# Dimensionality-reduction Procedure for the Capacitated p-Median Transportation Inventory Problem

## Abstract

**:**

## 1. Introduction

## 2. The Capacitated p-Median Transportation Inventory Problem with Heterogeneous Fleet

_{DL}) does not change when a facility is chosen to be a DC, however, in a real-life problem, the σ

_{DL}changes when a facility is assigned to be a DC; third, in their model product enters the network through only one supplier facility, but in a real-life problem, product could enter through different facilities. In this paper, the LIRP proposed in Carmona-Benitez et al. [14] is modified to overcome these limitations in Section 2.3, and it is called CLITraP-HTF.

#### 2.1. Strategic Decisions Assumptions

#### 2.2. Definition and Notations

#### Objective Function

_{ij}is derived explicitly.

#### 2.3. Mixed Integer Programming Model (MIP)

_{i}calculates the total demand of facility i ϵ {O ∪ V}, as the sum of its demand (λ

_{i}) plus the sum of the demands of the facilities j’s ϵ V it is assigned to supply (λ

_{j}) (Equation (14)). σ

^{2}

_{DLi}calculates the lead time variance of facility i ϵ {O ∪ V}, which is equal to the sum of its variance (s

^{2}

_{DLi}) plus the sum of the variance of the facilities j´s ϵ V it is assigned to supply (s

^{2}

_{DLj}) (Equation (15)). The amount of product supplied to facility j ϵ V during T

_{j}, is equal to the multiplication of q

_{ijw}by n

_{ijw}. The Cap

_{j}of facility j ϵ V is compose by CapN

_{j}plus CapI

_{j}, the left-over capacity is equal to Cap

_{j}minus IOp

_{j}. The total quantity of product to supply from all the facilities i´s ϵ {O ∪ V}, with a certain number of shipments n

_{ijw}, using different types of vehicles w´s ϵ W, to facility j ϵ V must be lower than facility j ϵ V remaining capacity (Equation (16)). The total amount of product supplied from all the facilities i´s ϵ {O ∪ V}, with a certain number of shipments n

_{ijw}, using different types of vehicles w´s ϵ W, to facility j ϵ V in time T

_{j}, must be higher than or equal to the demand of facility j ϵ V in time T

_{j}(Equation (17)). The total K offered by external facility l ϵ O must be higher than or equal to the total amount of product demanded from all the facilities j´s ϵ V it supplies, daily (Equation (18)). The amount of product to be shipped from facility i ϵ {O ∪ V} to facility j ϵ V with a vehicle type w ϵ W must be less than or equal to the capacity of vehicle type w ϵ W (Equation (19)). n

_{ijw}is an integer variable that must be higher than or equal to zero (Equation (20)). Equation (16) to Equation (20) are for the inventory management and product transportation, vehicles visit no more than one facility j ϵ V per trip, and they use different types of vehicles w´s ϵ W. Equation (21)–(23) are binary constraints. Equation (24) are nonnegativity constraints different from zero, and Equation (25) indicate that the level of services of each facility i ϵ V is between 0 and 1.

## 3. Dimensionality-Reduction Procedure

#### 3.1. Dimensionality-Reduction Procedure on the Inventory-Transport and Investment Decisions Variables

_{j}> 0. From Equation (10), it is possible to compute the optimal value of T

_{j}by taking the derivate of the objective function with respect to T

_{j}(Equation (10)).

^{*}

_{j}depends on finding the optimal values of the variables n

_{ijw}and Y

_{ijw}.

_{j}is calculated as follows:

_{ij}is an integer variable different from zero because the amount of product to transport from facility i ϵ {O ∪ V} to facility j ϵ V every T

^{*}

_{j}is equal to Λ

_{j}T

^{*}

_{j}and to n

_{ij}q

_{ij}(Equation (28)). Λ

_{j}T

^{*}

_{j}is different from zero because Equations (24) and (14) indicate that T

^{*}

_{j}and Λ

_{j}are positive and different from zero respectively.

_{ij}is when n

_{ij}= 1. Therefore, only one shipment using a vehicle of type w must be used to transport product from facility i ϵ {O ∪ V} to facility j ϵ V every T

^{*}

_{j}. Even if the Λ

_{j}increases or decreases, n

_{ij}is equal to 1, it does not matter if facility j ϵ V is chosen to be a DC or not.

_{ij}= 1 and when w = 1 (homogeneous fleet), T

^{*}

_{j}is equal to:

_{ij}in terms of q

_{ij}when n

_{ij}= 1 by substituting Equation (29) into Equation (27):

_{ij}the lower the TCp

_{ij}what is consistent with the theory of economies of scale. The value of q

_{ij}must be as large as possible to minimize TCp

_{ij}, and it is restricted by VCap

_{w}and the storage capacity at facility j ϵ V (Cap

_{j}). Since, Cap

_{j}can increase from CapN

_{j}to CapN

_{j}+ CapI

_{j}whether an investment is done, the value of q

_{ij}also depends on the decision investment variable δ

_{j}. The decision variables q

_{ij}and δ

_{j}are solved as follows:

- Whether $\mathrm{V}\mathrm{C}\mathrm{a}{\mathrm{p}}_{\mathrm{w}}\le \mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{N}}_{\mathrm{j}}\text{}\mathrm{then}\text{}{\mathrm{q}}_{\mathrm{i}\mathrm{j}}=\mathrm{V}\mathrm{C}\mathrm{a}{\mathrm{p}}_{\mathrm{w}}\text{}\mathrm{and}\text{}{\mathsf{\delta}}_{\mathrm{j}}=0$; otherwise,
- Whether$\mathrm{V}\mathrm{C}\mathrm{a}{\mathrm{p}}_{\mathrm{w}}\ge \mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{N}}_{\mathrm{j}}\text{}\mathrm{a}\mathrm{n}\mathrm{d}\text{}\mathrm{V}\mathrm{C}\mathrm{a}{\mathrm{p}}_{\mathrm{w}}\le \left(\mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{N}}_{\mathrm{j}}+\mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{I}}_{\mathrm{j}}\right)\text{}\mathrm{and}\text{}{\mathrm{C}}_{\mathrm{i}\mathrm{j}}\frac{{\Lambda}_{\mathrm{j}}}{\mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{N}}_{\mathrm{j}}}\le \left[{\mathrm{C}}_{\mathrm{i}\mathrm{j}}\frac{{\Lambda}_{\mathrm{j}}}{\mathrm{V}\mathrm{C}\mathrm{a}{\mathrm{p}}_{\mathrm{w}}}+\frac{\mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{I}}_{\mathrm{j}}\mathrm{I}\mathrm{n}\mathrm{v}{\mathrm{u}}_{\mathrm{j}}}{\mathrm{T}\mathrm{H}}\right]$$\mathrm{then}\text{}{\mathrm{q}}_{\mathrm{i}\mathrm{j}}=\mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{N}}_{\mathrm{j}}\text{}\mathrm{and}\text{}{\mathsf{\delta}}_{\mathrm{j}}=0$; Otherwise,
- Whether$\mathrm{V}\mathrm{C}\mathrm{a}{\mathrm{p}}_{\mathrm{w}}\ge \mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{N}}_{\mathrm{j}}\text{}\mathrm{a}\mathrm{n}\mathrm{d}\text{}\mathrm{V}\mathrm{C}\mathrm{a}{\mathrm{p}}_{\mathrm{w}}\le \left(\mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{N}}_{\mathrm{j}}+\mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{I}}_{\mathrm{j}}\right)\text{}\mathrm{and}\text{}{\mathrm{C}}_{\mathrm{i}\mathrm{j}}\frac{{\Lambda}_{\mathrm{j}}}{\mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{N}}_{\mathrm{j}}}\ge \left[{\mathrm{C}}_{\mathrm{i}\mathrm{j}}\frac{{\Lambda}_{\mathrm{j}}}{\mathrm{V}\mathrm{C}\mathrm{a}{\mathrm{p}}_{\mathrm{w}}}+\frac{\mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{I}}_{\mathrm{j}}\mathrm{I}\mathrm{n}\mathrm{v}{\mathrm{u}}_{\mathrm{j}}}{\mathrm{T}\mathrm{H}}\right]$$\mathrm{then}\text{}{\mathrm{q}}_{\mathrm{i}\mathrm{j}}=\mathrm{V}\mathrm{C}\mathrm{a}{\mathrm{p}}_{\mathrm{w}}\text{}\mathrm{and}\text{}{\mathsf{\delta}}_{\mathrm{j}}=1$; Otherwise,
- Whether$\mathrm{V}\mathrm{C}\mathrm{a}{\mathrm{p}}_{\mathrm{w}}\ge \left(\mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{N}}_{\mathrm{j}}+\mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{I}}_{\mathrm{j}}\right)\mathrm{then}\text{}{\mathrm{q}}_{\mathrm{i}\mathrm{j}}=\left(\mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{N}}_{\mathrm{j}}+\mathrm{C}\mathrm{a}\mathrm{p}{\mathrm{I}}_{\mathrm{j}}\right)\text{}\mathrm{and}\text{}{\mathsf{\delta}}_{\mathrm{j}}=1$

_{ij}and q

_{ij}, Equation (29) calculates the value of T

^{*}

_{j}.

_{j}is calculated as follows:

^{*}

_{j}using vehicle w ϵ W is equal to Λ

_{j}T

^{*}

_{j}and to n

_{ijw}q

_{ijw}. Λ

_{j}T

^{*}

_{j}is different from zero because Equations (24) and (14) indicate that T

^{*}

_{j}and Λ

_{j}are positive and different from zero respectively. Hence, only one shipment using one type of vehicle from the heterogeneous fleet of vehicles w ϵ W is used to transport product from facility i ϵ {O ∪ V} to facility j ϵ V every T

_{j}, even if Λ

_{j}in facility j ϵ V increases or decreases, or whether facility j ϵ V is chosen to be a DC or not.

^{*}

_{j}. Equation (34) chooses the vehicle based on the minimum TCp

_{ij}(Equation (29)) calculated for each vehicle w ϵ W when n

_{ij}= 1.

_{j}using the vehicle that achieves the lowest TCp

_{ij}(Equation (32)) from the heterogeneous fleet of vehicles w ϵ W.

#### 3.2. Dimensionality-Reduction Procedure on the Level of Service Decision Variables

_{ij}can be solved prior to starting the solution method, reducing the degree of computational difficulties.

^{−1}(α)). Figure 1 demonstrates that F(ROP

_{ij}) = α

_{ij}or ROP

_{ij}= F

^{−1}(α

_{ij}), and it is computed as:

_{ij}is the lead time demand also known as order placed or order fulfillment.

_{ij}is the shortage or unfulfilled demand at facility j ϵ V when its supplier is facility i ϵ {O ∪ V}, then:

_{ij}is computed in terms of α

_{ij}during T

_{j}, as it is expressed in Equation (7).

_{ij}means the trade-off between the different costs, TrC

_{ijw}(Equation (1)), INV

_{i}(Equation (2)), FLC

_{i}(Equation (3)), OpC

_{ij}(Equation (4)), IC

_{ij}(Equation (5)) and Pc

_{ij}(Equation (7)), among a facility i ϵ {O ∪ V} and facility j ϵ V. Therefore, the proposed optimization approach requires the existence of an equilibrium condition between TrC

_{ijw}, INV

_{i}, FLC

_{i}, OpC

_{ij}, IC

_{ij}and Pc

_{ij}(Equation (37)) for the distribution of a product between a facility i ϵ {O ∪ V} and a facility j ϵ V,

_{ij}, OpC

_{ij}, FLC

_{i}, INV

_{i}, TrC

_{ijw}and Pc

_{ij}in terms of T

_{i}, X

_{i}, Y

_{ij}, q

_{ijw}, n

_{ijw}, δ

_{i}and α

_{ij}. These derivatives are very tough. However, Section 3.1 mathematically demonstrates that T

_{i}, q

_{ijw}, n

_{ijw}, and δ

_{i}can be solved prior to starting the optimization methodology, and for the case of the distribution of a product between facility i ϵ {O ∪ V} and facility j ϵ V, the decision variables X

_{i}and Y

_{ij}does not exist. So, the complexity of them is avoided.

_{ij}before the solution method is applied. Their approach optimizes the costs in terms of T

_{i}and α

_{ij}simultaneously. This is possible because these variables are mutually dependent, and because an optimum value of α

_{ij}exists for every value of T

_{i}. Knowing the optimum value of T

_{i}, it is possible to find the equilibrium condition in terms of α

_{ij}for each T

_{i}. In this paper, their approach is explained in detail to demonstrate the optimal solution of α

_{ij}because it is part of the DRP proposed in this paper.

_{i}, q

_{ijw}, n

_{ijw}, and δ

_{i}).

_{ij}, and inside a specific neighborhood of these values. Equation (40) explains that this equality is caused by the evenness of the network configuration in the declared neighborhood:

_{ij}in terms of T

_{j}:

_{ij}can be calculated when T

_{j}is known.

## 4. Results

^{12}or 15.75 trillion; for a medium size instance with 20,000 facilities, the total number of variables is equal to 400 million. Equation (45) shows how high are the dimensionality of decision variables of large and medium size CLITraP-HTF.

#### Dimensionality-Reduction Proceedure Results

## 5. Discussion and Conclusions

## Funding

## Conflicts of Interest

## Appendix A. Shortage Cost

_{ij}(Equation (8)), σ

_{ij}(Equation (9)), ROP

_{ij}(Equation (36)), y

_{ij}(Equation (37)), p(y

_{ij}> 0) (Equation (38)), and Pc

_{ij}(Equation (39)).

_{ij}per day:

_{ij}) = α

_{ij}or ROP

_{ij}= F

^{−1}(α

_{ij}) in Equation (A7), and Equation (7) calculates Pc

_{ij}in terms of α

_{ij}during a period T

_{j}.

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Set | Definition |
---|---|

O {1, …, l} | is the set of all external suppliers’ facilities that supply product to the system or supply chain |

V {l + 1, …, i} | is the set of all facilities that can be DC or not, and |

V {l + 1, …, j} | is also the set of customer facilities |

W {1, …, w} | is the set of types of vehicles |

Variable | Definition | Type | Units |
---|---|---|---|

X_{i} | is equal to 1 if facility i ϵ V is operated as DC, 0 otherwise | Boolean | - |

X_{l} | is equal to 1 for all external supplier facilities l ϵ O | Boolean | - |

Y_{ij} | is equal to 1 if the link connecting facility i ϵ {O ∪ V} with facility j ϵ V is used to transport product from facility i ϵ {O ∪ V} to facility j ϵ V | Boolean | - |

δ_{j} | is equal to 1 if facility i ϵ V requires an investment | Boolean | - |

TH | is the time horizon (i.e., annual, semi-annual, monthly) | Integer | year |

p | determines the maximum number of DCs that can be located | Integer | - |

n_{ijw} | is the number of shipments per order or per replenishment period using vehicle w ϵ W to transport product from facility i ϵ {O ∪ V} to facility j ϵ V | Integer | shipments/order |

T_{j} | is the replenishment period in which facility j ϵ V must be supplied | Continuous | days/order |

q_{ijw} | is the replenishment amount of product shipped from facility i ϵ V to facility j ϵ V using vehicle type w ϵ W | Continuous | unit/shipment |

α_{ij} | is the inventory service level at facility j ϵ V when it is supplied by facility i ϵ {O ∪ V} | Continuous | - |

**Table 3.**CLITraP-HTF parameters [17].

Parameter | Definition | Units |
---|---|---|

Q_{ij} | is the total replenishment amount of product shipped per order from facility i ϵ {O ∪ V} to facility j ϵ V | unit/order |

C_{ijw} | is the costs of shipping product from facility i ϵ V to facility j ϵ V using vehicle type w ϵ W | $/shipment |

c | is the purchase cost | $/unit |

ct_{i} | is the ordering cost also known as setup cost | $/order |

ce_{i} | is the carrying or holding cost at facility i ϵ V; Equal to (c ir) | $/unit/year |

Cu_{i} | is the penalty cost per shortage or unfulfilled demand unit at facility i ϵ V | $/unit |

y_{ij} | is the shortage or unfulfilled demand at facility j ϵ V when its supplier is facility i ϵ {O ∪ V} | units |

TI_{i} | is the total inventory for facility i ϵ V during time T_{i} | units day/order |

λ_{i} | is the average daily demand at facility i ϵ V | units/day |

Λ_{i} | is the sum of average daily demand of facility i ϵ V and the demands of the facilities j ϵ V supplied by facility i ϵ V | units/day |

IOp_{i} | is the minimum number of days in inventory required to operate facility i ϵ V | units |

ROP_{ij} | is the reorder point at facility j ϵ V when its supplier is facility i ϵ {O ∪ V} | units |

ss_{ij} | is the safety stock at facility j ϵ V when its supplier is facility i ϵ {O ∪ V} | units |

L_{ij} | is the order lead time at facility j ϵ V when its supplier is facility i ϵ {O ∪ V} | days |

σ_{Lij} | is the L_{ij} standard deviation at facility j ϵ V when its supplier is facility i ϵ {O ∪ V} | days |

s_{DLij} | is the standard deviation of demand over the L_{ij} at facility j ϵ V when its supplier is facility i ϵ {O ∪ V} | units |

σ_{DLi} | is the standard deviation of demand over the lead calculated from the sum of the variances at facility i ϵ V and the variances of the facilities j ϵ V supplied by facility i ϵ V | units |

$\overline{{\mathrm{x}}_{\mathrm{ij}}}$ | is the mean of lead time demand of facility j ϵ {O ∪ V} when its supplier is facility i ϵ {O ∪ V} | units |

σ_{ij} | is the standard deviation of lead time demand of facility j ϵ {O ∪ V} when its supplier is facility i ϵ {O ∪ V} | units |

ir | is the capital cost rate (Timme and Williams-Timme, 2003) | 1/year |

Cap_{i} | is the total storage capacity at facility i ϵ V | units |

CapN_{i} | is the current storage capacity at facility i ϵ V | units |

CapI_{i} | is an extra storage capacity at facility i ϵ V that can be available only if an investment is made | units |

Parameter | Definition | Units |
---|---|---|

VCap_{w} | is the carrying capacity of vehicle type w ϵ W | units |

K_{l} | is the quantity of product offered at external supplier facility l ϵ O | units |

P | is the product price per unit | $/unit |

Invu_{i} | is the investment unit cost required at facility i ϵ V to increased storage capacity CapI_{i} | $/unit |

FC_{i} | is the location costs of a DC at facility i ϵ V | $ |

TrC_{ijw} | is the total shipping cost from facility i ϵ {O ∪ V} to facility j ϵ V with vehicle type w ϵ W in TH | $ |

Pc_{ij} | is the stock-out or shortage cost at facility j ϵ V because of facility i ϵ V | $ |

IC_{i} | is the total inventory cost for facility i ϵ V in TH | $ |

OpC_{i} | is the opportunity cost for facility i ϵ V in TH | $ |

INV_{i} | is the investment cost for facility i ϵ V in TH | $ |

FLC_{i} | is the facility location cost of a DC on facility i ϵ V (payable only once per TH) | $ |

TC | is the total cost in TH | $ |

**Table 5.**CLITraP-HTF dimension of decision variables for scenarios with different number of facilities.

Decision Variables | Number of Facilities (Small Size Instances) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

2 | 22 | 42 | 62 | 82 | 102 | 122 | 142 | 162 | 182 | |

X | 2 | 22 | 42 | 62 | 82 | 102 | 122 | 142 | 162 | 182 |

Δ | 2 | 22 | 42 | 62 | 82 | 102 | 122 | 142 | 162 | 182 |

T | 2 | 22 | 42 | 62 | 82 | 102 | 122 | 142 | 162 | 182 |

A | 2 | 22 | 42 | 62 | 82 | 102 | 122 | 142 | 162 | 182 |

Y | 2 | 462 | 1722 | 3782 | 6642 | 10,302 | 14,762 | 20,022 | 26,082 | 32,942 |

q | 6 | 1386 | 5166 | 11,346 | 19,926 | 30,906 | 44,286 | 60,066 | 78,246 | 98,826 |

n | 6 | 1386 | 5166 | 11,346 | 19,926 | 30,906 | 44,286 | 60,066 | 78,246 | 98,826 |

p | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Total | 23 | 3323 | 12,223 | 26,723 | 46,823 | 72,523 | 103,823 | 140,723 | 183,223 | 231,323 |

Number of Facilities (Small Size Instances) | |||||||||
---|---|---|---|---|---|---|---|---|---|

2 | 22 | 42 | 62 | 82 | 102 | 122 | 142 | 162 | |

Scenarios | 28 | 1.33 × 10^{17} | 1.05 × 10^{30} | 3.74 × 10^{42} | 9.55 × 10^{54} | 2.03 × 10^{67} | 3.82 × 10^{79} | 6.63 × 10^{91} | 1.08 × 10^{104} |

Decision Variables | Number of Facilities (Small Size Instances) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

2 | 22 | 42 | 62 | 82 | 102 | 122 | 142 | 162 | 182 | |

X | 2 | 22 | 42 | 62 | 82 | 102 | 122 | 142 | 162 | 182 |

α | 2 | 22 | 42 | 62 | 82 | 102 | 122 | 142 | 162 | 182 |

Y | 2 | 462 | 1722 | 3782 | 6642 | 10,302 | 14,762 | 20,022 | 26,082 | 32,942 |

p | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Total | 7 | 507 | 1807 | 3907 | 6807 | 10,507 | 15,007 | 20,307 | 26,407 | 33,307 |

Decision Variables | Number of Facilities (Small Size Instances) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

2 | 22 | 42 | 62 | 82 | 102 | 122 | 142 | 162 | 182 | |

X | 2 | 22 | 42 | 62 | 82 | 102 | 122 | 142 | 162 | 182 |

Y | 2 | 462 | 1722 | 3782 | 6642 | 10,302 | 14,762 | 20,022 | 26,082 | 32,942 |

p | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Total | 5 | 485 | 1765 | 3845 | 6725 | 10,405 | 14,885 | 20,165 | 26,245 | 33,125 |

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**MDPI and ACS Style**

Carmona-Benítez, R.B. Dimensionality-reduction Procedure for the Capacitated p-Median Transportation Inventory Problem. *Mathematics* **2020**, *8*, 471.
https://doi.org/10.3390/math8040471

**AMA Style**

Carmona-Benítez RB. Dimensionality-reduction Procedure for the Capacitated p-Median Transportation Inventory Problem. *Mathematics*. 2020; 8(4):471.
https://doi.org/10.3390/math8040471

**Chicago/Turabian Style**

Carmona-Benítez, Rafael Bernardo. 2020. "Dimensionality-reduction Procedure for the Capacitated p-Median Transportation Inventory Problem" *Mathematics* 8, no. 4: 471.
https://doi.org/10.3390/math8040471