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A company should supply 2 products namely A and B. The company can buy A and B in advance or in real-time, however at different prices. The demand is uncertain and prices are considered to be deterministic. Therefore, I developed a stochastic model in which uncertain demand and prices are the input and its revenue for selling the products is the output.

LA : The price of product A when buying before the delivery time.
LB : The price of product B when buying before the delivery time.

CA: The price for selling A to the customer
CB: The price for selling B to the customer

LAA : The price of product A when buying at the delivery time.
LBB : The price of product B when buying at the delivery time.

R   : Revenue of the company

Now I want to apply a sensitivity analysis based on market prices. My assumption is to gradually increase each price while keeping the others constant and see it's the effect on the revenue. Therefore, I will get 6 tables regarding the relationship between the increase in each price wrt the revenue.

For example: Let say LA = 30 while R = 50. Thus I change LA to 31 and feed it to the model. Let say the R' will be 48. I continue this routine until LA = 40. Then I can see the relationship between LA and R for a range of prices.

However, this method only considers the impact of the change in one input variable on the output. Therefore, it doesn't consider the combined variation of inputs (LA, LB). How can I perform a valid sensitivity analysis for this problem where a subset of the inputs are changed while the others are fixed? Moreover, the prices in the model are somehow correlated in reality.

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  • $\begingroup$ What do you mean by "correlation of prices"? Your problem description indicates that prices are deterministic. Are you assuming a relationship among them, such that if one price changes others also will change? $\endgroup$ – prubin Aug 21 at 15:58
  • $\begingroup$ I have considered the mean value of prices in the model. However I am asked to apply a sensitivity analysis on market prices wrt the revenue. But, I know that market prices depends on each others as well. Anyhow still don't know how they are correlated with each others @prubin $\endgroup$ – SAH Aug 21 at 16:20
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    $\begingroup$ it sounds like you are concerned that the price elasticity of demand is correlated across products. I.e., price elasticity of demand for product A depends on the price of product B. Is that correct? If so, you may need a multivariate model,and multivariate analysis of it. @prubin The prices may be deterministic (set by the seller), but the demand is not deterministic. $\endgroup$ – Mark L. Stone Aug 22 at 14:20
  • $\begingroup$ Yes. How can I do so? $\endgroup$ – SAH Aug 23 at 9:28
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Disclaimer: I did my PhD on the topic below and was the author of one of the toolboxes I mention below.

This sounds a lot like you are interested in multi-parametric programming, i.e. solving your optimization problem as a function of the market prices. Each price is then a parameter and you can then solve the problem as a function thereof.

The only tools that allow you to do this (to my knowledge) are the MATLAB toolboxes MPT and POP.

The main caveat is that this is fairly restrictive in terms of number of parameters etc., because the number of optimal active sets can explode. But it is impossible to know a priori whether this will be the case. The problem with the most number of parameters I've ever solved was 67 parameters, but I've also had cases where a problem with 5 parameters was really, really difficult.

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    $\begingroup$ I don't think this addresses the question unless you show how the solution to the question can be formulated as a Multi-Parametric Program. In particular (see my comment on the question), how to address the correlation of price elasticity of demand across products/ $\endgroup$ – Mark L. Stone Sep 7 at 12:00

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