1. INTRODUCTION
A supply chain is the connected network of individuals, organizations, resources, activities, and technologies involved in the manufacture and sale of a product or service. A supply chain starts with the delivery of raw material from a supplier to a manufacturer, and ends with the delivery of the finished product or service to the end consumer. SCM oversees each touch point of a company's product or service, from initial creation to final sale. With so many places along the supply chain that can add value through efficiencies or lose value through increased expenses, proper SCM can increase revenues, decrease costs, and impact a company’s bottom line.
Healthcare supply chain, recently noted by researchers and practitioners as an area with significant and increasing impact on GDP, has great costs associated with it which requires attention in order to improve performance efficiency (Kwon et al., 2016). However, this problem contains levels of risk which is an important issue in the literatures of supply chain management (Ceryno et al., 2013).
Most risk analysis simulation software products offer Latin hypercube sampling (LHS). It is a method for ensuring that each probability distribution in your model is evenly sampled which at first glance seems very appealing. The technique dates back to 1980 (Iman et al., 1980) when computer was very slow, very modest number of distributions in a simulation model took hours or days. It was attractive back then because it allowed one to obtain a stable output with far fewer examples of simple Monte Carlo simulation.
Improving and handling uncertainty in sampling is a vital component for effective decision making. Uncertainty is insufficiently and explicitly communicated to random sampling methods (Nadjafi et al., 2014). The quantification and propagation of uncertainty become essential in precisely those situations where quantitative modeling cannot draw upon extensive historical, statistical, or measurement data (Kurowicka and Cooke, 2006). Fuzzy logic systems constitute a powerful tool for coping with ubiquitous uncertainty in many engineering applications (Linda and Manic, 2010). The most important point in assessing uncertainty of gained population via sampling is to recognize that all uncertainties are not quantifiable, and therefore they should be separated from the sampling characteristics (Verdonck et al., 2007). In sampling methods, if we are willing to give up some features of random sampling, notably serial independence, then variance reduction techniques may be invoked (Kwakernaak, 1978). A suitable mathematical model for random variables which assume fuzzy values are called fuzzy random variables (Zadeh, 1978) and use linguistic variables to encounter fuzzy sets (Zadeh, 1975).
However, desktop computers are now at least 1,000 times faster than the early 1980s, and the value of LHS has disappeared as a result. LHS does not deserve a place in modern simulation software. We are often asked why we don’t implement LHS in our model risk software, since nearly all other Monte Carlo simulation applications do. Hence, we thought it would be worthwhile to provide an explanation here (Vose, 2014).
Design optimization usually requires a large number of potentially expensive simulations. Translational propagation algorithm (Kwon et al., 2016) is use to obtain optimal or near optimal Latin hypercube designs without using formal optimization (Viana et al., 2010).
Latin hypercube sampling is generalized in terms of a spectrum of stratified sampling (SS) designs referred to as partially stratified sample (PSS) designs. The variance of PSS estimates is derived along with some asymptotic properties. PSS designs are shown to reduce variance associated with variable interactions, whereas LHS reduces variance associated with the main effects. Several highdimensional numerical examples highlight the strengths and limitations of the method (Shields and Zhang, 2016).
2. DEFININTIONS
One of the stratified sampling types is Latin hypercube sampling. This type of random sample is generated based on a probability distribution. Probability distributions can be described by a cumulative curve. If you want to take N samples from this distribution, you can split the scale into N equal probability ranges:
$\begin{array}{l}\text{[01\xd7}\left(\text{100/N}\right)\text{\%}\left]\text{,}\right[\text{1\xd7}\left(\text{100/N}\right)\text{2\xd7}\left(\text{100/N}\right)\\ \text{\%],\u2026,}\left[\left(\text{N1}\right)\text{\xd7}\left(\text{100/N}\right)\text{N\xd7}\left(\text{100/N}\right)\text{\%}\right]\text{.}\end{array}$
Now you should take one random sample within each range and calculate the variable value that has this cumulative probability. In a model that contains just one variable, the distribution can be stratified into the same number of partitions as there are samples: so, if you want N samples you can have N stratifications to be guaranteed that there will be precisely 1 sample in each 0.1% of the cumulative probability range.
However, risk analysis models can use many types of distributions. LHS controls the sampling of each distribution separately to provide even coverage for each distribution individually, but does not have control over numbers generated from the combinations of distributions. This means that extra precision offered by LHS over standard Monte Carlo sampling rapidly becomes imperceptible as the number of distributions increases. However, we want to increase LHS performance in both generations of numbers and also precision.
3. COMPARISIONS LHS WITH MONTE CARLO
We have chosen normal distribution and LHS offers the greatest improvement in precision over Monte Carlo sampling when sample number is small. There is essentially no difference between the LHS and Monte Carlo simulation results. The increase in precision offered by LHS is extremely modest, even when one applies it to simulations where it offers the greatest benefit (i.e. few distributions and few samples) and this increase in precision is trivial in comparison to the imprecision of the results achieved from running few samples.
3.1. Precision of Results
The more samples (iterations) one performs in a simulation model, the closer the results approach the theoretical distribution that would be given with an infinite number of samples. The proximity of the actual results to the theoretical is called the level of precision. There are statistical tests for determining the level of precision that one has achieved by running a Monte Carlo simulation. However, no such statistical tests are available if one uses LHS.
LHS is useful in a couple of particular circumstances (Vose, 2014). First, you have only one or two distributions in your model and you need the answers very fast. In this situation, for most models the mean will stabilize more quickly with LHS, though the spread and shape of the output distribution will not stabilize much more quickly than with Monte Carlo sampling, and it is the spread and the tails of the output distribution that we are most concerned about in risk analysis. Model risk can be simulated using Monte Carlo sampling, a model with two distributions 100,000 times in under 11 seconds, or 10,000 times in a second, by which time the precision of the results will be essentially indistinguishable from LHS. Second, you are using Monte Carlo sampling to perform a numerical integration. A very specific technical situation that mostly applies in scientific and engineering work, but if you need to do a numerical integration there are better methods than simulation which will give a far greater precision than is possible to achieve with simulation, and performs the calculation in a fraction of a second. Table 1
3.2. LHS types and Comparisons LHS with other Important Works
In other related references such as (Cochran, 1977; Davis, 1987; Iman and Conover, 1982; Iman, 1992; McKay et al., 1979; Pebesma and Heuvelink, 1999; Stein, 1987; Wyss and Jorgensen, 1998; Zhang and Pinder, 2003) you can find some improved algorithms that efficiently computed LHS results.
There are many proposed algorithms for improving simple LHS. In the following sections, we introduce an algorithm that uses fuzzy approach which shows a good accuracy and precision. However, for comparison purposes we use the best precision of the available algorithms in LHS to compare the obtained results from the proposed algorithm (Table 2).
4. FUZZY LHS SAMPLING
In the literature of statistical methods in sampling approaches, some methods have significant importance from the point of view of accuracy, e.g. Clustering and Stratified sampling, but as an uncertainty resource, we must reduce missed accuracy due to classification, So that we proposed fuzzy approach to encounter with this problem. Our method and formulas deducted from fuzzy contribution for LHS (type of Stratified Sampling) method (Viertl, 2011). Statistical data indicates variation in numbers. In other words, observations are not always precise numbers, or vectors, or categories or objects. Some types of data are frequently called fuzzy. Examples where this fuzziness is obvious are quality of healthcare data, environmental, biological, medical, sociological, and economeconomics data. Also the results of measurements can be best described by using fuzzy numbers and fuzzy approaches. Statistical analysis methods have to be adapted for the analysis of fuzzy data. The foundations of the description of fuzzy data are explained, including methods on how to obtain the characterizing function of fuzzy measurement results. Furthermore, statistical methods are then generalized to the analysis of fuzzy data and fuzzy apriori information. Key Features provides basic methods for the mathematical description of fuzzy data, as well as statistical methods that can be used to analyze fuzzy data. This work is aimed at statisticians working with fuzzy logic (Viertl, 2011).
In the explained method each element just belongs to one category, but in our approach each element includes in all groups with one exception that membership values are different. For example, consider age ranges (Figure 1) i.e. a man with 30 years old belongs to five groups by the following membership values:
When we categorize such samples with LHS method, precision for computing measures such as mean and variance could improve. In fact, each element is a member of every category with a probability between 0 and 1. Our new approach is a combination of fuzzy and LHS. In this approach, we assume that linguistic variables comprised as five parts and modeled in trapezoid.
4.1. Proposed Algorithm
Like stratified and LHS method we choose some random elements from each stratum for sampling, and then we identify the neighborhood of LHS to model in fuzzy context. Let n_{1}, n_{2}…, nm be the number of elements in each of the m sampled LHS. Let ${\overline{\text{X}}}_{1},\text{\hspace{0.17em}}{\overline{\text{X}}}_{2},\text{\hspace{0.17em}}\cdots ,\text{\hspace{0.17em}}{\overline{\text{X}}}_{m}$, be the means of the sampled LHS. The relative uncertainty itself can also be used without the subsequent statistical testing particularly if enough is known about the parameter being evaluated. Variance reduction techniques are methods that attempt to reduce the variance, i.e., the dispersion associated with the variations of the parameter being evaluated. This can result in one of two outcomes. Either the variance is reduced for the same number of sampling or the number of sampling can be reduced for the same variance, the comparison of both being made when no variance reduction techniques are used. This either increases the confidence in the results or reduces the computational burden. There are many forms of variance reduction niques and a specialized text should be consulted if full details of all techniques are needed. This algorithm provides reduced variances and standard errors comparison with other traditional methods. Schematic of proposed method is shown in Figure 2.
Steps that are used for programming with the use of fuzzyLHS method are mentioned in the following algorithm (Figure 2):

Determine the number of dimensions for problem

Select m (No. of points) random elements from each set of LHS

Determine neighborhood of selected point between sets of LHS

Calculate membership values for each LHS

Calculate mean value, variance, max confidence and min confidence from formulas

Calculate values for each LHS with effect of LHS neighbors

Calculate values with effect of element neighbors in each LHS

Calculate values with effect of both element neighbors and LHS neighbors
Formulations and relations that are used for computing the parameters such as mean, variance, maximum and minimum confidence intervals (lower and upper bound of mean estimator) are the same as LHS and Stratified Sampling method formulations (Cochran, 1977), in addition with fuzzy membership functions. Hence, for mean estimator of this method we have to compute the overall sample mean and we need to compute the sample means for each dimensions. We used R.viertl definition for representing fuzzy numbers (chapter 2) and then computing Mean value of fuzzy numbers.(1)
Then(2)
where Χ_{i} and n_{i} are the mean and the number of elements for LHS, respectively. The parameters μ_{i+1} and μ_{i1} are membership values of two neighboring LHS. An estimator of samples variance is given by
where $\overline{n}=\raisebox{1ex}{$\sum}_{i=1}^{m}{n}_{i$}\!\left/ \!\raisebox{1ex}{$q$}\right.$ and $q=m({\mu}_{i+1}+{\mu}_{i1})$(4)
where N and N_{i} are the number of observations in the population for LHS, respectively. Margin of error (ME) is defined as(5)
The range of the confidence interval is defined by the sample statistic ± margin of error and the uncertainty is denoted by the confidence level, using the preceding information. We construct a 95% confidence interval for the mean as(6)
Example 1:
The steering committee has decided to sample five grades of health care services in north of Iran, Mazandaran province. Thus the study will include hospitals, health centers, home visits and ambulance services, etc. For the purpose of this document “health facilities” will refer to this full range of services.
In order to provide the best chance for generalizability, health care facilities should be categorized. LHS allows the researcher to increase precision by grouping categories within the sample into more homogeneous sets. Standardized definitions of categories are important regarding generalizability and comparability of results.
Assume that we have 50000 available facilities that each set has 10000 facilities, this year. A proportionate Latin Hypercube sample was used to select 500 facilities for testing. Because the population has equal facilities for every stratum, each stratum consisted of 100 facilities. We modeled the problem using this method, LHS and exhaustive method (real mean and variance) in a system (win 7, mat lab R2010a, 2.30 GHz, 2 GB), and compared results given in Table 1.
Results show that significantly error values for fuzzy method is less than LHS and also approximations of mean value for fuzzy is close to populations mean (Figure 3). Fuzzy method such as LHS is changing the standard error, As a result, statistical methods can be seen clearly that this is an efficient method to calculate the statistical values is due to accuracy and ME. Standard error deviation in various implementation of fuzzy and LHS shown in Figure 4.
Also, Figure 5 shows the mean values for the exact, LHS, and fuzzy sampling. As obviously, the fuzzyLHS method has less deviation from the mean of exact solution rather than simple LHS method. So, this method shows less error from the exact method and we can conclude that accuracy rate for fuzzy method is higher than LHS.
Example 2:
Assume the following information:
Total iterations = 1000;
Number of Samples = 100;
Number of Variables = 6;
Real Mean = [10 5 4 3 20 10];
Standard Deviation = [0.1 1 0.1 1 1 1];
In the mentioned literature in section 4, we can conclude that correlation matrix and duplication items have important rule in gaining better results, i.e. correlation matrix must be positive definitive matrix because in these algorithms one important part of the solution procedure is transposing to lower triangle matrix and this restriction for finding sufficient correlation matrix can be challengeable from mathematical point of view. From other perspective, these approaches prove that by increasing duplication they can provide better results. However, both of these items have an overloaded cost for simulations. As you can see, we don`t need correlation matrix and duplication factor in the proposed FLHS algorithm and it is an improvement to other algorithms, especially you can consider an example that has the following properties:
Number of variables = 10,000; with duplication factor 100; without correlation matrix and other parameters introduced randomly. As a results in LHS procedure concluded matrix has 1,000,000×100 dimension that means for gaining better results we need a big space for middle computations. For gaining better and more precise results to decrease uncertainty, we added a step to overall algorithms called total iterations and finally the average of results showed in Table 2.
5. PROVING OPTIMALITY
We can prove optimality of our approach in two different aspects:
5.1. Mathematical View
Of course for decreasing variance, as we can see in formula (3), an important and effective variable factor is q where Membership values used to determine q, we look at our fuzzy model and in worst case scenario we selected a point between two trapezoid sums such that two neighbor membership values are greater than 1. This means our proposed algorithm always has better (less than) variance from simple one.
5.2. Experimental Results View
We put a variable in our implemented code and measured that for the worst case scenario but even in an example that our algorithm chooses points close to the edges of sets we see that we have better results than other mentioned algorithms.
In the above table, results show that using fuzzy approach leads to significant improvement in comparison to other methods.
Figure 6 and Figure 7 show an example of output coordination using FLHS respectively for 3D and 4D results for 10 points. You can enable data curser in Matlab figure view for higher dimension example to check that points are independent.
The main contribution of the proposed method for displaying statistical variance reduction and the approximate average is more accurate in general. View app for healthcare supply chain as it is used to reduce the risk of problems.
6. CONCLUSIONS
With the set of given numbers, elements or objects in the statistical approaches we can select the elements in a random way to compute values of many subjects like mean and variance. There are some sampling methods for this purpose discussed in many publications. In the entire study, we assumed that the input data is precise, i.e. each data belongs to a set. For example, in the LHS we assumed that the population of N units may be divided into m groups and the m LHS are none overlapping, and then samples selected from within each groups. However, in real world, examples and elements may belong to any group. Even in simple random sampling or systematic way we can see some errors lead to equal probabilities of elements. In this paper we introduce new approach base on fuzzy theory combined with sampling methods. The process is to calculate the relative error or uncertainty as the simulation proceeds and, using appropriate statistical tests generally based on the fuzzy theory, calculate the confidence interval after each sampling. This is compared with the prespecified example.
The case study results show that the proposed method provides better measure of uncertainty than the existing methods as unlike traditional sampling method. In this paper, in order to decrease the uncertainty of sampling, the variance reduction techniques were used. Variance reduction techniques are methods that attempt to reduce the variance, i.e., the dispersion associated with the variation of the parameter being evaluated.