Mario Guajardo

A review on cost allocation methods in collaborative transportation

Collaboration in transportation between two or more agents is becoming an important approach to find efficient solutions or plans. Efficiency can be measured in, for example, lower cost or more flexibility. An important aspect of the collaboration is to decide on how to share the benefits—for example, cost, profit, or resources. There are many sharing mechanisms or cost allocations proposed in the literature. Some are based on simple proportional rules and others are based on theoretical concepts found in game theory. We provide a survey on cost allocation methods found in the literature on collaborative transportation, including problems on planning, vehicle routing, traveling salesman, distribution, and inventory. A total of 55 scientific articles compose the main part of the survey, most of them published between 2010 and 2015. We identify more than 40 cost allocation methods used in this stream of literature. We describe the theoretical basis for the main methods as well as the cases where they are used. We also report savings from the collaborations when they are based on industrial data. Some directions for future research are discussed.

Operations research models for coalition structure in collaborative logistics

Given a set of players and the cost of each possible coalition, the question we address is which coalitions should be formed. We formulate mixed integer linear programming models for this problem, considering core stability and strong equilibrium. The objective function looks for minimizing the total cost allocated among the players. Concerned about the difficulties of managing large coalitions in practice, we also study the effect of a maximum cardinality constraint per coalition. We test the models in two applications. One is in collaborative forest transportation and the other one in inventory of spare parts for oil operations. In these situations, collaboration opportunities involving significant savings exist, but for several reasons, it may be better to group the players in different sub-coalitions rather than in the grand coalition. The models we propose are thus relevant for deciding how to partition the set of players. We also prove that if the strong equilibrium model is feasible, its optimal cost is equal to the optimal cost of the core stability model and, consequently, a coalition structure that solves one problem also solves the other problem. We present results that illustrate this property. We also present results where the core stability problem is feasible and the strong equilibrium problem is infeasible. Setting an upper bound on the maximum cardinality of the coalitions, allows us to study the marginal savings of enlarging the cardinality of the coalitions. We find that the marginal savings of allowing one more player significantly decreases as the bound increases.

Common Mistakes in Computing the Nucleolus

Despite linear programming and duality have correctly been incorporated in algorithms to compute the nucleolus, we have found mistakes in how these have been used in a broad range of applications. Overlooking the fact that a linear program can have multiple optimal solutions and neglecting the relevance of duality appear to be crucial sources of mistakes in computing the nucleolus. We discuss these issues and illustrate them in mistaken examples collected from a variety of literature sources. The purpose of this note is to prevent these mistakes propagate longer by clarifying how linear programming and duality can be correctly used for computing the nucleolus.

Cost allocation in inventory pools of spare parts with service-differentiated demand classes

Holding inventory of spare parts is critical to assure safety and production. In order to save costs, different producers may collaborate through an inventory pool. We consider an inventory pool of spare parts, subject to a service level constraint, where the members of the pool may have different target service levels, so that they represent different demand classes. The pool is implemented either by round-up or rationing policies. The members should agree on how to share the costs. Based on cooperative game theory concepts, we show the important effects that different targets can have in the core stability for this problem. We perform a computational study in a large number of instances, providing insights on the emptiness of the core and the performance of seven allocation methods. We also propose the novel Minimum Deviation from Service Level Referential Cost Method (MIND). This method looks for a stable allocation such that the maximum difference between a cost allocated to a player and its service level referential cost is minimised. The MIND allocation is the solution to a linear programming model and is core guaranteed, in the sense that if the core is not empty, the allocation belongs to the core.

Speciality Oils Supply Chain Optimization: From a Decoupled to an Integrated Planning Approach

We study a problem of tactical planning in a divergent supply chain. It involves decisions regarding production, inventory, internal transportation, sales and distribution to customers. The problem is motivated by the context of a company in the speciality oils industry. The overall objective at tactical level is to maximize contribution and, in order to achieve this, the planning has been divided into two separate problems. The first problem concerns sales where the final sales and distribution planning is decentralized to individual sellers. The second problem concerns production, transportation and inventory planning through refineries, hubs and depots and is managed centrally with the aim of minimizing costs. Due to this decoupling, the solution of the two problems needs to be coordinated in order to achieve the overall objective. In the company, this is pursued through an internal price system aiming at giving the sellers the incentives needed to align their decisions with the overall objective. We propose and discuss linear programming models for the decoupled and integrated planning problems. We present numerical examples to illustrate potential effects of integration and coordination and discuss the advantages and disadvantages of the integrated over the decoupled approach. While the total contribution is higher in the integrated approach, it has also been found that the sellers’ contribution can be considerably lower. Therefore, we also suggest contribution sharing rules to achieve a solution where both the company and the sellers attain a better outcome under the integrated planning.

Inventory management of spare parts in an energy company

We address the problem of how to determine control parameters for the inventory of spare parts of an energy company. The prevailing policy is based on an (s, S) system subject to a fill rate constraint. The parameters are decided based mainly on the expert judgment of the planners at different plants. The company is pursuing to conform all planners to the same approach, and to be more cost efficient. Our work focuses on supporting these goals. We test seven demand models using real-world data for about 21 000 items. We find that significant differences in cost and service level may appear from using one or another model. We propose a decision rule to select an appropriate model. Our approach allows us to recommend control parameters for 97.9% of the items. We also explore the impact of pooling inventory for different demand sources and the inaccuracy arising from duplicate item codes.

Operations Research Techniques for Scheduling Chile's Second Division Soccer League

In this paper, we use operations research (OR) techniques to schedule the Second Division of the Chilean professional soccer league. The solution must satisfy a series of conditions requested by league officials. Because the teams generally travel long distances by bus, geographical restrictions are particularly important. We specify the scheduling problem and solve it using an integer linear programming (ILP) model that defines when and where each match is played, subject to constraints. For the most difficult instances, we formulate a second ILP model that generates home-away patterns and assigns them to the teams; we then run the model, which determines the match schedule. Chilean league officials have successfully used the models to schedule all five Second Division tournaments between 2007 and 2010, replacing the random scheduling methodology that they used previously. Since 2007, the two formulations have been adapted to various formats with which the Second Division has experimented; these include a quadruple round robin and a two-phase tournament with zonal and national phases. The application we present is one of a number of such projects that the authors and their colleagues developed over the past few years, and it represents an expansion of the use of OR techniques for managing tasks in Chilean soccer.

Referee assignment in the Chilean football league using integer programming and patterns

This article uses integer linear programming to address the referee assignment problem in the First Division of the Chilean professional football league. The proposed approach considers balance in the number of matches each referee must officiate, the frequency of each referee being assigned to a given team, the distance each referee must travel over the course of a season, and the appropriate pairings of referee experience or skill category with the importance of the matches. Two methodologies are studied, one traditional and the other a pattern-based formulation inspired by the home-away patterns for scheduling season match calendars. Both methodologies are tested in real-world and experimental instances, reporting results that improve significantly on the manual assignments. The pattern-based formulation attains major reductions in execution times, solving real instances to optimality in just a few seconds, while the traditional one takes anywhere from several minutes to more than an hour.

Joint optimization of pricing and planning decisions in divergent supply chain

Process industries has for long been important for the development of Swedish industry and society. All industries face different conditions that affect how to best run their operations. This thesis aims to describe some of the conditions that characterize process industries compared to other industries. Further one of these characteristics has been studied more closely.One of the traits of process industries is that they are positioned at the start of the transformation process close to the raw material mixing, separating or forming it into products often used for further transformation. Process industries hence become dependent on the properties of these materials. One of the most prominent characteristics inherent from the raw material properties is the divergent bill of material. The divergent bill of material originates from the fact that a given raw material is made up of different components that will yield several products with different characteristics when processed. When splitting the raw material into the desired products the yielded products from a certain raw material usually have different value to the producer, some more desired than others. These multiple products generated poses a challenge from a planning perspective raising questions like “How should we balance the supply and demand of all the products produced?”, “What shall we do with the excess products produced?”.The first paper in this thesis describe the production planning in four Swedish process industries with the ultimate aim to connect their planning to the supply chain characteristics they face as process industries. The study concludes that the industry specific conditions mainly affect planning at short time ranges when planning becomes more detailed. In general the use of planning or decision support systems is low, stemming from a, warranted or not, belief that general decision support systems do not fit process industries. Another finding is that the case companies mainly operate in niche markets. This study also highlighted that the planning complexity arising from characteristic of co- and by-product generation in combination with the lack of decision support systems requires further studies.The subsequent two papers focus on supply chain planning and coordination with a divergent bill of material. They present a mathematical model over the supply chain planning in a real case company in the specialty oils industry. The second paper investigates transfer pricing as a coordination tool by comparing decentralised supply chain planning with centralized planning in an integrated model. Transfer pricing is found to have a potential positive effect on supply chain planning while simultaneously creating problems in terms of uneven distribution of the contribution margin among supply chain partners.Finally the third paper more closely investigates different ways to set transfer prices and comparing them to the optimal transfer prices. Setting optimal transfer prices with a divergent bill of material has proven to be less straightforward than the case with no dependencies between products. Some optimal transfer prices could even be set lower than the marginal cost for producing them due to the dependency between the products in the divergent bill of material. This indicates that there is an opportunity cost for a product that is dependent on the demand of other products.

Scheduling the South American Qualifiers to the 2018 FIFA World Cup by integer programming

Every four years, the 10 national teams members of the South American Football Confederation (CONMEBOL) compete for one of the South American slots in the final phase of the FIFA World Cup. The qualifying competition consists of a double round robin tournament. The matches are scheduled in 9 closely spaced pairs known as double rounds. Every team plays twice in each double round. The tournament is spread over 2 years, so the double rounds are months apart. After using the same mirrored schedule for about twenty years, and persistent complaints from its members, CONMEBOL decided to change the schedule for the 2018 World Cup. Supported by one of CONMEBOL’s members, we used integer programmming to construct schedules that overcome the main drawbacks of the previous approach. After exploring many design criteria, we proposed a candidate schedule based on a French scheme. The main feature of the proposed schedule is that every team plays once at home and once away on each double round, a departure from traditional symmetric (mirrored) schemes. This proposal was unanimously approved by CONMEBOL members and is currently being used in the qualifier tournament for the 2018 FIFA World Cup in Russia.