Discrete Optimization
Lot sizing with carbon emission constraints

https://doi.org/10.1016/j.ejor.2012.11.044Get rights and content

Abstract

This paper introduces new environmental constraints, namely carbon emission constraints, in multi-sourcing lot-sizing problems. These constraints aim at limiting the carbon emission per unit of product supplied with different modes. A mode corresponds to the combination of a production facility and a transportation mode and is characterized by its economical costs and its unitary carbon emission. Four types of constraints are proposed and analyzed in the single-item uncapacitated lot-sizing problem. The periodic case is shown to be polynomially solvable, while the cumulative, global and rolling cases are NP-hard. Perspectives to extend this work are discussed.

Highlights

► We introduce four new environmental constraints in lot-sizing problems. ► Mathematical programming models for the single-item lot-sizing problem are introduced. ► A polynomial dynamic program is proposed for the periodic carbon emission constraint. ► It is shown that the problem is NP-hard with any of the three other constraints.

Introduction

Legislation is evolving in order to enforce control on carbon emissions. This will probably be done by constraining companies to emit less than a given amount of carbon dioxide by product unit that is produced and transported. Along with legislation and norms, some companies may volunteer in this direction for marketing reasons and to get a competitive advantage. The amount of carbon emission will probably appear on item packages in the near future. Companies will face new constraints that will force them to reduce carbon emissions while still minimizing production and transportation costs. There are few papers addressing production planning and transportation problems that take into account environmental constraints. Generally, environmental issues are integrated as cost components in the objective function, and the resulting problems are solved using multi-criteria approaches (see [13], [1]). Note that classical cost components (production and transportation costs) have the same behavior as environmental cost components (e.g. reducing the number of vehicles or the total distance).

One of the main objectives of green logistics is to evaluate the environmental impact of different distribution and production strategies to reduce the energy usage in logistics activities. Although the interest in green logistics has grown in the last decades, current logistics practice still rarely complies with environmental constraints. One of the objectives of the Kyoto protocol is to stabilize and then reduce greenhouse gas emissions in order to limit global warming. Carbon dioxide being one of the most important greenhouse gases, countries will have to reduce their carbon emissions, and will require companies to trim down their carbon emissions. A quota of carbon emission per company will probably be fixed (e.g. California). As a result, some companies have already started to monitor their carbon footprint to evaluate the environmental impact of their activities. The classical production and distribution models focus on the minimization of costs subject to operational constraints. Considering green logistics objectives and constraints will lead to new problems resulting in novel combinatorial optimization models.

Green supply chain management (see [23]) has been extended to include green inventory models that link inventory and ordering behavior and emissions. Sbihi and Eglese [21] present a short state-of-the-art on green logistics and combinatorial optimization. They describe some of the problems that arise in green logistics which can be formulated as combinatorial optimization problems. They focus on the topics of reverse logistics, waste management and vehicle routing and scheduling. Dekker et al. [9] provides a recent overview of issues and challenges in green logistics and operations research, focusing on supply chain management and design for transportation, inventory and production. Reverse logistics is an important part of Green supply chain management and a lot of research has been conducted in this field. Teunter et al. [24] address this issue for lot-sizing problems, and propose two models. In the first model, they assume that manufacturing and remanufacturing operations are carried out in the same factory. They model these operations with a joint setup cost. In the second model, they assume that manufacturing and remanufacturing operations are done on two separate lines. Several works on the integration of remanufacturing in a closed loop supply chain can be found in the literature [17], [15], [19].

Depending on the objectives of a company, the integration of carbon emission constraints can be considered at different decision levels (strategic, tactical and operational). At the strategic level, designing supply chain flows or locating a factory or a warehouse impact green constraints and objectives [8], [26], [18]. At the tactical level, carbon emissions can be considered in production and distribution planning decisions. Recently, some authors [14], [6] study the integration of carbon emission constraints in classical inventory management models. At the operational level, carbon emission constraints can be related to vehicle routing or production scheduling decisions [3], [11]. Our paper focuses on the tactical decision level of a supply chain.

There is little research addressing the introduction of carbon emission constraints in production and/or distribution planning models. In lot-sizing, we only found the work of Benjaafar et al. [4] that integrates carbon emission constraints. The authors insist on the potential impact of operational decisions on carbon emissions and the need for Operations Management research that incorporates carbon emission concerns. The authors also point out that the contribution of operational research in this area is almost absent. Benjaafar et al. [4] add a new capacity constraint that links and limits all carbon emissions related to production and storage over the planning horizon. The weakness of this constraint is that producers can create large carbon emissions at the beginning of the horizon by producing large quantities, and balance the total carbon emission by producing nothing at the end of the horizon.

As previously mentioned, the need of companies to monitor their carbon emissions is growing. This monitoring must be consistent with production and distribution planning models that must take into account carbon emission constraints. More precisely, if the monitoring of the carbon footprint is aggregated according to the type of vehicles and their consumption, there is no need to consider more detailed information on each vehicle in production and distribution planning models.

There are several methodologies to calculate carbon emissions (Greenhouse Gas protocol [12], ARTEMIS [2], EcoTransIT [10], etc.). Greenhouse Gas protocol is the most commonly used, since it is easy to use and its scope is worldwide. The unitary carbon emission of a product can be calculated using a linear function that depends on the distance traveled (in kilometers) and on the carbon emission of the vehicle used (in grams of CO2 per kilometer). This carbon emission model is adopted in this paper where, for a given supplying mode, the carbon emission is proportional to the number of product units that are shipped. We define a supplying mode as a combination of a transportation mode (combining one or more types of vehicles) and a production facility.

In this paper, we study multi-sourcing lot-sizing problems with carbon emission constraints. These new constraints are induced from a maximum allowed carbon dioxide emission coming from legislation, green taxes or the initiatives of companies. Contrary to Benjaafar et al. [4], where a global limit of carbon emission is studied, we consider a maximum environmental impact allowed on average per item. We study four types of carbon emission constraints: (1) a Periodic carbon emission constraint, (2) a Cumulative carbon emission constraint, (3) a Global carbon emission constraint and (4) a Rolling carbon emission constraint. The global carbon emission constraint has the same drawbacks than the constraint introduced in [4].

The main contribution of this paper is twofold. First, we propose new lot-sizing models that take into account different carbon emission constraints. Second, we determine the complexity status of these new models. We propose a polynomial dynamic programming algorithm for the problem with periodic carbon emission constraint, and show that the three other problems are NP-hard. The outline of the paper is as follows. In Section 2, we provide different mathematical formulations to model the four types of carbon emission constraints. In Section 3, we show that the uncapacitated multi-sourcing lot-sizing problem with the periodic carbon emission constraint can be solved using a polynomial dynamic programming algorithm. In Sections 4 The uncapacitated single-item lot-sizing problem with the cumulative carbon emission constraint, 5 The uncapacitated single-item lot-sizing problem with the global carbon emission constraint or rolling carbon emission constraint, we show that the uncapacitated multi-sourcing lot-sizing problem with the cumulative carbon emission constraint, global carbon emission constraint or rolling carbon emission constraint is NP-hard. We conclude and discuss some perspectives of this work in Section 6.

Section snippets

Mathematical programming models

Consider a multi-sourcing lot-sizing problem faced by a company that must determine, over a planning horizon of T periods, when, where and how much to produce of an item to satisfy a deterministic time-dependent demand. Different production locations and transportation modes are available to satisfy a given demand. Let us consider M different supplying modes, where a mode corresponds to the combination of a production facility and a transportation mode. To each mode are associated classical

The uncapacitated single-item lot-sizing problem with the periodic carbon emission constraint

We want to establish that the multi-sourcing Uncapacitated Lot-Sizing problem with the Periodic Carbon emission constraint (ULS-PC) problem can be solved in polynomial time. More precisely, we show that we can reformulate the ULS-PC problem as a standard lot-sizing problem, i.e. without carbon emission constraints, using a pre-computation step in O(M2T). Thus, standard lot-sizing combinatorial algorithms can be used to solve the problem.

The uncapacitated single-item lot-sizing problem with the cumulative carbon emission constraint

In this section, we study the multi-sourcing Uncapacitated Lot-Sizing problem with the Cumulative Carbon emission constraint (ULS-CC): For each period t, the average amount of carbon emission per product ordered from the first period up to t should not exceed an impact limit Etmax. As in the case of the periodic carbon emission constraint, it is dominant to use at most two modes per period.

Theorem 4

There exists an optimal solution for the ULS-CC problem that uses at most two modes in each period: One

The uncapacitated single-item lot-sizing problem with the global carbon emission constraint or rolling carbon emission constraint

The multi-sourcing Uncapacitated Lot-Sizing problem with the Global Carbon emission constraint (ULS-GC) is a relaxation of the ULS-CC problem, where (T  1) constraints are removed to only keep Constraint (9). Although the ULS-GC problem is simpler, it remains NP-hard. The proof uses the same reduction as Theorem 5 and is omitted.

Recall that Constraint (10) imposes a maximum per unit carbon emission Etmax on every interval of R consecutive periods. It is still dominant to use at most two modes

Conclusion and further research directions

We believe the integration of carbon emission constraints in lot-sizing problems leads to relevant and original problems. This paper is a first step to model such problems from which several new lot-sizing problems could arise. We tried to define and categorize these new constraints. We proposed and studied four types of carbon emission constraints: (1) periodic carbon emission constraint, (2) cumulative carbon emission constraint, (3) global carbon emission constraint and (4) rolling carbon

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