Keywords

1 Introduction

The rolling horizons procedure is a frequent tool in industry and academic environments [7, 35]. Practical application is identified in inventory management, production planning, scheduling/sequencing, plant location, machine replacement, cash management, capacity expansion and wheat trading/storage [5]. Its use can support decisions in uncertain environments and transform the resolution of a long-horizon problem into a sequential resolution of short-horizon problems [9, 17, 26, 30, 31, 39]. But it is a heuristic method, where it should be said that the best planning operations proposals that are obtained on each rolling horizon are not necessarily the same planning operations proposals that would be found in the solution of the entire time horizon [4].

The rolling horizons are a representation of the industrial reality [7]. Companies make decisions about the operations planning from forecasts and orders, their ongoing situation, and the available capacity [16]. The information is usually updated in the following periods along with the results of the planned planning. The new information available may have variations within the expected ranges (stochastic tools, fuzzy, sensitivity analysis can be used to simulate it), or totally unexpected (require contingency plans for accidents or catastrophes) or higher than forecast ranges but expected (reactive plan in real time together with preventive actions could be used). This new information allows updating the planning for the following periods, by modifying the previous plan or launching a new planning recalculation [24]. The companies recalculate their planning according to information updates, although they try not to make changes in the near periods, in order to reduce the nervousness or planning instability and costs [32]. The competitiveness of the company lies in the balance between its operating costs and its ability to react to changes.

These problems frequently have symmetries in their mathematical programming models. Alternative solutions can be generated with similar results in the objective function. Variables that can be exchanged without changing the structure of the problem [1]. These symmetries slow down the search algorithms of the best solution to the objective function that complies with the modeling constraints. The symmetries increase the options that the branch and bound algorithms must solve [33]. The symmetry in a problem increases the search space size for the algorithms. Equivalent solutions can be exchanged [3], where different solutions are proposed for the same objective value [38]. This may make it more difficult to demonstrate the optimality of the problem solutions and, therefore, increase the computation time [25]. Jans [13] showed that eliminating the formulation symmetry can be useful to accelerate computational times. It reduces the amount of search needed to solve the problem [10]. Margot [18] comments that breaking symmetry can turn a computationally intractable problem into one that is easily solved.

In the procedure of rolling horizons in mathematical programming models, symmetry can become especially relevant. The different solutions within the allowed tolerance or calculation time can imply large differences in the following planning horizons. The proposed solutions for the model can vary significantly from one computer equipment to another.

To the best of our knowledge, we have not found a review of the literature on the treatment of symmetry when using the rolling horizons procedure. Therefore, a systematic review of the literature on measures to avoid symmetry in models with the rolling horizons procedure is proposed.

The rest of the paper is structured as follows: first, a short description of the review methodology is introduced; second, a brief discussion of the results is presented; finally, the paper ends with the conclusion and future works.

2 Review Methodology

The review has been carried out following the systematic literature review protocol presented and used by Marín et al. [19] and Rius-Sorolla et al. [28, 29].

The steps proposed are as follows: set the goal, select type of reference and database, search filter and manage reference, extract information of selected reference, and write the report.

Therefore, it uses a transparent procedure, to find, evaluate, and synthesize the results of relevant research. The procedures are explicitly defined in advance to ensure that the exercise is transparent and can be replicated.

Our goal is to identify actions to avoid symmetry in the application of the rolling horizons procedure.

The search was carried out on selecting the references that contain the terms “symmetry” and “rolling horizon” in the database of Scopus and Web Of Science (WoS) with access from Universitat Politècnica of València. The extracted works have been those that deal with both concepts. A snowball strategy has also been applied to the first identified references in order to include other related jobs. In Table 1, the publication of the reviewed references can be identified.

Table 1 Publication of the selected references
Full size table

3 Results

According to Margot [18], the main approaches to deal with symmetries in integer linear programming are symmetry breaking inequalities, perturbation, fixing variables, and pruning of the enumeration tree. It also can be reformulated the problem so that it has a reduced amount of symmetry, or even none at all [10].

The actions identified to break the symmetry are:

  • Add new restrictions to order equivalent elements [12, 14, 22, 25]. If a product can be done on several machines, it is established that it should be done in the first machine. Lexicographic perturbation Eq. (1),

    $$x_{i} \le x_{i + 1} \,\forall i$$
    (1)

  • Sort the machines according to some natural logic, such as varying the setup costs per machine, or decreasing the total costs per machine or decreasing resources capacity, or with a weighting factor in the objective function increasing parameter values with product index [8, 13].

  • Set new restrictions that avoid those undesired solutions, without eliminating feasible solutions [6]. One way is by setting variables to reduce the feasible solution space [21, 34, 36, 37]. A binary variable is fixed on some periods, as can be seen in Eq. (2).

    $$\delta_{t} = 1,\,\forall t \ge t_{s}$$
    (2)

  • Give continuity with the previous period, like if the oven was working in the previous period, the setup is eliminated [23], as can be seen in Eq. (3).

    $$Z_{yt} \le Z_{{y\left( {t - 1} \right)}} \,\forall y,t > 1$$
    (3)

The actions identified in the systematic literature review attempt to break the symmetries produced by equivalent demands in different products [3], by the existence of equivalent resources in the models, such as parallel machines [2, 8, 11, 13] or a combination of both [38] or routes with equal costs [15].

4 Conclusion and Future Work

This work presents the results of a systematic literature review on actions to break the symmetry in models that apply the rolling horizons procedure.

It highlights the importance of breaking symmetry in order to avoid unwanted solutions or speed up the resolution process. The identified symmetries focus on demands related to equivalent products and operations that can be performed on equivalent resources. The identified symmetries relate to product, resources, operations and time indexes in the mathematical programming models [27].

The actions identified to break the symmetry have been grouped into four categories. And they are those that give a lexicographic order, varying the parameters of the model to establish a differentiation and setting variables in certain conditions to eliminate undesired solutions and restriction between periods to give an order.

Future research should be done to extend the symmetry breaking action to another index as the time relative index of the rolling horizons procedure. In addition, a specific pruning option could be developed for the rolling horizons procedure.