# Sensitivity Analysis of the Optimal Inventory-Pooling Strategies According to Multivariate Demand Dependence

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Costs and Inventory Pooling Stetting

#### 2.1. Total Cost in Consolidation Setting

#### 2.1.1. The Cycle Stock Cost

- D
_{i}: is the mean demand during one time period at decentralized location i, - P
_{f}: is the fixed order processing cost at centralized location f ($ per order), - h
_{f}: is the unitary inventory holding cost per time period at centralized location f ($ per day), and - W
_{i,f}: is the proportion of mean demand during one time period transferred from decentralized location i to centralized location f (where $0\le {W}_{i,f}\le 1,$ for all i and f, and ${\sum}_{f=1}^{m}{W}_{i,f}=1,$ for all i).

#### 2.1.2. The Safety Stock Cost

- LT
_{f}: is the mean lead time at centralized location f;S_{LT, f}: is the standard deviation of lead time at centralized location f; - k: is the safety stock factor;
- ρ
_{ij}: is the correlation of demand between decentralized locations i and j; - S
_{D, i}: is the standard deviation of demand during one period at decentralized location i; - σ
_{f}: is the standard deviation of demand at centralized location f during the lead time period; - E(): random variable expectation; and
- V(): random variable variance.

_{f}is the standard deviation of a compound distribution while k represents the quantile of this distribution. The final expression of σ

_{f}is shown in Equation (5).

#### 2.1.3. The Distribution Cost

_{f,i}is the distribution cost rate, representing the unitary transportation of a single item from a centralized location f to a decentralized location i.

#### 2.2. Demand Allocation Rules and Inventory Pooling Models

- -
- Either all centralized locations offer the same proportion of demand at each decentralized location [3];
- -

#### 2.2.1. Tyagi and Das’ Allocation Rule (Inventory Centralization)

#### 2.2.2. Ballou and Burnetas’ Allocation Rule (Regular Transshipments or Independent Systems)

#### 2.2.3. Linking Allocation Rules to Inventory Pooling Models

_{1}and D

_{2}, with a cumulative distribution functions F

_{D}

_{1}and F

_{D}

_{2}, respectively. The centralized decision maker places an order at the manufacturer before observing demand.

## 3. The Proposed Approach

#### 3.1. Optimal Decision Search

_{1}+ D

_{2}), the Monte Carlo simulation technique is used. The simulated demands (D

^{′}

_{1}, D

^{′}

_{2}) are generated using the software R and the packages DEoptim [27], e1071, tseries, and msm. Multivariate random vectors can be generated using copulas in two steps. The first step is about the generation of pairs of uniformly distributed random numbers u

_{1}, u

_{2}using the chosen copula C, and the dependence level (Kendall’s) τ. The second step is the obtention of D′

_{1}and D′

_{2}by transforming them into the desired marginal distributions using ${D}_{1}^{\prime}={F}_{{D}_{1}}^{-1}\left({u}_{1}\right)$, ${D}_{2}^{\prime}={F}_{{D}_{2}}^{-1}\left({u}_{2}\right)$, respectively.

#### 3.2. Operational Aspects of Dependence Structure and Distributions

#### 3.2.1. Copulas and Dependence

_{1}and X

_{2}with marginal distributions U

_{1}and U

_{2}can be described by F (X

_{1}, X

_{2}) = C (U

_{1}, U

_{2}), with an adequate copula C (U

_{1}, U

_{2}) [28]. As a measure of dependence, a well-known rank correlation, Kendall’s τ belonging to [−1, 1] is used. The Kendall tau is a statistic that measures the association and rank correlation between two variables X

_{1}and X

_{2}. Since this measure of concordance is invariant to any strictly increasing transformation, it can be used to measure the non-linear dependence that cannot be measured by Pearson’s linear correlation coefficient. It is possible to express the Kendall tau in term of copula C which joins the variables X

_{1}and X

_{2}as described in Equation (9).

#### 3.2.2. Distributions Used to Construct Bivariate Data

- -
- If α = β, this distribution is symmetric (null skewness);
- -
- if α < β, this distribution is asymmetric shifted left (with negative skewness); and
- -
- if α > β, this distribution is asymmetric shifted right (with positive skewness).

#### 3.3. Influence Degree of the Variables on the Decision

#### 3.4. Comparison between Decisions and Detection of Convergence and Divergence Cases

## 4. Results

#### 4.1. Input Data and Hypthesis

**Hypothesis**

**1.**

**Hypothesis**

**2.**

**Hypothesis**

**3.**

**Hypothesis**

**4.**

#### 4.2. Effect of Marginal Demand Distribution When the Demands Follow to a Symmetric Distribution

#### 4.2.1. Case of the Marginals Each Following a Truncated Normal Distribution

_{1}but accepted for the demand D

_{2}. It indicates that these two factors do not generally have a considerable effect on the chosen policy.

#### 4.2.2. Case of Marginals Following (Truncated Normal Distribution (100, 30)/Beta (5, 5))

_{1}and hypothesis H

_{2}. For example, if the dependence is modeled by a Clayton copula, then there is a convergence in (51 + 1341 + 8392 = 9784 cases) while there are 216 divergence cases. There are 73 cases where the hypothesis H

_{1}favors a transshipment policy whereas the hypothesis H

_{2}favors an independence policy. Moreover, there are 34 cases where the transshipment is preferable under H

_{1}but the centralization is adequate under H

_{2}. This shows that the assumption of a normal demand when, in reality, the demand is distributed according to another symmetric distribution other than normal increases the risk of error in the decision to 2.2% assuming an adequate copula choice.

_{1}or H

_{2}) favors an independence policy, then the other hypotheses cannot opt for a centralization policy whatever the selected copula.

#### 4.3. Effect of Marginal Demand Distribution When the Demands Follow to a Asymmetric Distribution

#### 4.3.1. Case of a Symmetrical Demand (beta (5, 5) and the Other Is Asymmetrical (beta (2, 8))

#### 4.3.2. Case of Two Asymmetrical Demands one Tailed to the Right and the Other Tailed to the Left

## 5. Discussion

- The majority of variables that enter into the total cost calculation are factors that influence the decision. The most influential factor is the dependence between demands. Generally, a rather high positive correlation is in favor of setting up an independent system while a weak correlation makes it possible to opt a transshipment policy. Other factors such as the standard deviation of demand and the average lead time can influence the decision.
- Results indicate, for all assumptions on marginals distribution, that the optimal pooling policy to face medium correlation demands of low holding cost products is to keep inventory decentralized and inventory centralized, as if they were independent and centralization systems. On the other hand, if the demands correlation is low and the holding cost is medium, then the transshipment system is preferable.
- For all assumptions on marginals distribution and for all considered copulas, the centralization policy, for convergence cases, is preferred followed by the independence system and then the transshipment policy.
- The degree of divergence in which each copula does not lead to the same conclusion (same optimal policy) at the optimum is in about 9% of cases. This percentage varies slightly according to the assumptions used on the marginal distributions of demands.
- The assumption of a normal demand when in reality the demand is distributed according to another symmetric distribution other than normal increases the risk of error in the decision to 2.2% assuming an adequate copula choice. The assumption of the normality of demands when in reality the demands are distributed according to another asymmetric distribution increases the risk of error in the decision to 4.3%. Further, the percentage of divergence between two symmetrical demands (identical or different) on the one hand and, a symmetrical demand (no normal) and an asymmetrical demand on the other hand is of the order of 4.5%. This shows that the assumption of a symmetrical demand whereas in reality the demand is asymmetric increases the risk of error in the decision from 2% to 4.5%.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notations

CSc | The cycle stock cost |

DC | The distribution cost |

D_{i} | The mean demand during one time period at decentralized location i |

d_{f,i} | The unitary transportation cost to move a single item from a centralized location f to a decentralized location i |

EOQ_{f} | The economic order quantity at a centralized location f |

h_{f} | The unitary inventory holding cost per time period at centralized location f ($ per day) |

iid | Independent and identically distributed |

k | The safety stock factor |

LT_{f} | The mean of the lead time at centralized location f |

m | Number of decentralized locations (Markets) |

n | Number of the centralized facilities (or stocking points that serve demand) |

S_{D,i} | The standard deviation of demand during one period at decentralized location i |

S_{LT, f} | The standard deviation of lead time at centralized location f |

SSc | The safety stock cost |

TC | The total cost |

P_{f} | The fixed order processing cost at centralized location f ($ per order) |

W_{i,f} | The proportion of mean demand during one time period transferred from decentralized location i to centralized location f |

σ_{f} | The standard deviation of demand at centralized location f during the lead time period |

ρ_{ij} | The correlation of demand between decentralized locations i and j |

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**Figure 1.**Different allocation rules [20].

Variables | Uniformly Distributed Parameters | |
---|---|---|

Min | Max | |

LT_{1} and LT_{2} (in days) | 1 | 5 |

S_{LT,}_{1} and S_{LT,}_{2} (in days) | 0.50 | 2 |

S_{D,}_{1} and S_{D,}_{2} (in days) | 3 | 30 |

Correlation ρ_{12} | −1 | 1 |

Kendall’s tau | 0 | 1 |

k | 1 | 3 |

P_{1} ($/ order) | 17 | 67 |

P_{2} ($/ order) | 20 | 140 |

h_{1} = h_{2} = h ($/ unit/ day) | 0 | 0.68 |

Unit distribution cost d_{1,1} and d_{2,2} ($/ unit) | 0.10 | 0.15 |

Unit distribution cost d_{1,2} et d_{2,1} ($/ unit) | 0.35 | 0.40 |

**Table 2.**Non-parametric tests for discrimination between groups’ product characteristics (Case where each demand is distributed according to a normal distribution truncated in the interval [0, 200]).

Variables | Convergence (91.93%) | Divergence (8.07%) | Kruskal–Wallis Test | |||
---|---|---|---|---|---|---|

Transshipment (0.16%) | Independent System (10.91%) | Centralization (80.86%) | Independence/Transshipment/ Centralization | Statistic | p-Value | |

Kendall tau | 0.1138 | 0.5471 | 0.5072 | 0.4243 | 52.977 ^{(2)} | 0.000 |

D_{1} | 75.8326 | 103.3217 | 99.3771 | 96.3395 | 18.059 ^{(7)} | 0.001 |

D_{2} | 92.9854 | 102.6983 | 99.8070 | 95.9804 | 21.597 ^{(5)} | 0.000 |

S_{D}_{1} | 32.9502 | 29.45726 | 30.0497 | 30.8193 | 11.908 ^{(8)} | 0.008 |

S_{D}_{2} | 33.1804 | 29.6666 | 29.9964 | 30.3597 | 5.8837 | 0.117 |

LT1 | 3.7589 | 2.9759 | 3.0153 | 2.9269 | 6.657 | 0.083 |

LT2 | 3.7711 | 2.9434 | 3.0037 | 2.9913 | 2.1612 | 0.539 |

S_{LT}_{1} | 1.2867 | 1.2934 | 1.2674 | 1.1592 | 21.784 ^{(4)} | 0.000 |

S_{LT}_{2} | 0.7516 | 1.3100 | 1.2497 | 1.3228 | 20.668 ^{(6)} | 0.000 |

k | 2.7407 | 1.9121 | 2.0036 | 2.0855 | 32.239 ^{(3)} | 0.000 |

P_{1} | 37.4917 | 42.0898 | 41.9275 | 41.8333 | 1.1059 | 0.776 |

P_{2} | 76.3640 | 77.3581 | 80.8451 | 77.7781 | 6.0931 | 0.1072 |

d_{11} | 0.1212 | 0.1248 | 0.1251 | 0.12493 | 0.75384 | 0.860 |

d_{12} | 0.3730 | 0.3755 | 0.3758 | 0.3745 | 1.08021 | 0.615 |

d_{21} | 0.3827 | 0.3745 | 0.3748 | 0.3752 | 1.0207 | 0.796 |

d_{22} | 0.1201 | 0.1246 | 0.1253 | 0.1242 | 1.0144 | 0.798 |

h | 0.5039 | 0.2917 | 0.3399 | 0.39924 | 83.729 ^{(1)} | 0.000 |

**Table 3.**Impact of tail dependence on the allocation rules for divergent results (case where each demand follows a truncated normal distribution (100, 30)).

Tail Dependence | Copula | Allocation Rules | ||
---|---|---|---|---|

Transshipment | Independent System | Inventory Centralization | ||

None | Normal | 141 (17.47%) | 349(43.25%) | 317(39.28%) |

Only upper tail | Gumbel | 149 (18.46%) | 338 (41.88%) | 320 (39.65%) |

Only lower tail | Clayton | 136 (16.85%) | 328 (40.65%) | 343 (42.50%) |

Both upper and lower | Frank | 148 (18.34%) | 330 (40.89%) | 329 (40.77%) |

**Table 4.**Non-parametric tests for discrimination between groups product characteristics (case where demand 1 and 2 are distributed according to a normal distribution (100, 30) truncated and beta (5, 5) respectively).

Variables | Convergence Results (92.72%) | Divergence Results (7.28%) | Kruskal–Wallis Test | |||
---|---|---|---|---|---|---|

Transshipment (0.26%) | Independent System (11.51%) | Centralization (80.95%) | Independence/ Transshipment/ Centralization | Statistic | p-Value | |

Kendall tau | 0.1130704 | 0.5697587 | 0.4950283 | 0.4116491 | 100.69 ^{(1)} | 0.000 |

D_{1} | 79.35873 | 99.85702 | 100.2379 | 99.1499 | 7.1567 | 0.067 |

D_{2} | 81.04929 | 101.4088 | 100.6575 | 100.2552 | 3.3183 | 0.3451 |

S_{D}_{1} | 29.9405 | 29.9405 | 30.07159 | 30.13745 | 17.572 ^{(6)} | 0.001 |

S_{D}_{2} | 34.86215 | 29.97824 | 29.99994 | 29.7123 | 9.8394 ^{(7)} | 0.019 |

LT1 | 3.399485 | 2.940081 | 2.995756 | 3.017965 | 4.2123 | 0.2394 |

LT2 | 3.48428 | 2.925191 | 3.020522 | 3.117185 | 2.2221 | 0.527 |

S_{LT}_{1} | 0.876977 | 1.252383 | 1.26419 | 1.176481 | 22.612 ^{(5)} | 0.000 |

S_{LT}_{2} | 0.9168764 | 1.248905 | 1.243123 | 1.360544 | 27.706 ^{(3)} | 0.000 |

K | 2.598716 | 1.980567 | 2.013901 | 2.068317 | 25.094 ^{(4)} | 0.000 |

P_{1} | 37.81694 | 41.69101 | 42.30205 | 41.34258 | 0.21993 | 0.974 |

P_{2} | 61.18521 | 78.55782 | 79.58111 | 79.37657 | 2.4177 | 0.490 |

d_{11} | 0.1185323 | 0.124659 | 0.1249411 | 0.1268122 | 1.1423 | 0.766 |

d_{12} | 0.3685662 | 0.3756773 | 0.374676 | 0.3756215 | 4.2429 | 0.2364 |

d_{21} | 0.3763664 | 0.374313 | 0.3751264 | 0.3745828 | 0.92412 | 0.8196 |

d_{22} | 0.12606 | 0.1243646 | 0.1246347 | 0.1263157 | 2.0969 | 0.5525 |

h | 0.6041329 | 0.3081451 | 0.3400904 | 0.3874453 | 66.312 ^{(2)} | 0.000 |

**Table 5.**Impact of tail dependence on the allocation rules for divergent results (case where demand 1 follows a truncated normal distribution (100, 30) and demand 2 follows beta (5, 5)).

Tail Dependence | Copula | Allocation Rules | ||
---|---|---|---|---|

Transshipment | Independent System | Inventory Centralisation | ||

None | Normal | 119 (16.35%) | 311 (42.72%) | 298 (40.93%) |

Only upper tail | Gumbel | 105 (14.42%) | 292 (40.11%) | 331 (45.47%) |

Only lower tail | Clayton | 122 (16.76%) | 281 (38.60%) | 325 (44.64%) |

Both upper and lower | Frank | 115 (15.80%) | 282 (38.74%) | 331 (45.47%) |

**Table 6.**Comparison between decisions of H1 and H2 (Normal/Normal versus Normal/Beta (5, 5) demands).

Copula | H2: Normal/Beta(5,5) H1: Normal/Normal /Normal | Transshipment | Independent System | Inventory Centralization |
---|---|---|---|---|

Normal | Transshipment | 45 | 74 | 36 |

Independence | 85 | 1363 | 0 | |

Centralization | 35 | 0 | 8362 | |

Divergent results | 230 | |||

Clayton | Transshipment | 51 | 73 | 34 |

Independence | 75 | 1341 | 0 | |

Centralization | 34 | 0 | 8392 | |

Divergent results | 216 | |||

Gumbel | Transshipment | 46 | 72 | 40 |

Independence | 86 | 1344 | 0 | |

Centralization | 34 | 0 | 8378 | |

Divergent results | 232 | |||

Frank | Transshipment | 50 | 73 | 35 |

Independence | 75 | 1343 | 0 | |

Centralization | 35 | 0 | 8389 | |

Divergent results | 218 |

**Table 7.**Non-parametric test for optimal results per variable (case of demands beta (5, 5)/beta (2, 8)).

Variables | Convergence (91.18%) | Divergence (8.82%) | Kruskal–Wallis Test | |||
---|---|---|---|---|---|---|

Transshipment (1.91%) | Independent System (7.73%) | Centralization (81.54%) | Independence/ Transshipment/ Centralization | Statistic | p-Value | |

Kendall tau | 0.2633759 | 0.6369554 | 0.4988388 | 0.4573843 | 139.84 ^{(2)} | 0.000 |

D_{1} | 30.33109 | 39.57752 | 36.45619 | 35.37287 | 22.628 ^{(7)} | 0.000 |

D_{2} | 97.27427 | 101.7792 | 100.2069 | 100.0294 | 5.2316 | 0.156 |

S_{D}_{1} | 32.76588 | 28.44602 | 30.1144 | 29.30128 | 39.045 ^{(6)} | 0.000 |

S_{D}_{2} | 29.74112 | 29.95576 | 30.01248 | 29.9597 | 1.619 | 0.655 |

LT1 | 3.372328 | 2.844004 | 3.02274 | 2.979948 | 17.84 ^{(8)} | 0.001 |

LT2 | 3.520158 | 3.03435 | 3.009818 | 0.9427624 | 12.576 ^{(9)} | 0.005 |

S_{LT}_{1} | 0.9427624 | 1.383117 | 1.258381 | 1.182262 | 83.409 ^{(3)} | 0.000 |

S_{LT}_{2} | 1.449199 | 1.121113 | 1.238151 | 1.354034 | 75.114 ^{(4)} | 0.000 |

K | 2.30984 | 1.825848 | 1.98886 | 2.032396 | 73.735 ^{(5)} | 0.000 |

P_{1} | 43.05813 | 43.70508 | 41.75749 | 40.03104 | 7.5397 | 0.056 |

P_{2} | 80.85732 | 73.00713 | 80.25319 | 79.45784 | 11.237 ^{(10)} | 0.011 |

d_{11} | 0.1271849 | 0.1254713 | 0.1249515 | 0.1260524 | 2.214 | 0.529 |

d_{12} | 0.3744417 | 0.3748704 | 0.3746136 | 0.3749082 | 1.0547 | 0.788 |

d_{21} | 0.3733928 | 0.3751038 | 0.375288 | 0.3749008 | 0.10973 | 0.991 |

d_{22} | 0.1277572 | 0.1237569 | 0.1255486 | 0.1250479 | 3.9424 | 0.268 |

h | 0.5138296 | 0.2265138 | 0.3384318 | 0.3950181 | 248.85 ^{(1)} | 0.000 |

**Table 8.**Impact of tail dependence on the allocation rules for divergent results (demands follow beta (5, 5)/beta (2, 8)).

Tail Dependence | Copula | Allocation Rules | ||
---|---|---|---|---|

Transshipment | Independent System | Inventory Centralisation | ||

None | Normal | 269 | 353 | 260 |

Only upper tail | Gumbel | 259 | 339 | 284 |

Only lower tail | Clayton | 230 | 366 | 286 |

Both upper and lower | Frank | 255 | 341 | 286 |

Copula | H3: Beta (5,5)/Beta (2,8) H1: Normal/Normal Normal | Transshipment | Independent System | Inventory Centralization |
---|---|---|---|---|

Normal | Transshipment | 77 | 41 | 37 |

Independence | 363 | 1085 | 0 | |

Centralization | 20 | 0 | 8377 | |

Divergences results | 461 | |||

Clayton | Transshipment | 77 | 48 | 36 |

Independence | 352 | 1062 | 0 | |

Centralization | 24 | 0 | 8402 | |

Divergences results | 459 | |||

Gumbel | Transshipment | 83 | 32 | 43 |

Independence | 323 | 1107 | 0 | |

Centralization | 15 | 0 | 8397 | |

Divergences results | 413 | |||

Frank | Transshipment | 75 | 48 | 35 |

Independence | 352 | 1066 | 0 | |

Centralization | 19 | 0 | 8405 | |

Divergences results | 454 |

**Table 10.**Comparison of results between H2 and H3 (Normal/beta (5, 5) versus beta (5, 5)/beta (2, 8)).

Copula | H3: Beta (5,5)/Beta (2,8) H2: Normal/Beta(5,5) Beta (5,5) | Transshipment | Independent System | Inventory Centralization |
---|---|---|---|---|

Normal | Transshipment | 81 | 49 | 35 |

Independence | 360 | 1077 | 0 | |

Centralization | 19 | 0 | 8379 | |

Divergence results | 493 | |||

Clayton | Transshipment | 83 | 41 | 36 |

Independence | 343 | 1071 | 0 | |

Centralization | 24 | 0 | 8402 | |

Divergence results | 444 | |||

Gumbel | Transshipment | 88 | 41 | 37 |

Independence | 318 | 1098 | 0 | |

Centralization | 15 | 0 | 8403 | |

Divergence results | 411 | |||

Frank | Transshipment | 83 | 41 | 36 |

Independence | 343 | 1073 | 0 | |

Centralization | 20 | 0 | 8404 | |

Divergence results | 440 |

**Table 11.**Non-parametric test for optimal results per variable (case of demands beta (8, 2)/beta (2, 8)).

Variables | Convergence (90.82%) | Divergence (9.18%) | Kruskal–Wallis Test | |||
---|---|---|---|---|---|---|

Transshipment (1.55%) | Independent System (7.76%) | Centralization (81.51%) | Independence/ Transshipment/ Centralization | Statistic | p-Value | |

Kendall tau | 0.2906627 | 0.6185432 | 0.4968518 | 0.470429 | 92.973 ^{(2)} | 0.000 |

D_{1} | 30.19045 | 37.79599 | 36.62742 | 36.09918 | 11.022 | 0.012 |

D_{2} | 157.2561 | 165.8269 | 163.5958 | 163.4406 | 1.937 | 0.5856 |

S_{D}_{1} | 32.76588 | 28.17479 | 30.11815 | 29.42868 | 49.176 ^{(6)} | 0.000 |

S_{D}_{2} | 30.57189 | 30.00948 | 30.02349 | 29.62515 | 2.1866 | 0.535 |

LT1 | 3.29439 | 2.852775 | 3.020513 | 3.026129 | 18.272 ^{(7)} | 0.000 |

LT2 | 3.628139 | 2.894886 | 3.034347 | 3.004923 | 18.053 ^{(8)} | 0.000 |

S_{LT}_{1} | 0.9358209 | 1.381382 | 1.258955 | 1.179125 | 86.827 ^{(4)} | 0.000 |

S_{LT}_{2} | 1.340071 | 1.099255 | 1.237596 | 1.395932 | 92.459 ^{(3)} | 0.000 |

K | 2.325581 | 1.828309 | 1.990122 | 2.030634 | 63.789 ^{(5)} | 0.000 |

P_{1} | 40.75418 | 43.83816 | 41.73683 | 40.28208 | 7.5988 | 0.055 |

P_{2} | 82.40695 | 73.34916 | 80.19862 | 79.77782 | 8.2998 ^{(9)} | 0.040 |

d_{11} | 0.1294936 | 0.125543 | 0.1249481 | 0.1255741 | 5.372 | 0.147 |

d_{12} | 0.3737614 | 0.3747455 | 0.3746526 | 0.3746703 | 0.64197 | 0.888 |

d_{21} | 0.3723548 | 0.3752034 | 0.375292 | 0.3747065 | 1.5818 | 0.663 |

d_{22} | 0.1298994 | 0.1240718 | 0.1255781 | 0.124536 | 4.8427 | 0.184 |

h | 0.5170642 | 0.2299275 | 0.33857 | 0.392985 | 223.49 ^{(1)} | 0.000 |

**Table 12.**Impact of tail dependence on the allocation rules for divergent results (demands follow beta (8, 2)/beta (2, 8)).

Tail Dependence | Copula | Allocation Rules | ||
---|---|---|---|---|

Transshipment | Independent System | Inventory Centralization | ||

None | Normal | 259 | 363 | 296 |

Only upper tail | Gumbel | 233 | 367 | 318 |

Only lower tail | Clayton | 258 | 353 | 307 |

Both upper and lower | Frank | 231 | 369 | 318 |

Copula | H1: Normal/Normal H4: Beta (8,2)/Beta(2,8) Beta (2,8) | Transshipment | Independence | Centralization |
---|---|---|---|---|

Normal | Transshipment | 70 | 290 | 18 |

Independence | 49 | 1158 | 0 | |

Centralization | 36 | 0 | 8379 | |

Divergences results | 393 | |||

Gumbel | Transshipment | 83 | 351 | 26 |

Independence | 39 | 1065 | 0 | |

Centralization | 36 | 0 | 8400 | |

Divergences results | 452 | |||

Clayton | Transshipment | 81 | 331 | 23 |

Independence | 34 | 1099 | 0 | |

Centralization | 43 | 0 | 8389 | |

Divergences results | 431 | |||

Frank | Transshipment | 83 | 350 | 20 |

Independence | 39 | 1068 | 0 | |

Centralization | 36 | 0 | 8404 | |

Divergences results | 445 |

**Table 14.**Comparison of results between H2 and H4 (beta (5, 5)/Normal versus beta (8, 2)/beta (2, 8)).

Copula | H2: Beta (5,5)/Normal H4: Beta (8,2)/ Beta (2,8) | Transshipment | Independence | Centralization |
---|---|---|---|---|

Normal | Transshipment | 66 | 294 | 18 |

Independence | 64 | 1143 | 0 | |

Centralization | 35 | 0 | 8380 | |

Divergences results | 411 | |||

Clayton | Transshipment | 90 | 344 | 26 |

Independence | 34 | 1070 | 0 | |

Centralization | 36 | 0 | 8400 | |

Divergences results | 440 | |||

Gumbel | Transshipment | 83 | 330 | 22 |

Independence | 47 | 1086 | 0 | |

Centralization | 36 | 0 | 8396 | |

Divergences results | 435 | |||

Frank | Transshipment | 90 | 343 | 20 |

Independence | 34 | 1073 | 0 | |

Centralization | 36 | 0 | 8404 | |

Divergences results | 433 |

**Table 15.**Comparison of results between H3 and H4 (beta (5, 5)/beta (2, 8) versus beta (8, 2)/beta (2, 8)).

Copula | H3: Beta (5,5)/Beta (2,8) H4: Beta (8,2)/ Beta (2,8) | Transshipment | Independence | Centralization |
---|---|---|---|---|

Normal | Transshipment | 241 | 118 | 19 |

Independence | 199 | 1008 | 0 | |

Centralization | 20 | 0 | 8395 | |

Divergences results | 356 | |||

Clayton | Transshipment | 287 | 148 | 25 |

Independence | 140 | 964 | 0 | |

Centralization | 23 | 0 | 8413 | |

Divergences results | 336 | |||

Gumbel | Transshipment | 268 | 144 | 23 |

Independence | 138 | 995 | 0 | |

Centralization | 15 | 0 | 8417 | |

Divergences results | 320 | |||

Frank | Transshipment | 286 | 147 | 20 |

Independence | 140 | 967 | 0 | |

Centralization | 20 | 0 | 8420 | |

Divergences results | 327 |

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**MDPI and ACS Style**

Derbel, M.; Hachicha, W.; Aljuaid, A.M.
Sensitivity Analysis of the Optimal Inventory-Pooling Strategies According to Multivariate Demand Dependence. *Symmetry* **2021**, *13*, 328.
https://doi.org/10.3390/sym13020328

**AMA Style**

Derbel M, Hachicha W, Aljuaid AM.
Sensitivity Analysis of the Optimal Inventory-Pooling Strategies According to Multivariate Demand Dependence. *Symmetry*. 2021; 13(2):328.
https://doi.org/10.3390/sym13020328

**Chicago/Turabian Style**

Derbel, Mouna, Wafik Hachicha, and Awad M. Aljuaid.
2021. "Sensitivity Analysis of the Optimal Inventory-Pooling Strategies According to Multivariate Demand Dependence" *Symmetry* 13, no. 2: 328.
https://doi.org/10.3390/sym13020328