Finding the optimum between two functions of cost

[Lurifax]

Hi there

Im having some issues to find the optimum of two expenses functions, regarding inventory costs vs. downtime cost.
The situation is like this:

The function dictating the holding cost is like this:
(Average inventory value on a yearly basis) x (a percentage cost, i.e. 0,2 of the inventory value) = inventory cost

The function giving the downtime cost is like this:
(the total downtime on a yearly basis in days) x (the downtime cost pr. day) = the yearly downtime cost

I want to want to find the optimum between these two functions, which gives me the lowest point of the total cost graph. Can someone please help me with this one please.

Best regards

 

[stapel]

In what sense are you looking for the minimum? From the picture, it looks like the minimum is where the two cost-lines intersect. But without equations for each cost, I see no way of finding anything specific.

 

[Lurifax]

In what sense are you looking for the minimum? From the picture, it looks like the minimum is where the two cost-lines intersect. But without equations for each cost, I see no way of finding anything specific.

This is exactly what i would like to find; the cost-lines intersection.

I've written the two functions with "words", but as i can remember, the two functions needs to have a common letter, to solve the intersection between the two of those? Or is there a work around on this matter, that could help me?

 

[stapel]

...the two functions needs to have a common letter, to solve the intersection between the two of those?

If, by "letter", you mean "variable", then yes, this is necessary. Lacking any sort of formula for the two lines, I see no way to proceed.

 

[HallsofIvy]

It would help if you told us what you are talking about! If all you have is what you have told us- the graph without any formulas or equations, then all we can say is "measure the value on your graph". You say "I've written the two functions with "words", but as i can remember, the two functions needs to have a common letter". What "words" were those? "Inventory Holding Costs" and "Unavailability Costs"? The key point is that those are both "costs"! So you vertical distance on the graph represents "costs". You are asking for the point where the same "Stock Level" gives the same "costs" for both of those functions.

 

[JeffM]

Hi there

Im having some issues to find the optimum of two expenses functions, regarding inventory costs vs. downtime cost.
The situation is like this:
View attachment 3526
The function dictating the holding cost is like this:
(Average inventory value on a yearly basis) x (a percentage cost, i.e. 0,2 of the inventory value) = inventory cost

The function giving the downtime cost is like this:
(the total downtime on a yearly basis in days) x (the downtime cost pr. day) = the yearly downtime cost

I want to want to find the optimum between these two functions, which gives me the lowest point of the total cost graph. Can someone please help me with this one please.

Best regards

First, please read the other responses carefully. None of us is sure about what you are asking. But this looks like a common kind of economic problem.

\(\displaystyle t(s) = h(s) + u(s),\ where\ h(0) = 0,\ u(0) > 0,\ u(s) \ge 0,\ h'(s) > 0,\ and\ u'(s) < 0.\) I assume that is what is given.

The object is to minimize t(s). That minimum will necessarily be at a value of s such that t'(s) = 0. It will not generally be true that t(s) is minimized when h(s) = u(s).

\(\displaystyle t'(s) = h'(s) + u'(s) \implies t'(s) = 0\ at\ s\ such\ that\ h'(s) = - u'(s).\)

The picture is deceptive.

Suppose that h(x) = u(x) and that h'(x) = 0.4 and u'(x) = -0.6

In that case \(\displaystyle t(x + 1) \approx t(x) + 0.4 - 0.6 = t(x) - 0.2.\) Total costs are lower at x + 1 than they are at x, where h(x) = u(x).

This idea of equating marginal values reappears in economics over and over again.

Of course this assume that I have guessed correctly what the problem is.

 

[Lurifax]

First, please read the other responses carefully. None of us is sure about what you are asking. But this looks like a common kind of economic problem.

\(\displaystyle t(s) = h(s) + u(s),\ where\ h(0) = 0,\ u(0) > 0,\ u(s) \ge 0,\ h'(s) > 0,\ and\ u'(s) < 0.\) I assume that is what is given.

The object is to minimize t(s). That minimum will necessarily be at a value of s such that t'(s) = 0. It will not generally be true that t(s) is minimized when h(s) = u(s).

\(\displaystyle t'(s) = h'(s) + u'(s) \implies t'(s) = 0\ at\ s\ such\ that\ h'(s) = - u'(s).\)

The picture is deceptive.

Suppose that h(x) = u(x) and that h'(x) = 0.4 and u'(x) = -0.6

In that case \(\displaystyle t(x + 1) \approx t(x) + 0.4 - 0.6 = t(x) - 0.2.\) Total costs are lower at x + 1 than they are at x, where h(x) = u(x).

This idea of equating marginal values reappears in economics over and over again.

Of course this assume that I have guessed correctly what the problem is.

Okay. First of all thanks you for your patience, I'm now aware that i missed some things out in the first message.

The case is about a production facility with a downtime cost of app. 11 millions DKK a day.
Furthermore i know the holding cost of the stocked spare parts.

The equation with letters would look somehow like this i think:
I_value x 0,2 = Annual_I_cost
DT_days x 11 millions DKK/day = Annual_DT_cost

I've now provided the equations with the information i know. I it possible to find the optimum now?

 

[Lurifax]

Okay. First of all thanks you for your patience, I'm now aware that i missed some things out in the first message.

The case is about a production facility with a downtime cost of app. 11 millions DKK a day.
Furthermore i know the holding cost of the stocked spare parts.

The equation with letters would look somehow like this i think:
I_value x 0,2 = Annual_I_cost
DT_days x 11 millions DKK/day = Annual_DT_cost

I've now provided the equations with the information i know. I it possible to find the optimum now?

I've been in the thinking box for a while, and found, according to what @HallsofIvy writes, it would be fair to call equally name both Annual_I_cost and Annual_DT_cost as the term Cost, as they are represented by the same Y-coordinate. Would the new step then be setting these equations equal 0 and then putting them together or what is it?

 

[JeffM]

Okay. First of all thanks you for your patience, I'm now aware that i missed some things out in the first message.

The case is about a production facility with a downtime cost of app. 11 millions DKK a day.
Furthermore i know the holding cost of the stocked spare parts.

The equation with letters would look somehow like this i think:
I_value x 0,2 = Annual_I_cost
DT_days x 11 millions DKK/day = Annual_DT_cost

I've now provided the equations with the information i know. I it possible to find the optimum now?

Whoa. First, are you writing from Denmark? You are using typographical conventions that are a bit strange in the US.

Second, what are you studying? What is your math background? Do you understand function notation? Do you understand derivatives?

Third, there is a horrible practice in economics of using acronyms for variable names. Let's do it this way.

Quantity in stock = S

Holding cost = H = 0.2 * S. Is this correct?

Downtime cost = D = what? The equation that you seem to imply in words involves days and has no relation to quantity in stock. If you have one relationship stated in terms of days and another in terms of stock, there is no possible way to get an answer involving two completely different units.

It would help a lot if you gave a complete and exact statement of the problem, but I am not sure anyone here knows Danish.

 

[Lurifax]

Whoa. First, are you writing from Denmark? You are using typographical conventions that are a bit strange in the US.

Second, what are you studying? What is your math background? Do you understand function notation? Do you understand derivatives?

Third, there is a horrible practice in economics of using acronyms for variable names. Let's do it this way.

Quantity in stock = S

Holding cost = H = 0.2 * S. Is this correct?

Downtime cost = D = what? The equation that you seem to imply in words involves days and has no relation to quantity in stock. If you have one relationship stated in terms of days and another in terms of stock, there is no possible way to get an answer involving two completely different units.

It would help a lot if you gave a complete and exact statement of the problem, but I am not sure anyone here knows Danish.

Yes I'm from Denmark, and its quite some time since i've used common calculus, which is the explanation of some of the stupid questions

No. The correct equation for holding cost would be : I x 0,2 = holding cost
annual downtime cost : D x 11 mill. DKK/day = downtime cost (or unavailability cost)
I = Inventory value i DKK, D = Downtime stated in days.

Does this make sense?
Im thinking that holding cost and downtime cost could be set equal in terms of both are "cost", and thereby, there are only two unknown letters: I x 0,2 = D x 11 mill. DKK/day <--- And this can't be solved, right ?

 

[JeffM]

Yes I'm from Denmark, and its quite some time since i've used common calculus, which is the explanation of some of the stupid questions

No. The correct equation for holding cost would be : I x 0,2 = holding cost
annual downtime cost : D x 11 mill. DKK/day = downtime cost (or unavailability cost)
I = Inventory value i DKK, D = Downtime stated in days.

Does this make sense?
Im thinking that holding cost and downtime cost could be set equal in terms of both are "cost", and thereby, there are only two unknown letters: I x 0,2 = D x 11 mill. DKK/day <--- And this can't be solved, right ?

I apologize for being dense, but I am still not understanding the question.

When you showed your graph, you implied that both holding cost and downtime cost were related to the amount in stock. That leads to a soluble problem. What you are now saying is that downtime cost is not related to the quantity in stock, but to number of days of downtime. If that is all you have, there is no solution possible. I am guessing that there is IN ADDITION some relationship between stock levels and days down so that it is possible to find a relationship between stock levels and cost of downtime. We need that relationship. (By the way, the English word that you mean by "letters" is "variable." We "assign" a unique "letter" to represent each "variable.")

I am also guessing that the objective is to minimize total cost, the variable I previously designated as T, and that
T = H + D, where H and D both represent millions of krone in annual cost and both have a relationship with stock level S.

How am I doing?

 

[Lurifax]

I apologize for being dense, but I am still not understanding the question.

When you showed your graph, you implied that both holding cost and downtime cost were related to the amount in stock. That leads to a soluble problem. What you are now saying is that downtime cost is not related to the quantity in stock, but to number of days of downtime. If that is all you have, there is no solution possible. I am guessing that there is IN ADDITION some relationship between stock levels and days down so that it is possible to find a relationship between stock levels and cost of downtime. We need that relationship. (By the way, the English word that you mean by "letters" is "variable." We "assign" a unique "letter" to represent each "variable.")

I am also guessing that the objective is to minimize total cost, the variable I previously designated as T, and that
T = H + D, where H and D both represent millions of krone in annual cost and both have a relationship with stock level S.

How am I doing?

Ok, sorry i get your confusion about my first post. The case is, that the the values i have access to, are as you also tells me, not directly related, by still related in a sense of, how much i stock, would PROBABLY decrease my downtime, and hereby my downtime cost. But i only know the stocking level in sense of a quantity and total value. This is also why the holding cost are calculated on basis of a still picture of the current inventory value. You could of cause take the average unit cost, but i don't know if that would help i any way, if the quantity should Does that makes sense?

 

[Lurifax]

I apologize for being dense, but I am still not understanding the question.

When you showed your graph, you implied that both holding cost and downtime cost were related to the amount in stock. That leads to a soluble problem. What you are now saying is that downtime cost is not related to the quantity in stock, but to number of days of downtime. If that is all you have, there is no solution possible. I am guessing that there is IN ADDITION some relationship between stock levels and days down so that it is possible to find a relationship between stock levels and cost of downtime. We need that relationship. (By the way, the English word that you mean by "letters" is "variable." We "assign" a unique "letter" to represent each "variable.")

I am also guessing that the objective is to minimize total cost, the variable I previously designated as T, and that
T = H + D, where H and D both represent millions of krone in annual cost and both have a relationship with stock level S.

How am I doing?

You are doing just fine, this is exactly what trying to find: the lowest possible cost.

 

[Lurifax]

I would be happy to provide some additional information, I'm just not sure what you are asking for?

 

[JeffM]

I would be happy to provide some additional information, I'm just not sure what you are asking for?

I take it that this is not class exercise, but some sort of real world problem you are coping with. Am I correct?

This may take some time as I know no Danish. Here is what I think you are telling me: you are confident that there is a relationship between quantity of stock and days of downtime, but you do not know what it is. Furthermore, you suspect that the relationship is probabilistic, which does make things more complicated.

Now our first (minor) difficulty (if I am following you) is that I think that you represent inventory by the letter I whereas I have been using S so we are miscommunicating.

The second, much more serious difficulty is that I propose to ignore the fact that the number of days of downtime will vary around an average for each different level of inventory. That is, I am going to ignore the probabilistic nature of the problem.

The third, and most important difficulty is that you do not seem to have a good estimate of the relationship between inventory level and the expected days of downtime. Maybe you can make some informed guesses as a starting point. The problem is not even crudely soluble without at least a crude estimate on that relation.

Please see whether the general structure of the problem is correctly phrased in mathematical language below. Notice that I have changed the letters representing the variables. Please use my letters when responding so we are being mutually intelligible.

P = annual production (measured in physical units) if there is no downtime.

A = fraction of annual production held in inventory on average.

I = average inventory = PA

D = days of downtime per year

C = cost of downtime per day of downtime

L = annual cost of downtime = C * D

H = annual holding cost = 0.2 * I

T = total annual cost = H + L.

Objective is to find the A that minimizes T. The problem cannot be solved unless we can estimate a relationship between D and I.

Your initial guess in the graph suggests that the relationship between D and I is NOT linear, which seems a highly plausible suggestion.

So let's say \(\displaystyle D = \dfrac{X}{I}.\) For values of I greater than 0 and far less than P, that too seems plausible as an approximation..

Does this make sense so far?

If so here is the solution:

\(\displaystyle T = H + L = (0.2 * I) + (C * D) = 0.2 * A * P + C * \dfrac{X}{I} = 0.2AP + \dfrac{CX}{P} * A^{-1}.\)

\(\displaystyle \dfrac{dT}{dA} = 0.2P - \dfrac{CX}{P} * A^{-2}.\)

\(\displaystyle \dfrac{dT}{dA} = 0 \implies 0.2P - \dfrac{CX}{P} * A^{-2} = 0 \implies 0.2P = \dfrac{CX}{P} * A^{-2} \implies A^2 = \dfrac{CX}{0.2P^2} \implies A = \sqrt{\dfrac{CX}{0.2P^2}}.\)

A is the optimum fraction of annual production to be held as average inventory assuming variance is not an issue and the relationship between D and I is approximately correct. All you need to do is to estimate X to get a crude approximation of optimum inventory because you presumably already know C and P.

Now you can get someone to do a properly scientific study, but I suggest you need a Danish applied mathematician to do so. I have solved this using calculus, but a solution addressing probability may be best addressed through Monte Carlo methods.

 

[Lurifax]

I take it that this is not class exercise, but some sort of real world problem you are coping with. Am I correct?

This may take some time as I know no Danish. Here is what I think you are telling me: you are confident that there is a relationship between quantity of stock and days of downtime, but you do not know what it is. Furthermore, you suspect that the relationship is probabilistic, which does make things more complicated.

Now our first (minor) difficulty (if I am following you) is that I think that you represent inventory by the letter I whereas I have been using S so we are miscommunicating.

The second, much more serious difficulty is that I propose to ignore the fact that the number of days of downtime will vary around an average for each different level of inventory. That is, I am going to ignore the probabilistic nature of the problem.

The third, and most important difficulty is that you do not seem to have a good estimate of the relationship between inventory level and the expected days of downtime. Maybe you can make some informed guesses as a starting point. The problem is not even crudely soluble without at least a crude estimate on that relation.

Please see whether the general structure of the problem is correctly phrased in mathematical language below. Notice that I have changed the letters representing the variables. Please use my letters when responding so we are being mutually intelligible.

P = annual production (measured in physical units) if there is no downtime.

A = fraction of annual production held in inventory on average.

I = average inventory = PA

D = days of downtime per year

C = cost of downtime per day of downtime

L = annual cost of downtime = C * D

H = annual holding cost = 0.2 * I

T = total annual cost = H + L.

Objective is to find the A that minimizes T. The problem cannot be solved unless we can estimate a relationship between D and I.

Your initial guess in the graph suggests that the relationship between D and I is NOT linear, which seems a highly plausible suggestion.

So let's say \(\displaystyle D = \dfrac{X}{I}.\) For values of I greater than 0 and far less than P, that too seems plausible as an approximation..

Does this make sense so far?

If so here is the solution:

\(\displaystyle T = H + L = (0.2 * I) + (C * D) = 0.2 * A * P + C * \dfrac{X}{I} = 0.2AP + \dfrac{CX}{P} * A^{-1}.\)

\(\displaystyle \dfrac{dT}{dA} = 0.2P - \dfrac{CX}{P} * A^{-2}.\)

\(\displaystyle \dfrac{dT}{dA} = 0 \implies 0.2P - \dfrac{CX}{P} * A^{-2} = 0 \implies 0.2P = \dfrac{CX}{P} * A^{-2} \implies A^2 = \dfrac{CX}{0.2P^2} \implies A = \sqrt{\dfrac{CX}{0.2P^2}}.\)

A is the optimum fraction of annual production to be held as average inventory assuming variance is not an issue and the relationship between D and I is approximately correct. All you need to do is to estimate X to get a crude approximation of optimum inventory because you presumably already know C and P.

Now you can get someone to do a properly scientific study, but I suggest you need a Danish applied mathematician to do so. I have solved this using calculus, but a solution addressing probability may be best addressed through Monte Carlo methods.

This is a real problem yes. Trying to find the best possible balance between a spare part level for a production facility, and the cost of downtime, when it is not running

You are completely right on your assumptions also. The point i think that you misunderstood me, is to start with, when writing "P = annual production (measured in physical units) if there is no downtime." I do also have this number, of the perfect production, but the case in this matter regarding inventory is the spare parts held as inventory. Do you get me?
And the issue is the same with "A = fraction of annual production held in inventory on average.". The ONLY information i got on the inventory level of spare parts, i the current level, in both value, and numbers (the number aren't though interesting, because of the fact that every part has it's own individual price, ok?
Furthermore, as you are suggesting at the end, i should know P? But the fact is that i know the current inventory level (which is only a small fraction, of the parts that the facility are using). Does it make sense?
This means that making an estimate of the whole whole spare parts inventory, would be a far long shot, because the variation of these parts value goes from 1 dollar to 100.000 dollars.

With the above misunderstandings, i still think we are on to something with your solution of my problem. But I'm no sure, that P and A just can be replaced by this:

P = the total value of the annual production of the facility in terms of value or in terms of "units" (it can both be units and value, because i do know the value of each unit, and i also know the annual production of how many units produced each year. OR P = the estimated value of the whole spare part inventory, if all spare parts were stocked. (the last one CAN be made, but as I'm telling, it would be a long shot because of the varying value of the spare parts.)

A = the current inventory level in terms of value, which may be my suggestion, because of the fact that the value of each spare part, are highly variable. It could be both though.

As you may have noticed I'm preferring going with the values in the equations, because this is what i have the best basis for estimating, and it is also the economics I'm seeking to optimize in the end

I hope that i hans't mate this matter completely confusing for you?

And again THANK YOU SO MUCH!

 

[JeffM]

This is a real problem yes. Trying to find the best possible balance between a spare part level for a production facility, and the cost of downtime, when it is not running

You are completely right on your assumptions also. The point i think that you misunderstood me, is to start with, when writing "P = annual production (measured in physical units) if there is no downtime." I do also have this number, of the perfect production, but the case in this matter regarding inventory is the spare parts held as inventory. Do you get me?
And the issue is the same with "A = fraction of annual production held in inventory on average.". The ONLY information i got on the inventory level of spare parts, i the current level, in both value, and numbers (the number aren't though interesting, because of the fact that every part has it's own individual price, ok?
Furthermore, as you are suggesting at the end, i should know P? But the fact is that i know the current inventory level (which is only a small fraction, of the parts that the facility are using). Does it make sense?
This means that making an estimate of the whole whole spare parts inventory, would be a far long shot, because the variation of these parts value goes from 1 dollar to 100.000 dollars.

With the above misunderstandings, i still think we are on to something with your solution of my problem. But I'm no sure, that P and A just can be replaced by this:

P = the total value of the annual production of the facility in terms of value or in terms of "units" (it can both be units and value, because i do know the value of each unit, and i also know the annual production of how many units produced each year. OR P = the estimated value of the whole spare part inventory, if all spare parts were stocked. (the last one CAN be made, but as I'm telling, it would be a long shot because of the varying value of the spare parts.)

A = the current inventory level in terms of value, which may be my suggestion, because of the fact that the value of each spare part, are highly variable. It could be both though.

As you may have noticed I'm preferring going with the values in the equations, because this is what i have the best basis for estimating, and it is also the economics I'm seeking to optimize in the end

I hope that i hans't mate this matter completely confusing for you?

And again THANK YOU SO MUCH!

I am no more than usually confused. This site is not the proper place to answer your question: it is staffed by unpaid volunteers primarily to help students with schoolwork. Many years ago, I was employed to solve this kind of problem (in the sense of finding a reasonable approximation) in the real world. Nevertheless, I do not think think that you should engage me as a paid consultant because (1) I should not use a non-commercial site to solicit business (in fact, doing so may be a violation of the rules of this site), (2) I have not done work in inventory management for decades, and (3) although your English is good, my Danish is non-existent.

I can tell you that the problem you have now described is much more complex than my simple model above can address. Here is the reason. Yes, the spare parts inventory can be assigned a single aggregated monetary value, but that aggregate can be achieved in many ways. The real issue is how much money to assign to stocking which parts. In other words, the problem must be disaggregated to get a meaningful answer. This involves getting information about each part kept in stock: cost per unit, probability of failure per unit of time, units of time required to obtain the part if it is not in inventory. It might take a week to get shims, which cost $0.15 per shim, and six weeks to get mercury switches, which cost $15 per switch. If you merely minimize dollars invested, you may end up with way too many shims and way too few mercury switches. A solution will likely require a mixture of applied math and computer programming. To put it a different way, a model that tells you to keep 10 million krone worth of spare parts is likely to be wrong if it does not also tell you which parts to stock and what the probability of prolonged downtime is with that inventory.

In the US, you could probably find a competent consultant in a university's department of industrial engineering or applied math or operations research or at a consulting firm. I have no idea how Danish universities are organized so I cannot advise you on where to look for proper help in Denmark. Sorry to be of so little help.

 

[Lurifax]

I am no more than usually confused. This site is not the proper place to answer your question: it is staffed by unpaid volunteers primarily to help students with schoolwork. Many years ago, I was employed to solve this kind of problem (in the sense of finding a reasonable approximation) in the real world. Nevertheless, I do not think think that you should engage me as a paid consultant because (1) I should not use a non-commercial site to solicit business (in fact, doing so may be a violation of the rules of this site), (2) I have not done work in inventory management for decades, and (3) although your English is good, my Danish is non-existent.

I can tell you that the problem you have now described is much more complex than my simple model above can address. Here is the reason. Yes, the spare parts inventory can be assigned a single aggregated monetary value, but that aggregate can be achieved in many ways. The real issue is how much money to assign to stocking which parts. In other words, the problem must be disaggregated to get a meaningful answer. This involves getting information about each part kept in stock: cost per unit, probability of failure per unit of time, units of time required to obtain the part if it is not in inventory. It might take a week to get shims, which cost $0.15 per shim, and six weeks to get mercury switches, which cost $15 per switch. If you merely minimize dollars invested, you may end up with way too many shims and way too few mercury switches. A solution will likely require a mixture of applied math and computer programming. To put it a different way, a model that tells you to keep 10 million krone worth of spare parts is likely to be wrong if it does not also tell you which parts to stock and what the probability of prolonged downtime is with that inventory.

In the US, you could probably find a competent consultant in a university's department of industrial engineering or applied math or operations research or at a consulting firm. I have no idea how Danish universities are organized so I cannot advise you on where to look for proper help in Denmark. Sorry to be of so little help.

Ok ok. Hold your horses now. When i Said that it was a Real problem it is in fact a Real problem, and i am also a Real student, working on this issue in a Real Company as part of my education.
So my goal is in fact to come with my best answer, to how spare parts should be managed. And ONE of my approaches to this matter is the inventory cost vs downtime cost. You should not be in doubt that im Aware of all the facts you are mentioning regarding generel spare parts issues.
This equation is just an idea among many others, that i am working on with this particular matter.

So once and for all, i am a student, just in a wonderful situation, working on a Real problem as part of my Education (which in fact is the most valuble Way of learning in my opinion, but that is another discussion)

 

[Lurifax]

So JeffM if you would still like to help me, i would highly appreciate that!

 

[JeffM]

So JeffM if you would still like to help me, i would highly appreciate that!

We are here to help you do the work. The wikipedia article on inventory models cites a number of textbooks on this topic in English. You might see whether any of them are available to you through a university library.

It would help us help you if we knew more about your education. How good is your probability theory, your differential calculus, your microeconomics, your computer programming skills? It is hard to give advice in a void.

I suggest that you read READ BEFORE POSTING before posting again. I further suggest, because you will be doing the work, that you start a new thread for each problem as it comes up. People tend not to look at threads with multiple postings because they assume that one of the posts has already given the answer.

OK Now to answer the questions you have already asked.

First, it is good that you have a number of ideas. It may be that the problem will require a number of solutions to deal with different aspects of the problem.

Second, you may want to break down spare parts into categories, such as by unit cost, by frequency of need, and by time from order to delivery. In other words, how best to handle a part that is needed daily, takes a week to get delivery, and has a low unit cost may differ from how best to handle a part that is needed two or three times a year, has a three month delivery period, and has a very high unit cost. That categorization is merely an example and may not apply to your situation. The point is that a solution that is appropriate for some parts may not be appropriate for others. That is something for you to think about.

Third, ultimately, a solution will have to apply to individual parts. A bolt is not money. At some point, you need to figure out how many bolts to keep in inventory on average, order point, and order quantity, and you must do so part by part. That is where the computer comes in. You might start by going to see how your company does that now. At the very least, you may learn some practical constraints that a solution will have to address.

Fourth, your aggregated model that you asked about initially may be too conceptual for application, but that aggregated model is certainly good for presentations to discuss the purpose of your solution. Explaining your solution well enough that it is adopted is every bit as important as developing the solution because, if it is not adopted, your development is in vain. Moreover, your aggregated model may have practical use in setting budget guidelines.

Fifth, total cost is NOT necessarily minimized where holding cost equals downtime cost. As I explained before, let

v = monetary value of average level of inventory

h(v) = holding cost per period as a function of v

d(v) = cost of downtime per period as a function of v

t(v) = total cost as a function of v = h(v) + d(v).

Conceptually this is fine. It likely is true in practice that h(v) = a * v, where a > 0. The practical problem is that it may be difficult or even impossible to specify d(v) numerically, particularly because it assumes that the monetary value is allocated to various parts optimally, and that optimum has yet to be found.

However, it is certainly true that d'(v) < 0 and it is almost certainly true that d''(v) > 0 for all non-negative v. Consequently, the optimum value of v is v such that d'(v) = - a.

Sixth, you may want to do a little research on Monte Carlo methods if you are not familiar with them. It may give you something new to think about.