Mathematical models of distribution networks

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Industry 4.0 Technologies Impact on Supply Chain Sustainability

3. Description of a cyber-physical distribution network

3.2 Mathematical models of distribution networks

Within the frame of this section, the mathematical models of a conventional and cyber-physical distribution network are described. The mathematical models are focusing on energy efficiency-related“green”objective functions, including energy consumption and greenhouse gas emission as the most important influencing factors of environmental impact. The mathematical model includes time, capacity, and energy-related constraints.

3.2.1 Mathematical model of a conventional distribution network

The objective function of the optimization problem is either the energy consumption or the real or virtual greenhouse gas emission. The energy consumption as objective function can be defined as follows in the case of conventional distribution:

EC^CONV ¼X^amax

a¼1

EC^MD_a þX^amax

a¼1

EC^DC_a ! min: (1)

whereEC^CONV is the energy consumption of the conventional distribution system, EC^MD_a is the energy consumption between the manufacturer clusteraand distribution clustera,EC^DC_a is the energy consumption between the distribution clusteraand the consumers,amaxis the number of independent distribution networks.

The energy consumption between the manufacturer clusteraand the distribution clusteracan be defined as follows:

EC^MD_a ¼ X

α_maxa

α¼1

ε_v

αaq_vαa

∙l^opt_αaðΘ_αaÞ (2) wherevαais the transportation vehicle assigned to route αof distribution systema in Tier1 between manufacturers and distribution centers,α_maxa is the number of distribution routes for distribution systema in Tier1 between manufacturers and distribution centers,ε_v

α is the energy consumption of transportation vehiclev_αa assigned to routeαof distribution systemain Tier1 between manufacturers and distribution centers, l^opt_αa is the length of the optimal distribution routeαof distribution systemain Tier1 between manufacturers and distribution centers,which is a function of theΘ_αaset of distribution centers assigned to routeαof distribution systemain Tier1 between manufacturers and distribution centers, andq_vαa is the current capacity of transportation vehicle assigned to routeαof distribution systemain Tier1 between manufacturers and distribution centers.

The energy consumption between the distribution clusteraand consumers assigned to distribution centeracan be defined as follows:

EC^DC_a ¼ X

β_maxa

β¼1

ε_v

βaq_vβa

∙l^opt_βaΘ_βa

(3) wherevβais transportation vehicle assigned to routeβof distribution systemain Tier2 between distribution centers and consumers,β_maxais the number of distribution routes for distribution systemain Tier2 between distribution centers and consumers, ε_v

βa is the energy consumption of transportation vehiclevβa assigned to routeβof distribution systemain Tier2 between distribution centers and consumers, l^opt_βa is the length of the optimal distribution routeβ of distribution systema in Tier2

distribution centers and consumers,which is a function of theΘ_βaset of distribution centers assigned to routeβ of distribution systema in Tier2 between distribution centers and consumers, andq_vβa is the current capacity of transportation vehicle assigned to routeβ of distribution systema in Tier2 between distribution centers and consumers.

The second objective function is the greenhouse gas emission, which can be defined for the following GHGs: carbon dioxide, methane, nitrous oxide, and fluori- nated gases. The GHG emission can be defined in the following way for the different GHGs in the case of conventional distribution:

EM^CONV_MGHG ¼X^amax

a¼1

EM^MDa_MGHGþX^amax

a¼1

EM^DCa_MGHG ! min: (4) whereEM^CONV_MGHG is theGHGemission of the conventional distribution system, EM^MDa_MGHGis the GHG emission of the vehicle assigned to the distribution operations between the manufacturer clusteraand distribution clustera, EM^DCa_MGHG is the GHG emission of vehicles assigned to distribution operations between the distribution clusteraand the consumers, andMGHG¼½CO₂,SO₂,CO,HC,NO_x,PMŠis the matrix of greenhouse gases to be taken into consideration.

The emission between the manufacturer clusteraand the distribution clusteracan be defined as follows:

EM^MDa_MGHG ¼ X

α_maxa

α¼1

ϑ^TIER1,vαa

MGHG ∙ε_v

αaq_vαa

∙l^opt_αaðΘ_αaÞ (5) whereϑ^TIER1,vαa

MGHG is the specific GHG emission in the case of transportation vehicle assigned to routeαof distribution systemain Tier1 between manufacturers and distribution centers.

The emission between the distribution clusteraand consumers assigned to distribution centeracan be defined as follows:

EM^DCa_MGHG ¼ X

β_maxa

β¼1

ϑ^TIER2,vβa

MGHG ∙ε_v

βaq_vβa

∙l^opt_βaΘ_βa

(6)

whereϑ^TIER2,v_MGHG^βa is the specific GHG emission in the case of transportation vehicle assigned to routeβ of distribution systema in Tier2 between distribution centers and consumers.

As constraints, we can take the following into consideration: capacity of vehicles, capacity of loading and unloading equipment, capacity of distribution centers, time window for manufacturer, time window for customers, time window for 3PL pro- viders in Tier1, time window for 3PL providers in Tier2, and available energy for electric vehicles.

Constraint 1a: We can define the upper limit of the loading capacity of transporta- tion vehicles. It is not allowed to exceed this upper limit of loading capacity while assigning distribution tasks to the routes and scheduling the delivery tasks:

∀α,a : Cv_αa≥ X^imax

i¼1

q_i∈Ψ_αa (7)

where C_vαa is the upper limit of loading capacity of the transportation vehicle assigned to routeαof distribution systema,i_maxis the upper limit of customers’ demands, q_iis the volume or weight (capacity unit) of customers’demandi,Ψ_αais the set of customers’demands assigned to routeαof distribution systema.

Constraint 2a: We can define the upper limit of the material-handling capacity of loading and unloading equipment. It is not allowed to exceed this upper limit of material handling capacity while assigning distribution tasks to the routes and scheduling the delivery tasks:

∀α,a: Czαa≥ X^imax

i¼1

ziq_i∈Ψ_αa

(8) where C_zαa is the upper limit of the material-handling capacity of the loading and unloading equipment assigned to delivery tasks of routeαof distribution systema, zi

is the required material handling capacity of customers demandi.

Constraint 3a: We can define the upper limit of the storage capacity of distribution centers. It is not allowed to exceed this upper limit by assigning manufacturers to distribution centers and distribution centers to customers:

∀a: CWa≥ X^imax

i¼1

X

α_max

α¼1

q_iα∈Ψ_αa (9)

where CW_ais the storage capacity of the distribution center of distribution system a, q_iαis the customers’demandiassigned to routeαof distribution systema.

Constraint 4a: We can define a time window for the potential manufacturing process for each demand of customers. It is not allowed to exceed this lower and upper limit while assigning customers’demands to manufacturers and scheduling them:

∀i,a: τ^MINm_ia ≤τ^m_ia≤τ^MAXm_ia (10) whereτ^MINm_ia is the lower limit of the time window for the manufacturing

process for customers’demand i at the manufacturer of the distribution system a, τ^MAXm_ia is the upper limit of the time window for the manufacturing process for customers’demand i at the manufacturer of the distribution system a,τ^m_ia is the scheduled manufacturing time for customers’demand i at the manufacturer of the distribution system a.

Constraint 5a: We can define a time window for the customers’demands. The manufactured products must be delivered within this predefined time window to the customers and it is not allowed to exceed this time window:

∀i,a : τ^MINcd_ia ≤τ^cd_ia≤τ^MAXcd_ia (11) whereτ^MINcd_ia is the lower limit of the time window for delivering the manufactured product to customeriin the distribution systema,τ^MAXcd_ia is the upper limit of the time window for delivering the manufactured product to customeriin the distribution systema,τ^cd_ia is the scheduled delivery of manufactured product to customer iin the distribution systema.

Constraint 6a: The material handling operations can be performed by third-party logistics providers in the case of Tier1 and Tier 2. We can define an available

time window of these 3PL providers and it is not allowed to exceed this time

window while assigning and scheduling material handling tasks performed by the 3PL providers:

∀i,a,μ : τ^MIN3PL_iaμ ≤τ^3PL_iaμ ≤τ^MAX3PL_iaμ (12) whereτ^MIN3PL_iaμ is the lower limit of the time window of availability of third-party logistics provider for customers’demandiin distribution systemain Tierμ,τ^MAX3PL_iaμ is the upper limit of the time window of availability of third-party logistics provider for customers’demandiin distribution systemain Tierμ,τ^3PL_iaμ is the scheduled logistics service for customers’demand iin distribution systemain Tierμ.

Constraint 7a: As a sustainability and energy efficiency-related constraint, we can define the available energy of transportation vehicles and other material handling equipment. For example, in the case of electric vehicles we can define the available capacity of batteries or the required reloading time:

∀a,α: ε_v

αaq_vαa∈Ψ_αa

∙l^opt_αaðΘ_αaÞ≤E^max_vαa (13) whereE^max_vαa is the upper limit of available energy (capacity of a battery in the case of electric vehicles).

3.2.2 Mathematical model of a cyber-physical distribution network

In the case of a cyber-physical distribution system, where Industry 4.0 technologies make it possible to integrate the operation of the different distribution system within and between tiers the energy consumption can be computed in the following way:

EC^CYB ¼EC^MDþEC^DC ! min: (14) whereEC^CYB is the energy consumption of the cyber-physical distribution system, which integrates all individual separated distribution systems of the conventional solu- tion,EC^MDis the energy consumption between manufacturers and distribution clusters, EC^DC is the energy consumption between the distribution centers and consumers.

The energy consumption between manufacturers and distribution centers in Tier 1 can be defined as follows:

EC^MD ¼X

α_max

α¼1

ε_v

α q_v

α

∙l^opt_α ðΘ_αÞ (15) wherevα is the transportation vehicle assigned to routeαin Tier1 between manu- facturers and distribution centers,α_maxis the total number of distribution routes for Tier1 between manufacturers and distribution centers,ε_v

α is the energy consumption of transportation vehiclevαassigned to routeαin Tier1 between manufacturers and distribution centers, l^opt_α is the length of the optimal distribution routeαin Tier1 between manufacturers and distribution centers,which is a function of theΘ_αset of distribution centers assigned to routeαin Tier1 between manufacturers and distribu- tion centers, andq_vαis the current capacity of transportation vehicles assigned to route αin Tier1 between manufacturers and distribution centers.

The energy consumption between the distribution centers and consumers in Tier2 can be defined as follows:

EC^DC ¼X

β_max

β¼1

ε_v

βq_vβ

∙l^opt_β Θ_β

(16) wherevβ is the transportation vehicle assigned to routeβin Tier2 between distri- bution centers and consumers,β_maxis the number of distribution routes in Tier2 between distribution centers and consumers,ε_v

β is the energy consumption of trans- portation vehiclevβ assigned to routeβin Tier2 between distribution centers and consumers, l^opt_β is the length of the optimal distribution routeβ in Tier2 between distribution centers and consumers,which is a function of theΘ_β set of distribution centers assigned to routeβ in Tier2 between distribution centers and consumers, and q_vβ is the current capacity of transportation vehicles assigned to routeβin Tier2 between distribution centers and consumers.

The transformation of the conventional distribution system into cyber-physical distribution is suitable from energy consumption point of view, if

X^amax

a¼1

EC^MD_a þX^amax

a¼1

EC^DC_a ≫EC^MDþEC^DC (17)

In the case of a cyber-physical distribution system, the emission of greenhouse gases can be computed in the following way:

EM^CYB_MGHG ¼EM^MD þEM^DC ! min: (18) whereEM^CYB_MGHG is the GHG emission of the cyber-physical distribution

system, which integrates all individual separated distribution systems of the conventional solution,EM^MD is the GHG emission between manufacturers and

distribution clusters,EM^DC is the GHG emission between the distribution centers and consumers.

The GHG emission between manufacturers and distribution centers in Tier 1 can be defined as follows:

EM^MD_MGHG ¼ X

α_maxa

α¼1

ϑ^TIER1,vα

MGHG ∙ε_v

αq_vα

∙l^opt_α ðΘ_αÞ (19) whereϑ^TIER1,vα

MGHG is the specific GHG emission in the case of transportation vehicles assigned to routeαin Tier1 between manufacturers and distribution centers.

The emission between the distribution centers and consumers in Tier2 can be defined as follows:

EM^DC_MGHG ¼ X

β_maxa

β¼1

ϑ^TIER2,v_MGHG^β∙ε_v

βq_vβ

∙l^opt_β Θ_β

(20)

whereϑ^TIER2,v_MGHG^β is the specific GHG emission in the case of transportation vehicles assigned to routeβin Tier2 between distribution centers and consumers.

The transformation of the conventional distribution system into cyber-physical distribution is suitable from GHG emission point of view, if

X^amax

a¼1

EM^MDa_MGHG þX^amax

a¼1

EM^DCa_MGHG≫EM^MD_MGHGþEM^DC_MGHG (21) As constraints, we can take the following into consideration: capacity of vehicles, capacity of loading and unloading equipment, capacity of distribution centers, time window for manufacturer, time window for customers, time window for 3PL pro- viders in Tier1, time window for 3PL providers for Tier2, available energy for electric vehicles.

Constraint 1b: We can define the upper limit of the loading capacity of transportation vehicles. It is not allowed to exceed this upper limit of loading

capacity while assigning distribution tasks to the routes and scheduling the delivery tasks. The difference between the constraints1aand1bis that, while in the case of a conventional distribution network, customer demand can only be assigned to the transport vehicles within the given distribution network, in the case of a cyber- physical distribution system, any customer demand can be assigned to any transportation vehicle:

∀α: C_vα≥ X^imax

i¼1

q_i∈Ψ_α (22)

whereΨ_αis the set of customers’demands assigned to routeαin the cyber-physical distribution network.

Constraint 2b: We can define the upper limit of the material handling capacity of loading and unloading equipment. It is not allowed to exceed this upper limit of material-handling capacity while assigning distribution tasks to the routes and sched- uling the delivery tasks. The difference between the constraints2aand2bis that, while in the case of a conventional distribution network, customer demand can only be assigned to the transport vehicles and related material handling equipment (loading and unloading equipment, packaging machines, labeling) within the given distribution network, in the case of a cyber-physical distribution system, any customer demand can be assigned to any material handling equipment:

∀α,a : Czα≥ X^imax

i¼1

ziq_i∈Ψ_α

(23) Constraint 3b: We can define the upper limit of the storage capacity of distribution centers. The difference between constraints 3a and 3b is that while in the case of the conventional distribution system the capacity of a distribution system depends on only the manufacturers and customers of the same distribution system, in the case of a cyber-physical distribution network all products produced by all manufacturers can be assigned to all distribution centers (warehouses):

∀a: CW_a≥ X^imax

i¼1

X

α_max

α¼1

q_iα∈Ψ_α (24)

where CW_ais the storage capacity of the distribution center of distribution system a, q_iαis the customers’demandsiassigned to routeαof distribution systema.

Constraint 4b: We can define a time window for the potential manufacturing process for each demand of customers. It is not allowed to exceed this lower and upper limit while assigning customers’demands to manufacturers and scheduling them. In this cyber-physical network, the time windows can be defined for all manufacturers of the whole network, while in the case of conventional distributions networks, the time windows are focusing on the manufacturers of separated distribution systems:

∀i,a: τ^MINm_ia ≤τ^m_ia≤τ^MAXm_ia (25) whereτ^MINm_ia is the lower limit of the time window for the manufacturing process for customers’demandiat the manufacturer of the distribution systema,τ^MAXm_ia is the upper limit of the time window for the manufacturing process for customers’demandiat the manufacturer of the distribution systema,τ^m_ia is the scheduled manufacturing time for customers’demandiat the manufacturer of the distribution systema.

Constraint 5b: We can define a time window for the customers’demands. The manufactured products must be delivered within this predefined time window to the customers and it is not allowed to exceed this time window. In this cyber-physical network, the time windows can be defined for all customers of the whole network, while in the case of conventional distribution networks, the time windows are focus- ing on the customers of separated distribution systems:

∀i,a : τ^MINcd_ia ≤τ^cd_ia≤τ^MAXcd_ia (26)

whereτ^MINcd_ia is the lower limit of the time window for delivering the manufactured product to customeriin the distribution systema,τ^MAXcd_ia is the upper limit of the time window for delivering the manufactured product to customeriin the distribution systema,τ^cd_ia is the scheduled delivery of manufactured product to customer iin the distribution systema.

Constraint 6b: The material handling operations can be performed by third-party logistics providers in the case of Tier1 and Tier 2. We can define an available

time window of these 3PL providers and it is not allowed to exceed this time window while assigning and scheduling material-handling tasks performed by the 3PL providers. In this case, the 3PL providers can perform all logistics operations in the cyber-physical distribution network, while in the case of conventional

distribution systems, the 3PL providers of separated distribution systems can work uncoordinated:

∀i,a,μ : τ^MIN3PL_iaμ ≤τ^3PL_iaμ ≤τ^MAX3PL_iaμ (27) whereτ^MIN3PL_iaμ is the lower limit of the time window of availability of third-party logistics provider for customers’demandiin distribution systemain Tierμ,τ^MAX3PL_iaμ is the upper limit of the time window of availability of third-party logistics provider for customers’demandiin distribution systemain Tierμ,τ^3PL_iaμ is the scheduled logistics service for customers’demand iin distribution systemain Tierμ.

Constraint 7b: As a sustainability and energy efficiency-related constraint, we can define the available energy of transportation vehicles and other material handling equipment. For example, in the case of electric vehicles we can define the available capacity of batteries or the required reloading time:

∀a,α: ε_v

αaq_vαa∈Ψ_αa

∙l^opt_αaðΘ_αaÞ≤E^max_vαa (28) whereE^max_vαa is the upper limit of available energy (capacity of a battery in the case of electric vehicles).

The decision variables of this NP-hard optimization problem are the followings:

• assignment of customers’demands to manufacturers (Tier 1),

• assignment of final products to 3PL providers in manufacturer–distribution center relation (Tier 1),

• assignment of customers’demands to distribution centers (Tier 2),

• scheduling of manufacturing of customers’demands (Tier 1),

• scheduling of logistics operations of 3PL provider in manufacturer–distribution center relation (Tier 1),

• assignment of 3PL providers to perform delivery operations from distribution centers to customers (Tier2),

• assignment of vehicles to routes and distribution networks (Tier1 and Tier2),

• scheduling of logistics operations of 3PL provider in distribution center– customer relation (Tier 2).

To solve this integrated assignment, scheduling and routing problem of the green distribution network and heuristic algorithms can be used. In the literature, we can find a wide range of heuristic solutions to integrated assignment, scheduling and routing problems [54–56].