RELOG/docs/src/problem.md

Mathematical problem definition

Overview and assumptions

The mathematical model employed by RELOG is based on three main components:

1. Products and Materials: Inputs and outputs for both manufacturing and recycling plants. This include raw materials, whether virgin or recovered, and final products, whether new or at their end-of-life. Each product has associated transportation parameters, such as costs, energy and emissions.

2. Manufacturing and Recycling Plants: Facilities that take in specific materials and produce certain products. The outputs can be sent to another plant for further processing, to a collection & distribution center for customer sale, or simply disposed of at landfill. Plants have associated costs (capital, fixed and operating), as well as various limits (processing capacity, storage and disposal limits).

3. Collection and Distribution Centers: Facilities that receive final products from the plants, sell them to customers, and then collect them back once they reach their end-of-life. Collected products can either be sent to a plant for recycling or disposed of at a local landfill. Centers have associated revenue and various costs, such as operating cost, collection cost and disposal cost. The amount of material collected by a center can either be a fixed rate per year, or depend on the amount of product sold at the center in previous years.

!!! note

- We assume that transportation costs, energy and emissions scale linearly with transportation distance and amount being transported. Distances between locations are calculated using either approximated driving distances (continental U.S. only) or straight-line distances.
- Once a plant is opened, we assume that it remains open until the end of the planning horizon. Similarly, once a plant is expanded, its size cannot be reduced at a later time.
- In addition to serving as a source of end-of-life products, centers can also serve as a source for virgin materials. In this case, the center does not receive any inputs from manufacturing or recycling plants, and it generates the desired material at a fixed rate. Collection cost, in this case, refers to the cost to produce the virgin material.
- We assume that centers accept either no input product, or a single input product.

Sets

Symbol	Description
C	Set of collection and distribution centers
P	Set of manufacturing and recycling plants
M	Set of products and materials
G	Set of greenhouse gases
M^+_u	Set of output products of plant/center u.
M^-_u	Set of input products of plant/center u.
T	Set of time periods in the planning horizon. We assume T=\{1,\ldots,t^{max}\}.
E	Set of transportation edges. Specifically, (u,v,m) \in E if m is an output of u and an input of v, where m \in M and u, v \in P \cup C.
E^-(v)	Set of incoming edges for plant/center v. Specifically, edges (u,m) such that (u,v,m) \in E.
E^+(u)	Set of outgoing edges for plant/center u. Specifically, edges (v,m) such that (u,v,m) \in E.

Constants

Symbol	Description	Unit
K^{\text{dist}}_{uv}	Distance between plants/centers u and v	km
K^\text{cap}_{p}	Capacity of plant p, if the plant is open	tonne
K^\text{disp-limit}_{mt}	Maximum amount of material m that can be disposed of (globally) at time t	tonne
K^\text{disp-limit}_{mut}	Maximum amount of material m that can be disposed of at plant/center u at time t	tonne
K^\text{em-limit}_{gt}	Maximum amount of greenhouse gas g allowed to be emitted (globally) at time t	tonne
K^\text{em-plant}_{gpt}	Amount of greenhouse gas g released by plant p at time t for each tonne of input material processed	tonne/tonne
K^\text{em-tr}_{gmt}	Amount of greenhouse gas g released by transporting 1 tonne of material m over one km at time t	tonne/km-tonne
K^\text{mix}_{pmt}	If plant p receives one tonne of input material at time t, then K^\text{mix}_{pmt} is the amount of product m in this mix. Must be between zero and one, and the sum of these amounts must equal to one.	tonne
K^\text{out-fix}_{cmt}	Fixed amount of material m collected at center m at time t	$/tonne
K^\text{out-var-len}_{cm}	Length of the K^\text{out-var}_{c,m,*} vector.	--
K^\text{out-var}_{c,m,i}	Factor used to calculate variable amount of material m collected at center m. See eq_z_collected for more details.	--
K^\text{output}_{pmt}	Amount of material m produced by plant p at time t for each tonne of input material processed	tonne
R^\text{collect}_{cmt}	Cost of collecting material m at center c at time t	$/tonne
R^\text{disp}_{umt}	Cost to dispose of material at plant/center u at time t	$/tonne
R^\text{em}_{gt}	Penalty cost per tonne of greenhouse gas g emitted at time t	$/tonne
R^\text{fix}_{ut}	Fixed operating cost for plant/center u at time t	$
R^\text{open}_{pt}	Cost to open plant p at time t	$
R^\text{rev}_{ct}	Revenue for selling the input product of center c at this center at time t	$/tonne
R^\text{tr}_{mt}	Cost to send material m at time t	$/km-tonne
R^\text{var}_{pt}	Cost to process one tonne of input material at plant p at time t	$/tonne

Decision variables

Symbol	JuMP name	Description	Unit
x_{pt}	x[p.name, t]	One if plant p is operational at time t	binary
y_{uvmt}	y[u.name, v.name, m.name, t]	Amount of product m sent from plant/center u to plant/center v at time t	tonne
z^{\text{collected}}_{cmt}	z_collected[c.name, m.name, t]	Amount of material m collected by center c at time t	tonne
z^{\text{disp}}_{umt}	z_disp[u.name, m.name, t]	Amount of product m disposed of at plant/center u at time t	tonne
z^{\text{input}}_{ut}	z_input[u.name, t]	Total plant/center input at time t	tonne
z^{\text{prod}}_{umt}	z_prod[u.name, m.name, t]	Amount of product m produced by plant/center u at time t	tonne
z^{\text{em-tr}}_{guvmt}	z_em_tr[g.name, u.name, v.name, m.name, t]	Amount of greenhouse gas g released at time t due to transportation of material m from u to v	tonne
z^{\text{em-plant}}_{gpt}	z_em_plant[g.name, p.name, t]	Amount of greenhouse gas g released by plant p at time t	tonne

Objective function

The goals is to minimize a linear objective function with the following terms:

• Transportation costs, which depend on transportation distance K^{\text{dist}}_{uv} and product-specific factor R^\text{tr}_{mt}:

\sum_{(u, v, m) \in E} \sum_{t \in T} K^{\text{dist}}_{uv} R^\text{tr}_{mt} y_{uvmt}

• Center revenue, obtained by selling products received from manufacturing and recycling plants:

- \sum_{c \in C} \sum_{(p,m) \in E^-(c)} \sum_{t \in T} R^\text{rev}_{ct} y_{pcmt}

• Center collection cost, incurred for each tonne of output material sent to a plant:

\sum_{c \in C} \sum_{(p,m) \in E^+(c)} \sum_{t \in T} R^\text{collect}_{cmt} y_{cpmt}

• Center disposal cost, incurred when disposing of output material, instead of sending it to a plant:

\sum_{c \in C} \sum_{m \in M^+_c} \sum_{t \in T} R^\text{disp}_{cmt} z^\text{disp}_{cmt}

• Center fixed operating cost, incurred for every time period, regardless of input or output amounts:

\sum_{c \in C} \sum_{t \in T} R^\text{fix}_{ct}

• Plant disposal cost, incurred for each tonne of product discarded at the plant:

\sum_{p \in P} \sum_{m \in M^+_p} \sum_{t \in T} R^\text{disp}_{pmt} z^\text{disp}_{pmt}

• Plant opening cost:

\sum_{p \in P} \sum_{t \in T} R^\text{open}_{pt} \left(
  x_{pt} - x_{p,t-1}
\right)

• Plant fixed operating cost, incurred for every time period, regardless of input or output amounts, as long as the plant is operational:

\sum_{p \in P} \sum_{t \in T} R^\text{fix}_{pt} x_{pt}

• Plant variable operating cost, incurred for each tonne of input material received by the plant:

\sum_{p \in P} \sum_{(u,m) \in E^-(p)} \sum_{t \in T} R^\text{var}_{pt} y_{upmt}

• Emissions penalty cost, incurred for each tonne of greenhouse gas emitted:

\sum_{g \in G} \sum_{t \in T} R^\text{em}_{gt} \left(
  \sum_{p \in P} z^{\text{em-plant}}_{gpt} + \sum_{(u,v,m) \in E} z^{\text{em-tr}}_{guvmt}
\right)

Constraints

• Definition of plant input (eq_z_input[p.name, t]):

\begin{align*}
& z^{\text{input}}_{pt} = \sum_{(u,m) \in E^-(p)} y_{upmt}
& \forall p \in P, t \in T
\end{align*}

• Plant input mix must have correct proportion (eq_input_mix[p.name, m.name, t]):

\begin{align*}
& \sum_{u : (u,m) \in E^-(p)} y_{upmt}
= K^\text{mix}_{pmt} z^{\text{input}}_{pt}
& \forall p \in P, m \in M^-_p, t \in T
\end{align*}

• Definition of amount produced by a plant (eq_z_prod[p.name, m.name, t]):

\begin{align*}
& z^\text{prod}_{pmt} = K^\text{output}_{pmt} z^\text{input}_{pt}
& \forall p \in P, m \in M^+_p, t \in T
\end{align*}

• Material produced by a plant must be sent somewhere or disposed of (eq_balance[p.name, m.name, t]):

\begin{align*}
& z^\text{prod}_{pmt} = \sum_{v : (v,m) \in E^+(p)} y_{pvmt} + z^\text{disp}_{pmt}
& \forall p \in P, m \in M^+_p, t \in T
\end{align*}

• Plants have a maximum capacity; furthermore, if the plant is not open, its capacity is zero (eq_capacity[p.name,t])

\begin{align*}
& z^\text{input}_{pt} \leq K^\text{cap}_p x_{pt}
& \forall p \in P, t \in T
\end{align*}

• Disposal limit at the plants (eq_disposal_limit[p.name, m.name, t]):

\begin{align*}
& z^\text{disp}_{pmt} \leq K^\text{disp-limit}_{pmt}
& \forall p \in P, m \in M^+_p, t \in T
\end{align*}

• Once a plant is built, it must remain open until the end of the planning horizon (eq_keep_open[p.name, t]):

\begin{align*}
& x_{pt} \geq x_{p,t-1}
& \forall p \in P, t \in T
\end{align*}

• Definition of center input (eq_z_input[c.name, t]):

\begin{align*}
& z^\text{input}_{ct} = \sum_{u : (u,m) \in E^-(c)} y_{ucmt}
& \forall c \in C, t \in T
\end{align*}

• Calculation of amount collected by the center (eq_z_collected[c.name, m.name, t]). In the equation below, K^\text{out-var-len} is the length of the K^\text{out-var}_{c,m,*} vector.

\begin{align*}
& z^\text{collected}_{cmt}
  = \sum_{i=0}^{\min\{K^\text{out-var-len}_{cm}-1,t-1\}} K^\text{out-var}_{c,m,i+1} z^\text{input}_{c,t-i}
  + K^\text{out-fix}_{cmt}
& \forall c \in C, m \in M^+_c, t \in T
\end{align*}

• Products collected at centers must be sent somewhere or disposed of (eq_balance[c.name, m.name, t]):

\begin{align*}
& z^\text{collected}_{cmt} = \sum_{v : (v,m) \in E^+(c)} y_{cvmt} + z^\text{disp}_{cmt}
& \forall c \in C, m \in M^+_c, t \in T
\end{align*}

• Disposal limit at the centers (eq_disposal_limit[c.name, m.name, t]):

\begin{align*}
& z^\text{disp}_{cmt} \leq K^\text{disp-limit}_{cmt}
& \forall c \in C, m \in M^+_c, t \in T
\end{align*}

• Global disposal limit (eq_disposal_limit[m.name, t])

\begin{align*}
& \sum_{p \in P} z^\text{disp}_{pmt} + \sum_{c \in C} z^\text{disp}_{cmt} \leq K^\text{disp-limit}_{mt}
& \forall m \in M, t \in T
\end{align*}

• Computation of transportation emissions (eq_emission_tr[g.name, u.name, v.name, m.name, t):

\begin{align*}
& z^{\text{em-tr}}_{guvmt} = K^{\text{dist}}_{uv} K^\text{em-tr}_{gmt} y_{uvmt}
& \forall g \in G, (u, v, m) \in E, t \in T
\end{align*}

• Computation of plant emissions (eq_emission_plant[g.name, p.name, t]):

\begin{align*}
& z^{\text{em-plant}}_{gpt} = \sum_{(u,m) \in E^-(p)} K^\text{em-plant}_{gpt} y_{upmt}
& \forall g \in G, p \in P, t \in T
\end{align*}

• Global emissions limit (eq_emission_limit[g.name, t]):

\begin{align*}
& \sum_{p \in P} z^{\text{em-plant}}_{gpt} + \sum_{(u,v,m) \in E} z^{\text{em-tr}}_{guvmt} \leq K^\text{em-limit}_{gt}
& \forall g \in G, t \in T
\end{align*}