A Tabu List-Based Algorithm for Capacitated Multilevel Lot-Sizing with Alternate Bills of Materials and Co-Production Environments

Abstract

Since 2013 [19,20], the functioning of GMOP has been improved with the implementation of mathematical relaxation methods and its efficiency in practical applications has been demonstrated (e.g., in the automotive sector) [21,22,23]. [...]conclusions and further research opportunities are highlighted in Section 7. [...]Equations (5)–(8) define the range and domain of the decision variables. [...]diversification strategies guide the search towards poorly-explored search spaces [32].

1. Introduction

The definition of lot sizes represents one of the most important decisions in production planning. Lot-sizing models aim to guarantee the fulfillment of the demand requirements, establishing a balance between holding and setup costs. Complex assembly systems usually require wide and robust product structures, which may involve the use of alternate bills of materials and co-production settings. In these cases, the complexity of lot sizing decisions increases along with flexibility. Depending on the problem size and the number of considered constraints, the use of exact solution models can be inefficient in terms of computational times, especially for operational planning purposes [1,2].

Exact solution approaches have been widely used for the NP-hard Capacitated Lot Sizing Problem (CLSP), including cut-generation [3] and redefinition techniques [4], supported by mathematical approaches as Branch and Bound, Lagrangian Relaxation, and Wagner-Within. In the case of multi-level CLSP, the use of linear programming (LP) methods showed good results [5].

On the other hand, the use of heuristic and meta-heuristic methods for solving lot-sizing problems has become increasingly frequent and implementations have flooded the scientific literature [2,6,7,8,9]. The advantages of these kinds of approximate procedures include reduced computational times, the ability to solve larger problems, higher flexibility, high-quality solutions, and reduced implementation costs (e.g., compared with commercial software licenses [10]).

Among the different heuristic approaches, local and guided-search algorithms, such as the Tabu Search (TS), have demonstrated efficiency in a variety of lot-sizing problems, especially with multi-product and multi-level planning structures [11,12,13]. Most implementations take initial solutions from constructive algorithms, which are later improved with iterative small modifications (called moves). The search is guided by long-term and short-term memory structures within intensification and diversification phases [9,14,15,16].

The Generic Materials and Operations Planning model (GMOP) is one of the most recent exact models for solving lot-sizing problems that considers alternative bills of materials, multi-level structures, multi-site production/packaging, and co-production settings. This model was proposed in 2011 [17,18] as a robust mixed integer programming (MIP) problem. Since 2013 [19,20], the functioning of GMOP has been improved with the implementation of mathematical relaxation methods and its efficiency in practical applications has been demonstrated (e.g., in the automotive sector) [21,22,23].

Developing heuristic approaches for solving this problem has been suggested as a research opportunity since 2012 [24,25], but these kinds of procedures have not been extensively explored [26]. Proposing alternative solution methods and data structures represents a contribution not only to further GMOP studies, but also to the wider research field of multi-level lot-sizing problems, employing alternative product structures instead of the generalized Gozinto structure.

The aim of this article is to propose an alternative heuristic algorithm for the GMOP, using the short-term memory principles of TS. The main considerations include: Capacity constraints, a multi-product/multi-level structure, a co-production environment, and the existence of alternate bills of materials for final products [27,28].

This paper is organized as follows. Section 2 shows the GMOP generalities. Section 3 shows a brief introduction to TS algorithms. Section 4 exposes the methodological framework and a functioning overview of the proposed algorithm. The obtained results and their discussion are shown, respectively, in Section 5 and Section 6. Finally, conclusions and further research opportunities are highlighted in Section 7.

2. The Generic Materials and Operational Planning Model (GMOP)

The Generic Materials and Operational Planning model was proposed by Garcia Sabater et al. [19] and Maheut et al. [18], as an alternative for modeling the existing relations between the processes and the materials needed for the elaboration of a product.

Unlike the Gozinto representation (where the priority falls on the final product and its components), this lot-sizing model focuses on the planning of operations (strokes) to be made for the manufacture, purchase, or transportation of a certain product or a group of products.

A stroke is defined as any activity or operation that allows for the transformation of a set of products or Stock Keeping Units (SKUs) into another set of SKUs, using or immobilizing a certain amount of resources. As shown in Figure 1, a Stroke can contain the following attributes [19]:

1. Outputs (Stroke Outputs): The product or set of products obtained from the stroke execution, as shown in Table 1.

2. Inputs (Stroke Inputs): The product or set of products consumed at the execution of the stroke, as shown in Table 2.

3. Lead times

4. Operation times and costs

5. Set-up times and costs

6. Resource usage: Resources can be, for example, machinery, workforce, and so on.

The GMOP model can easily include capacity constraints, as well as direct, inverse, and alternate bill of materials, multi-site production, resource requirements, by-products, transportation modes, and packaging processes [19].

The problem is presented as a mixed integer programming model, whose parameters and variables are shown in Table 3. The objective function (1) aims to minimize the total planning cost Z, which includes storage, operation, and set up costs generated by the execution of strokes.

Equation (2) represents the inventory constraint. It considers the stock levels from the previous period, demand requirements, purchased items, and the quantities generated and consumed by the strokes in every time period.

Equation (3) ensures the inclusion of a setup cost when a stroke is used: If _zk,t is larger than zero, then _δk,t must be 1 in order to satisfy the constraint.

Equation (4) is a capacity constraint that limits the use of resources by considering both setup and operations times.

Finally, Equations (5)–(8) define the range and domain of the decision variables.

Z=Min∑t∑i(_hi,t∗_Xi,t)+∑t∑k(C_Sk,t∗_δk,t+C_Ok,t∗_zk,t)+∑t∑i(C_Bi,t∗_wi,t)

_Xi,t=_Xi,t−1−_Di,t+_wi,t−∑k(S_Ii,k∗_zk,t)+∑k(S_Oi,k∗_zk,t−LTk)∀(i,t)

_zk,t−M∗_δk,t≤0∀(k,t)

∑k(T_Sk,r∗_δk,t)+∑k(T_Ok,r∗_zk,t)≤KA_Pr,t∀(r,t)

_Xi,t≥0

_wi,t≥0

_zk,t∈^Z+

_δk,t∈0,1.

3. Tabu Search Algorithms

The Tabu Search (TS) method is a metaheuristic method proposed by Glover in 1989 [29] and 1993 [30]. It is an iterative procedure which explores a set of problem solutions, making moves from one solution x to another solution ^x′ inside a neighborhood V(x) . Moves aim to find optimal or near-optimal solutions, evaluating some objective function that is to be minimized.

TS involves adaptive memory principles, the creation of constrained search spaces, and the utilization of short-term and long-term learning mechanisms [31].

The Tabu list is the main short-term memory mechanism and keeps a record of the most recently adopted moves, keeping the algorithm from re-evaluating previously-considered solutions and getting stuck in local optima. The algorithm starts with an initial solution _Z0 as the current solution S. Moves allow the algorithm to make changes to the solution array, generating candidate solutions _Sc . These changes can be done using an insertion, mutation, combination, or by crossing strategies.

A set V is generated with _Ns candidate solutions. After evaluation, only the best solution per iteration is adopted ( S←_Sc ) and the moves ( _Sk to _Sc ) and ( _Sc to _Sk ) are registered in the Tabu list, being forbidden for a specific number of iterations (the so-called Tabu tenure).

The algorithm stops when a completion criterion is fulfilled. The most frequent finalization conditions include a maximum number of iterations, a minimum value for the objective function, or a specific number of iterations without substantial improvements.

TS also allows for the implementation of aspiration, diversification, and intensification mechanisms. Aspiration criteria allow the algorithm to improve the solution by considering moves included in the Tabu list. Intensification mechanisms are short-term or long-term memory structures that allow a deep exploration of promising search spaces. Finally, diversification strategies guide the search towards poorly-explored search spaces [32].

TS algorithms have been implemented in a variety of lot-sizing and scheduling problems. This approach frequently offers high-quality solutions and has been able to outperform other heuristics and relaxation methods [12,33,34,35]. One of the main motivations for adopting a TS approach lies in the advantages of local search, especially its efficiency for managing hard constraints in large scale problems [36]. TS principles are especially useful when reducing search neighborhoods and guiding the search into feasible solutions.

The reference [9] shows one of the first TS implementations for a multi-level lot-sizing problem. The authors compared the performance between a pure Linear Programming (LP) method and two LP-based heuristics (LP combined with Simulated Annealing (SA) and TS) when solving a multi-level capacitated lot-sizing problem (MCLSP) in an assembly production system with bottleneck constraints. An initial solution was obtained with a modified greedy algorithm and the search was guided according to non-restricted moves with higher improvements in the objective function. No diversification, intensification, or aspiration mechanisms were specified. The results showed better performance in the LP-based approaches.

In [15], two different heuristic methods were proposed to solve a capacitated multi-level lot-sizing and scheduling problem for a multiple-item, single machine system. The first method was based on a “randomized regrets” heuristic, and the second was a TS-based heuristic. A Gozinto product structure was represented using a disjunctive arcs method, and moves were performed according to the existence of adjacent nodes with larger improvements in the objective function. The computational results were similar for both heuristics and the inclusion of multiple resources, setup times, and back-orders were proposed as future work opportunities.

Other well-known heuristic approaches, such as SA and Genetic Algorithms (GA), were tested and compared with TS in [12]. The obtained results showed that TS outperformed the SA and GA methods, especially when the problem involved confirmed order demand.

TS is usually combined with other methods in order to improve results. A hybrid algorithm TS–SA was proposed in [14] for solving a multi-level lot-sizing problem with general product structures. TS mechanisms were used to guide the search with the help of SA components. The results demonstrated that the inclusion of the TS method led to an improvement in cost performance, when compared with CPLEX–LP solutions.

In the case of single-level lot-sizing problems, TS has been tested within a wide variety of configurations and constraints: [37] showed a capacitated, single-item problem with dynamic demand. The multi-item variation for this problem was shown in [38].

In [39], the problem defined was multi-item with setup carry-over. The TS approach included dynamic Tabu lists and penalty constraints.

A dynamic lot-sizing problem with product returns and re-manufacturing constraints was presented in [35]. The initial solutions were generated using a blockchain-based algorithm and the TS algorithm was able to obtain satisfactory results in at least 96% of test instances.

4. Methods 4.1. Algorithm Overview

The proposed algorithm is an approximation to a TS method [29,40]. An outline of the Tabu list-based algorithm (TLBA) is shown in Algorithm 1.

Algorithm 1: Tabu List-based algorithm outline

[Image omitted. See PDF.]

Figure 1.Bill of materials example representation in the Generic Materials and Operational Planning (GMOP) model [28].

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Figure 2.The identification system for test instances [28]. SKU, Stock Keeping Unit.

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Figure 3.Main effects plot for total planning cost in large instances. Source: Minitab®.

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Figure 4.Main effects plot for total computational time in large instances. Source: Minitab®.

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Figure 5.Solution generation process.

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Figure 6.Box chart for the Tabu List-Based Algorthm (TLBA) average gaps.

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Figure 7.Box chart for the average TLBA computational times.

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Table 1.Stroke outputs matrix, according to Figure 1.

_SOi,k k1 k2 k3 k4 k5

SKU A 1 1 0 0 0

SKU B 0 0 1 0 0

SKU C 0 0 0 1 1

SKU D 0 0 0 0 0

SKU E 0 0 0 0 0

SKU F 0 0 0 0 0

SKU G 0 0 0 0 0

SKU H 0 0 0 0 0

[Image omitted. See PDF.]

Table 2.Stroke inputs matrix, according to Figure 1.

_SIi,k k1 k2 k3 k4 k5

SKU A 0 0 0 0 0

SKU B 1 0 0 0 0

SKU C 0 1 0 0 0

SKU D 0 0 3 0 0

SKU E 0 0 2 0 0

SKU F 0 0 0 5 4

SKU G 0 0 0 3 0

SKU H 0 0 0 0 5

[Image omitted. See PDF.]

Table 3.Parameters and decision variables for the GMOP model [19].

Symbol Description

i Index set of products (includes product, packaging, and site)

t Index set of planning periods

r Index set of resources

k Index set of strokes

_Di,t Demand of product i for period t

hi,t Cost of storing a unit of product i in period t

C_Ok,t Cost of stroke k in period t

C_Sk,t Cost of the setup of stroke k in period t

C_Bi,t Cost of purchasing product i in period t

S_Oi,k Number of units i that generates a stroke k

S_Ii,k Number of units i that stroke k consumes

L_Tk Lead time of stroke k

KA_Pr,t Capacity availability of resource r in period t (in time units)

M A sufficiently large number

T_Ok,r Capacity of the resource r required for performing one unit of stroke k (in time units)

T_Sk,r Capacity required of resource r for setup of stroke k (in time units)

_zk,t Amount of strokes k to be performed in period t

_δk,t =1 if stroke k is performed in period t (and 0 otherwise)

_wi,t Purchase quantity for product i in period t

_Xi,t Stock level of product i on hand at the end of period t

Z Total Planning Cost

[Image omitted. See PDF.]

Table 4.Parameter settings for randomly-generated test cases [28].

Parameter Value

i 50, 100, or 200 SKU

i∗ 15, 30, or 60 final products

t 25, 50, or 75 planning periods

r 5, 10, or 20 resources

k 50, 100, or 200 strokes

_Di,t Uniform [1000, 2000]

_hi,t Uniform [10, 20]

C_Ok,t Uniform [5, 8]

C_Sk,t Uniform [5, 10]

S_Oi,k Uniform [35, 50]

S_Ii,k Uniform [4, 8]

L_Tk Uniform [1, 2]

KA_Pr,t Uniform [2000, 6000]

Uniform [4000, 8000]

Uniform [12,000, 14,000]

T_Ok,r Uniform [2, 5]

T_Sk,r Uniform [5, 10]

[Image omitted. See PDF.]

Table 5.Experimental factors and levels.

Response Variables Experimental Factor Experimental Levels

Total Planning Cost

Total Computational Time Number of Iterations 25

50

75

Tabu Tenure 1/4 of Number of Iterations

1/2 of Number of Iterations

3/4 of Number of Iterations

Number of Candidate Solutions

per Iteration 5

10

15

[Image omitted. See PDF.]

Table 6.ANOVA for total planning cost in small problems.

Source DF Adj. MS F p

Tabu_Tenure 2 5.68 ×¹⁰¹² 5.13 0.037

Iterations 2 1.65 ×¹⁰¹⁴ 149.4 0.000

Candidate_Sol. 2 3.99 ×¹⁰¹³ 36.04 0.000

Tabu_Tenure *Iterations 4 3.21 ×¹⁰¹² 2.9 0.094

Tabu_Tenure *Candidate Sol. 4 8.33 ×¹⁰¹¹ 0.75 0.583

Iterations *Candidate Sol. 4 6.64 ×¹⁰¹² 6 0.016

Error 8 1.11 ×¹⁰¹²

Total 26

S = 1,051,813 R-Sq = 98.13% R-Sq(adj) = 93.92%

* DF, Degrees of Freedom; Adj MS, Adjusted Mean of Squares; F, F statistic; p, p value.

[Image omitted. See PDF.]

Table 7.ANOVA for total planning cost in large problems.

Source DF Adj. MS F p

Tabu_Tenure 2 1.28 ×¹⁰¹⁴ 4.24 0.055

Iterations 2 4.91 ×¹⁰¹⁴ 16.24 0.002

Candidate_Sol. 2 3.13 ×¹⁰¹⁴ 10.34 0.006

Tabu_Tenure *Iterations 4 3.64 ×¹⁰¹³ 1.2 0.380

Tabu_Tenure *Candidate Sol. 4 3.06 ×¹⁰¹³ 1.01 0.456

Iterations *Candidate Sol. 4 2.63 ×¹⁰¹³ 0.87 0.523

Error 8 3.02 ×¹⁰¹³

Total 26

S = 5,499,026 R-Sq = 90.24% R-Sq(adj) = 68.29%

* DF, Degrees of Freedom; Adj MS, Adjusted Mean of Squares; F, F statistic; p, p value.

[Image omitted. See PDF.]

Table 8.ANOVA for total computational time in small problems.

Source DF Adj. MS F p

Tabu_Tenure 2 4894 1.09 0.382

Iterations 2 1,400,810 311.45 0.000

Candidate_Sol. 2 1,772,038 393.99 0.000

Tabu_Tenure *Iterations 4 5545 1.23 0.370

Tabu_Tenure *Candidate Sol. 4 11,985 2.66 0.111

Iterations *Candidate Sol. 4 119,395 26.55 0.000

Error 8 4498

Total 26

S = 67.0647 R-Sq = 99.48% R-Sq(adj) = 98.31%

* DF, Degrees of Freedom; Adj MS, Adjusted Mean of Squares; F, F statistic; p, p value.

[Image omitted. See PDF.]

Table 9.ANOVA for total computational time in large problems.

Source DF Adj. MS F p

Tabu_Tenure 2 21,268 0.11 0.900

Iterations 2 27,135,914 135.58 0.000

Candidate_Sol. 2 27,235,975 136.08 0.000

Tabu_Tenure *Iterations 4 126,682 0.63 0.653

Tabu_Tenure *Candidate Sol. 4 237,283 1.19 0.387

Iterations *Candidate Sol. 4 1,891,696 9.45 0.004

Error 8 200,144

Total 26

S = 447.374 R-Sq = 98.66% R-Sq(adj) = 95.64%

* DF, Degrees of Freedom; Adj MS, Adjusted Mean of Squares; F, F statistic; p, p value.

[Image omitted. See PDF.]

Table 10.Results for total planning cost and total computational times.

Test Instance B&B GMOP Initial Solution TLBA

Optimal Cost (COP) Time (secs) GAP Min. GAP Avg. GAP Min. Comp. Time (secs) Avg. Comp. Time (secs)

TI1 10,326,117 969.4 14.54 0.2666 0.2952 1424.9 1482.2

TI2 2,683,534 972.5 23.7 0.2757 0.2895 1258.3 1279

TI3 4,890,778 972.8 72.96 0.1659 0.1722 1321.5 1350.2

TI4 106,247,444 994.56 30.23 0.1058 0.1148 7227.1 7260.2

TI5 74,585,367 999.25 10.84 0.1705 0.1773 7741.5 7792.7

TI6 43,278,154 993.24 21.84 0.2871 0.3041 8030.7 8042.5

TI7 1,585,713,234 1001 2.62 0.2186 0.2328 24,959.6 25,002.3

TI8 765,235,068 1020.1 4.66 0.2309 0.3322 27,000.4 27,145.6

TI9 252,626,622 1017.2 16.08 0.1301 0.1486 23,741.6 23,741.6

Author Contributions

Conceptualization, J.R.C.-H., G.R.-S., and J.P.G.-S.; Formal analysis, A.R.R.-C.; Methodology, A.R.R.-C.; Project administration, A.R.R.-C.; Software, A.R.R.-C.; Supervision, J.R.C.-H., G.R.-S., and J.P.G.-S.; Validation, A.R.R.-C.; Writing-original draft, A.R.R.-C.; Writing-review & editing, J.R.C.-H., G.R.-S., and J.P.G.-S.

Funding

This research received no external funding.

Acknowledgments

This paper shows the results of the project entitled "Algoritmo heurístico basado en listas tabú para la planificación de la producción en sistemas multinivel con listas de materiales alternativas y entornos de coproducción" supported by Universidad de la Costa and Universitat Politècnica de València.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:

TLBA Tabu-List Based Algorithm

GMOP Generic Materials and Operational Planning Model

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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