Almost without words. The profit P depends on the production volume x. The sale price is p(x). The production cost of x units is c(x).

P(x) = xp(x)-c(x).

Assume differentiability. The necessary conditions to maximize P is:

p(x) + xp'(x) - c'(x) = 0. In this general version such a condition characterizes the so called "monopoly equilibrium".

Under "perfect competition" the price p(x) is constant:: p(x) = k for every x.

In this case p'(x) = 0 and the necessary condition for the profit maximization boils down to k = c'(x).

There is only one case in which "max profit" is equivalent to "min cost".

This is the case of unitary demand elasticity: p(x) = a/x.

In this case:

P(x) = x a/x - c(x) = a - c(x).

The maximization of P is equivalent to the minimization of c.

Best.

Yes, there is a difference. Profit includes revenue which depends on the type of the market you model. Thus, maximizing profit includes additional decision on the production output.

Dear Carlos,

Maximizing profit can be done by maximizing revenues and/or minimizing costs. The equation is:

Profit = Revenues - Costs

But because, usually Revenues and Costs are related, minimizing costs may also minimize revenues and therefore will not maximize the profit.

I hope I answered your question.

Hello, you can increase profits by innovation in performance level of the product, so the the value of the product can increase for an constant cost, allowing price increases. Another option is to decrease product performance, the cost and price, in purpose to gaining profit.

Since the outcome in each case might be different, are there cases in which each approach is preferred?

For example, does it makes sense minimize costs, instead of maximize profit, in the case of nonprofit companies? Minimizing costs might lead to a better result for the customers.

Almost without words. The profit P depends on the production volume x. The sale price is p(x). The production cost of x units is c(x).

P(x) = xp(x)-c(x).

Assume differentiability. The necessary conditions to maximize P is:

p(x) + xp'(x) - c'(x) = 0. In this general version such a condition characterizes the so called "monopoly equilibrium".

Under "perfect competition" the price p(x) is constant:: p(x) = k for every x.

In this case p'(x) = 0 and the necessary condition for the profit maximization boils down to k = c'(x).

There is only one case in which "max profit" is equivalent to "min cost".

This is the case of unitary demand elasticity: p(x) = a/x.

In this case:

P(x) = x a/x - c(x) = a - c(x).

The maximization of P is equivalent to the minimization of c.

Best.

Minimizing costs is an internal function to an organization and is definitely possible. Maximizing profits is subject to market demand and sales, which is external to the organization. Thus, it is not certain.

Maximizing profit is starkly different then minimizing cost only when the producer can affect the market price. If the producer is a price taker, profit maximization is merely equating marginal cost (first derivative of cost fn) to the fixed price.

Not the same! Please see what Andrey Krasovskii, Michel Owayjan and Akiner Tuzuner said above. I have nothing to add.

I am agree with most the previous answers I just want to add : That depends on what is your problem, what you want modelize.

As it has been said previously Profit = Revenues - Costs. Then there is 3 cases according to the influence of the variables on the Revenues and on the costs...

1- If in your problem, you have variables that change both revenues and costs,

maximizing the Profit is different that minimizing the costs.

2- If in your problem, Revenues are fixed i.e. you can not make decision that change the revenues, maximizing the profit is equal to minimizing the costs.

3- If the costs are fixed because your optimization problem does not include variables that change the cost, but include vairables that change revenues then maximizing the Profit is equivalent to maximizing the revenues...

Dear Bertrand,

I do agree with you on the three points you made. If the costs and revenues are dependent, then minimizing costs may also decrease the revenues and not maximize profits.

Maximizing profit can be achieved by increasing price (cost is fixed) or by reducing cost (price is fixed). In some situation increasing price is not an option. So, in order to maximize profit we need to minimize cost. It means efficiency needs to be improved.

I also endorse yours

When we say 'maximizing profits', we aim at increasing the Volume of Sales, keeping cost of production factors constant. But 'minimizing costs' mean reducing the wastes, unnecessary costs involved in the manufacturing of a product. Analysis using various tools - lean,Six Sigma, Value analysis etc are applied to identify the areas of wastes such as in rawmaterial, high defect rate , idle man and machining hours etc . and unnecessary costs involved in cost of poor quality (COPQ), cost of labour in correcting the defects, maintenance etc. So, optimizing the wastes and unnecessary costs, we can reduce the cost of production. That is to strive to produce cheap and sell cheap keeping the utility and quality of the product intact. This helps to provide a high value to the product which promotes the sales. Thus we can conclude that minimizing cost of production keeping the utility and quality intact, would definitely raise the sales and aid in maximizing profit (profit = sales - cost of production).

Good

The difference between the two namely minimizing the cost and maximizing the profit are well brought out by several researchers in this forum. In solving linear pogramming problems the two are shown as mathematically equaivalent in some cases. But the most important point to remember is: minimizing the cost is an internal activity and can be achieved with proper techniques discussed earlier. But maximizing the profit is uncertain because profit is generated only if the sales occur. Hence it is dependent on the external environment and that too over an uncertain period of time. While one can minimize the cost, they cannot say that they have maximized the profit unless the profit is realized., because profit accrues post-sales.

Yes, they are not necessarily the same thing but my be two different things:

Maximizing profits may not be linked to cost, but to reducing overheads, and dealing with other assets of the organization that give us returns e.g. investments

Reducing cost increases margins; but the organization may still have losses due to enhanced overheads.

Dear Carlo,

The key to this solution is optimization. And to define the objective of the issue.

Dear Ali

I agree that using MCDM you can reach a compromise solution, but not optimize, because they are opposite aspects. Try to optimize Life and Death, Good and Bad, Rich and Poor

You can optimize one or the other but not both, or can get a result in between, reaching a compromise solution. You can do that using heuristic methods but they will never optimize.

I am preparing data to test the theory. I am not an expert in optimziation so would politely like to ask for your advice on which method is the best to find the 'Optimized Nurse Staffing." LP? Non-linear programming? MCDM? Data Envelopement Analysis? I know that it depends on my research question. Even so, here, I would like to get your farsighted view on my work. The topic is a hot issue in my discipline. I should pursue for the best.

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One note can be added here. Both profit and cost can be considered in one optimization model. In production systems there is a term called most productive scale size (MPSS). For example, when we are evaluating some peer companies which all use cost as input and profit as output, the MPSS maximizes the average of the difference between profit and cost. In other words, MPSS model maximizes the profit and minimizes the cost of the evaluated company at the same time.

For more information on MPSS, readers can refer to this paper:

https://www.researchgate.net/publication/324761124_Most_productive_scale_size_decomposition_for_multi-stage_systems_in_data_envelopment_analysis?_sg=Tl-fosjGvAoQGUZI6g_oLlHYGS9Fde2ZqaQMSMNHBCvmzA7pumoxu4q47oT16XRG5ceyZc8upQwvxfkzSdP4tQveHxEZDBZJCiJ1wpR7.uvLZlB-pnw9MnMGzgck8wxwEZ8Swa1iO8p8_X45mOsXaC3LPoJ_VvuMW9ekygSqUX5D08hbg_9tiMj8ilKY7iA

Dear sir nolberto,

thank you for clarification, and then I mean optimization within a specific context, for example: In the financial crisis, the least loss companies are optimal in the environmental context, in life and death too, the penalty itself is a life.

Thank Saeed Assani for your insightful comment. Let me read your paper carefully.