15 votes
Accepted

Convexity of Variance Minimization

It holds $$ \begin{array}{rcl} \operatorname V(x) &= &\dfrac1N\left\| x-\dfrac{e^\top x}{N} e \right\|^2 \\ & = & \dfrac1N\left(x^\top x+\dfrac{(e^\top x)^2 e^\top e}{N^2}-2\dfrac{...
ErlingMOSEK's user avatar
  • 3,101
13 votes
Accepted

Normal demand and normal lead time; is lead-time demand normal?

I had the same doubt, and I arrived at the conclusion that the formula given in the textbooks is, at best, a practical approximation. The lead-time demand, in fact, is not normally distributed. Let $...
Alberto Santini's user avatar
12 votes
Accepted

Standard cumulative distribution function with optimization model variable

For strictly increasing CDFs, you can invert: $$x \le \Phi^{-1}(b)$$
RobPratt's user avatar
  • 31.5k
12 votes
Accepted

Loss functions for specific probability distributions?

There is indeed a paper titled Loss Distributions that provides the limited expected value functions $L(x)$ for several probability distributions (on page 15). It is directly related to the first-...
TheSimpliFire's user avatar
  • 5,382
12 votes

Modeling the uncertainty of the input parameters

In reference to the first question, I think it often comes down to the information you have about the underlying uncertainty. If you only have intervals or ranges, robust is the way to go. If you have ...
E. Tucker's user avatar
  • 1,317
10 votes

Modeling the uncertainty of the input parameters

The following papers discuss this extensively with numerical experiments, but they tackle specific examples. Emphasis is mine. Kazamzadeh et al. (2017) This is a comparison of the two techniques using ...
TheSimpliFire's user avatar
  • 5,382
9 votes
Accepted

Modeling the uncertainty of the input parameters

Regarding your first question, I think other answers have summed it up pretty good. Two things I would add are as follows: Stochastic programming models (besides chance constraint/probabilistic ...
Ehsan's user avatar
  • 2,448
9 votes
Accepted

How to fit a Beta distribution to three estimates from an "expert"?

Let $a = \hat a$ and $b = \hat b$. Denote $r = \frac{\hat b - \hat m}{\hat m - \hat a}$. Then choose the shape parameters to be $\alpha_1 = \frac{4 + 3r + r^2}{1 + r^2}$ and $\alpha_2 = \frac{...
SecretAgentMan's user avatar
7 votes

Queuing models in R, $\lambda$ Little

Not directly answering your question of how to code it manually but for discrete simulation of queues in R I would strongly recommend the simmer package. The ...
CMichael's user avatar
  • 1,333
7 votes

Loss functions for specific probability distributions?

Article: https://www.researchgate.net/publication/369926923_Loss_functions_for_inventory_control Let $L_1(r)$, $L_c(r)$ and $L_2(r)$ be the first-order, complementary and second-order loss functions ...
Steven01123581321's user avatar
6 votes
Accepted

Safety stock for log-normal distribution demand

To calculate the base-stock to meet a 99% type-1 service level, we need the 0.99 fractile of the demand distribution. The safety stock level is the base-stock level minus the mean demand. For the ...
LarrySnyder610's user avatar
5 votes
Accepted

Convexity of the variance of a mixture distribution

In order to find the best upper bound for variance, for given input values of $u_i$ and $\sigma_i^2$, you should globally maximize variance with respect to the $w_i$, subject to the constraints $w_i \...
Mark L. Stone's user avatar
4 votes

Model or State Uncertainty in Queueing Model due to uncertain arrival rate

After having read Chapter 5.3 of Decision Making Under Uncertainty by Mykel J. Kochenderfer, I have come to some conclusions. We are dealing with model uncertainty, in which case we can formulate a ...
Dylan Solms's user avatar
4 votes

Normal demand and normal lead time; is lead-time demand normal?

I tried simulating lots of normally distributed lead times and the normally distributed demand in each. The lead time demand sure looks normal: But a normality test gives $p = 0$ to at least 9 ...
LarrySnyder610's user avatar
3 votes

How to calculate Cycle & Safety Stock

Cycle stock is used to meet the mean demand. Safety stock is used to protect against randomness in demand. So, I would use your historical data to calculate the mean demand over time and the inventory ...
LarrySnyder610's user avatar
3 votes

What is the meaning of monotone hazard rate (MHR) distribution?

For a Poisson process the rate of events is constant. The distribution of time between events in the Poisson process is exponential with $F(v)=1-e^{-\lambda v}$ for $v\ge 0$ which gives the hazard ...
kjetil b halvorsen's user avatar
3 votes

Binary variable to indicate zero probabilities

You are going to bump into a limitation of finite-precision floating point arithmetic. Basically, a small enough probability value will be indistinguishable from rounding error. Assuming you are using ...
prubin's user avatar
  • 38.7k
3 votes

Normal demand and normal lead time; is lead-time demand normal?

I've thought about this for a bit, and I now believe that leadtime demand in most common situations is not normally distributed, although it may be as usual a good approximation. Of course, we know ...
alerera's user avatar
  • 1,542
3 votes

Question about a queueing problem

This is a $M/M/1$ queue with Poisson arrival distribution with $\lambda =1/10$ and Exponential service distribution with $\mu=1/3$. The proportion of the time that the system is busy can be calculated ...
Oguz Toragay's user avatar
  • 8,642
2 votes

Modeling the uncertainty of the input parameters

Stochastic Optimization (SO) requires the probability distributions (PDF) of the uncertain variables which are usually hard to fit. Then, a large number of scenarios are required to be sampled from ...
shady mamdouh's user avatar
2 votes

Normal demand and normal lead time; is lead-time demand normal?

I have done extensive analysis of procurement lead time distribution across industries. In my experience, I found most of the distribution are heavily skewed towards the left. Yes, you are right! ...
Lal's user avatar
  • 81
2 votes
Accepted

Generating numbers that should add up to a fixed value while they follow a known distribution

I'm still not sure I understand the question, but I'll suggest an answer to what I think is being asked. I'm going to assume that an a priori upper bound $O$ exists for $o_t$. To keep what follows ...
prubin's user avatar
  • 38.7k
2 votes

What is the hazard-rate of a truncated probability distribution?

The blog entry Defining Hazard Rate at a Point Mass. "Applied Probability and Statistics in Actuarial Science and Financial Economics", contains what is needed to answer the question. Define ...
Mark L. Stone's user avatar
1 vote

Optimizing Safety Stock in a Multi Echelon Optimization

If you would like to take into account the inventory model directly in an optimization model, your problem is more likely classified in the following areas: Multi-Depot Location Routing Problem ...
A.Omidi's user avatar
  • 8,642
1 vote
Accepted

lsqcurvefit error in simulation of demands in MATLAB

My professor helped me for solving this error like below: ...
ramin's user avatar
  • 55
1 vote

How to calculate Cycle & Safety Stock

Safety stock for a demand stream (assuming it is normally distributed without recurring lumpiness) depends on a number of factors: Target service level Demand accuracy Lead time Supplier reliability ...
Julian Elwood's user avatar
1 vote

Queuing models in R, $\lambda$ Little

So far this is what I have come up with lambda <- 2 interarrivals <- rexp(5000,lambda) ## (2 items per minute) Provided the $\mu$ we expect that the ...
pulselaserman_quantum's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible