% 
    or sell it at a Loss of -7%

    Sell P2 for a profit of 11%
    Or sell it at a loss of -6%

    Sell P3 for a profit of 7%
    or sell for a loss of -1%

Considering the profit margins provided above and max loss rate, at which they need to be cleared by the end of month.

As a seller, stocking which of the above is more profitable to the business. Assuming all products will be sold at the mentioned P/L levels.

If I consider PL ratio,

  • For P1, it would be 14:7 ~ 2 : 1
  • For P2, it would be 11:6 ~ 1.9 : 1
  • For P3, it would be 7:1 ~ 7 : 1

Clearly the higher ratios isnt going to aid in determining the ideal product.

How could I identify the right product?

  • Looking for a solution. I am stuck on this. Any help is highly appreciated. Atleast any recommendations on which Math/Stats concepts should i be looking at? –  Dec 16 '19 at 05:50
  • May I use my reputation to assign some bounty to this. I am fine with deductions from other sub forums too. –  Dec 16 '19 at 07:10
  • May i use my reputation to seek an answer for this question. I am stuck on this. Thanks in advance. –  Dec 17 '19 at 06:54
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    Do you know the likelihood of selling each product? I mean do they all have the same chance of being sold or you have some info on that as well? – passerby51 Dec 23 '19 at 19:54

2 Answers2

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Here is a simple model. Assume that we buy $n$ units of a given product and each unit is sold by the end of the month with probability $p$, independent of all other units. Then the number of sold items $X$ has a Binomial$(n,p)$ distribution. If we have a profit of $a > 0$ and a loss of $b > 0$ for each unit, our "total profit per unit" by the end of the month is $$ \frac{a X - b(n-X)}{n} $$ This converges to the expectation $a p - b(1-p)$, as $n \to \infty$, by the law of large numbers.

In the example above, for product P1, we get $f(p_1) = 14p_1 - 7(1-p_1) = 21p_1 - 7$ and for P2 we have $g(p_2)=11p_2 - 6(1-p_2) = 17p_2 - 6$. The functions $f$ and $g$ are two straight lines. Assume that $p_1 = p_2 = p$, i.e., the products have equal chance of being sold. This is the plot of the two functions:

enter image description here

The intersection is at $p=1/4$. So if you expect to sell less than 25% of your inventory you should go with P2. If you expected to sell more that 25%, you should go with P1. A better choice is to not sell either if you expected to sell $\le 33\%$ (i.e. where $21p - 7 = 0$), and sell P1 otherwise. Below $\approx 33\%$, both products have negative expected return.

Two other ideas for more realistic modeling:

  • You can also view the problem as portfolio optimization, if you have expectation and variances of the returns on each product. You can use the variance as a measure of risk in that case and try to maximize the expectation of a portfolio subject to a given threshold on the risk of the portfolio.
  • You can model the problem as a multi-armed bandit. Each product is an arm that you can pull and get a reward (think of a slot machine). You try to simultaneously figure out the best arm to pick (i.e., estimate the expected return of each arm and find out which one has the largest return) as well as maximize your overall earning during the exploration part. There is the so-called exploration-exploitation trade-off. A famous algorithm to solve the problem is the upper confidence bound (UCB) algorithm. How to incorporate your known data so far (e.g., the profit/loss figures you have) is an interesting problem. There might be variants of the bandit problem that consider the notion of risk as well, otherwise it is worth thinking about.
passerby51
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  • Sir, Loss will always be a negative number. The profit or loss % define the net values. the price of the product is already being considered in the profit or loss % provided. I will look into the binomial distribution again, to reevaluate again. Thanks for your inputs. –  Dec 23 '19 at 20:36
  • I am using "b" as the absolute value of the loss. So I will have a negative profit of $-b$ or a loss of $b$. It is just a matter of notation. So when you say a loss of -7%, you can set $b= 0.07\times \text{price}$ in my notation. – passerby51 Dec 23 '19 at 20:45
  • Got it sir. But still trying to figure out, which product would give me the best returns for a considered risk level. I mean, Between P1 and P2, P1 gives an additional return of 3% for 1 unit of additonal risk over P2. Clearly P1 > P2, wrt returns. Not sure how to identify such products amongst thousands. Clearly dont see Binomial disnt being a solution. Am i missing something? –  Dec 23 '19 at 20:55
  • Suppose that noone wants to buy P1, but everyone wants to buy whatever amount of P2 they can get. Then, it doesn't matter if P1 is more profitable because you are not selling any. In that case, in the above model one would set p = 0 for P1 and p = 1 for P2. You need some estimate/expectation of how much of each product you will sell to be able to say anything. I will add some details. – passerby51 Dec 23 '19 at 21:18
  • I agree with everything you say. we have considered all those scenarios and hence, we are ready to sell the product at -ve value. Say, there are 100 different types of chocolates of different sizes, colours, costs and other attributes. Of all these, some are more profitable and some aren't. We are trying to identify those products based on risk to rewards ratio (Max gain per unit of investment with minimal risk), so a part of earnings would be invested on marketing these. Essentially, more profit with less risk. –  Dec 23 '19 at 21:46
  • Is selling chocolate more profitable than selling a soap? if yes, then we could organize the store to present more chocolates and fewer soaps. The math runs deep. we also calculate the rate of movement of goods. i.e. the day they were stocked and the rate at which they move out. Many other factors. One of them, for which, i am seeking alternative solutions is this. –  Dec 23 '19 at 21:48
  • You might be able to factor some of the data you have (on the movements of goods, etc.) into the problem. For example, if you consistently sell ~ 75% of your inventory for a particular chocolate month after month, then $p=0.75$ is a good estimate for that item. – passerby51 Dec 23 '19 at 23:32
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    Only based on the information you gave in the problem statement, it is hard to compare (if at all possible.) You can also view the problem as portfolio optimization, if you have expectation and variances of the returns on each product. You can use variance as a measure of risk in that case and try to maximize the expectation of a portfolio subject to a given threshold on the risk of the portfolio. – passerby51 Dec 23 '19 at 23:39
  • Thanks for your input sir. Going through these concepts now. will update you on the progress made. My initial understanding is, the portfolio optimization concept deals with maximizing returns for a specific risk rate. It doesn't discuss which is more beneficial. –  Dec 24 '19 at 05:16
  • @kris, added some more ideas on realistic modeling of the problem. – passerby51 Dec 31 '19 at 19:21
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You should just calculate the expected revenue of each of the items, and take the one with the highest expected revenue. All with a fixed investment. Either $N$ or $100$ to your liking.

Then the probability of loss/profit comes into play, if I understand you correctly, you mean $p_1$, the probability that $P_1$ will be sold with profit it $2/3$ from your PL ratio?!

Then you end up with $100 \cdot (p_1*1.14 + (1-p_1)*0.93)$, etcetera.

Then just take the maximum for all $3$ cases.

Pieter21
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