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Mid-term planning optimization model with sales contracts under demand uncertainty
Abstract
Uncertainty modeling is a challenging topic in supply chain and operation management. When planning material purchase and stock levels, demand uncertainty could have an important impact on the plan results and its feasibility. Additionally, uncertainty could greatly affect customer satisfaction, inventory costs and company profits. From a modeling perspective, problems considering uncertainty are difficult to tackle and lead to complex optimization approaches. This work proposes a mid-term planning model dealing with sales contracts to diminish the effect of uncertainty. Another interesting feature is given by the selection of different price levels. Price elasticity functions are introduced for each customer in order to jointly decide demand targets and prices. A linear generalized disjunctive programming model is developed. Short execution time shows that this model can be applied to analyze several real scenarios to decide material purchase plan, inventory levels, sales strategies, prices and demand levels in a medium term horizon planning.
... Here, quantity-based contracts were modelled using logical and generalised disjunctive programming, and a decision tree approach was used to model the uncertain amount supplied by the suppliers. Another research paper was published on contracts optimisation by Rodríguez and Vecchietti (2012). They proposed a mid-term planning optimisation model with sales contracts. ...
Selection of the best supply chain contract plays a significant role as a strategic feature and factor of better coordination of all stages in a supply chain. As a variety of uncontrollable and unpredictable factors affect the evaluation and decision-making process at different levels, identification of the most appropriate contract is usually very complex and unstructured. In this paper, the Delphi-based analytic hierarchy process (AHP) approach has been proposed to tackle the problem. The sole purpose of the paper is to analyze the existing contracts of the organization and recommend the best one. Necessary data of powdered milk item were collected from a renowned super shop in Bangladesh, namely, Swapno superstore, Dhaka. In order to demonstrate the applicability of the proposed approach, an illustrative example is presented; the result is analyzed at the end of this paper. This work can be a guide for Bangladeshi planners as well as other researchers to identify the most suitable agreement for their supply chain.
... GDP has been developed for modeling various applications of process systems ranging from supply chain (Rodriguez and Vecchietti, 2012), passing by plant design (Moreno et al., 2009;García-Ayala et al., 2012;Caballero et al., 2014), to scheduling problems (Knudsen and Foss, 2013;Castro and Marques, 2015). GDP models are formulated through continuous and Boolean variables, algebraic equations, disjunctions and logical clauses. ...
A Generalized Disjunctive Programming (GDP) model for the optimal multi-period production planning and stock management is proposed in this work. The formulation is applied to a polyurethane foam manufacturing plant that comprises three stages: a first step that produces pieces with certain characteristics, a second process that involves the location of these pieces in a limited area and a third stage where pieces are stored in dedicated spaces. This article shows the GDP capabilities to provide a qualitative framework for representing the problem issues and their connections in a natural way, especially in a context where decisions integration is required. Due to the multi-period nature of the planning problem, a rolling horizon approach is suitable for solving it in reasonable computing time. It serves as a tool for analyzing the trade-offs among the different costs. Through the examples, the capabilities of the formulation and the proposed resolution method are highlighted.
... When the expense of satisfying a product is too high for the company, it may lose a customer. Rodríguez and Vecchietti (2012) propose a midterm planning model dealing with sales contracts to diminish the effect of uncertainty. Anupindi and Bassok (1999, Chapter 7) also present quantitative model for supply contracts under uncertainty. ...
... Quantity-based contracts were modeled with disjunctions and logic restrictions, and the uncertain amount delivered by the suppliers is modeled with a decision tree approach that represents supplier failure discrete probability distributions. Rodríguez & Vecchietti (2012) proposed a mid-term planning model with sales contracts. Piece-wise linear price-response models were used to model demand as a function of selling price. ...
In this work, we propose extending the production planning decisions of a chemical process network to include optimal contract selection under uncertainty with suppliers and product selling price optimization. We use three quantity-based contract models: discount after a certain purchased amount, bulk discount, and fixed duration contracts. We propose the use of general regression models to describe the relationship between selling price, demand, and possibly other predictors, such as economic indicators. For illustration purposes, we consider three demand-response models (i.e., selling price as a function of demand) that are typically encountered in the literature: linear, constant-elasticity, and logit. We develop a mixed-integer nonlinear two-stage stochastic programming that accounts for uncertainty in both supply (e.g., raw material spot market price) and demand (random nature of the residuals of the regression models) for the planning of the process network. The proposed method is illustrated with two numerical examples of chemical process networks.
Product design undergoes various stages and each stage requires valid inputs to obtain the desired output(s). Many times, the inputs that are provided are inaccurate or uncertain. Further, the design of a new product normally starts with a limited knowledge and not properly defined targets to convert an idea or concept into a marketable product. The complex and dynamic nature of design leads to uncertainty, which makes taking the right decision critical. Thus, it is felt that there is a serious need to handle uncertainties during product design. This work analyses the sources of uncertainty that may creep in the product design process during different design stages. Besides, it also illustrates the taxonomy of uncertainty and the tools that might be used to mitigate the uncertainty during each design stage. The paper validates the approach with the help of a design example. It is felt that the designers may benefit from this work as they can identify what type of uncertainty may come upon during a particular stage of the design process, along with the tool that can be used to handle the uncertainty in an effective manner.
actually decreased due to their discount parameters. In this paper, we study a multi-period model, with a buyer facing stochastic end-item demand and a supplier oering an all-units quantity discount to him, to understand better the dynamics of such systems. We provide guidelines and insights on how to set eective discount parameters, and when not to expect much from them. We derive the optimal policy of the buyer, develop insights as to why the policy is complex, study the supplier's profit as a function of her oered quantity discount scheme (accommodating the buyer's optimal policy), and discover a new phenomenon that is distinct and structurally dierent from the well-known bullwhip eect.
Recent increases in energy prices, the rise in natural gas demand due to the concerns over CO2 emissions and a possible carbon tax, the development of a low-cost and high-capacity liquefied natural gas (LNG) value chain, the emergence of new suppliers with large gas reserves, the flow of uncommitted LNG capacity to the market, and the disappearance of conventional clauses such as destination are stimulating global LNG trade. Moreover, natural gas liberalization is resulting in the emergence of new buyers with variable demands, which is increasing the competitiveness and dynamicity of the LNG market. The LNG contracts are thus diversifying in price formulations, flexibility, duration, quality, quantity, commitment, discount, and other terms and conditions. Finding a combination of contracts and suppliers, which trades off various cost factors in an optimal manner, is becoming more and more challenging, where systematic optimization-based approaches can be very useful. In this study, we address contract selection from the perspective of an LNG buyer company. We develop a mixed-integer linear programming formalism that helps the buyer select the best combination of suppliers and contracts in an integrated manner that addresses various aspects, such as contract timings and lengths, demands, price formulations, volume discounts, delivery terms, shipment costs, purchase commitments, etc. We minimize the sum of purchase and transport costs, and we illustrate our approach using three examples with diverse and realistic features.
This paper utilizes the framework of mid-term, multisite supply chain planning under demand uncertainty to safeguard against inventory depletion at the production sites and excessive shortage at the customer. A chance constraint programming approach in conjunction with a two-stage stochastic programming methodology is utilized for capturing the trade-off between customer demand satisfaction (CDS) and production costs. In the proposed model, the production decisions are made before demand realization while the supply chain decisions are delayed. The challenge associated with obtaining the second stage recourse function is resolved by first obtaining a closed-form solution of the inner optimization problem using linear programming duality followed by expectation evaluation by analytical integration. In addition, analytical expressions for the mean and standard deviation of the inventory are derived and used for setting the appropriate CDS levels in the supply chain. A three-site example supply chain is studied within the proposed framework for providing quantitative guidelines for setting customer satisfaction levels and uncovering effective inventory management options. Results indicate that significant improvement in guaranteed service levels can be obtained for a small increase in the total cost.
Supply processes play an important role in customer satisfaction and company costs. The main characteristics of this problem are given by several decisions that follow a hierarchical structure and a very uncertain context, conditioning the success of the solutions proposed. Two significant sources of uncertainty are considered in this work, namely, provision and demand, both modeled as exogenous variables with random behavior. An optimization model is formulated to reduce the effects of the uncertainty in the company supply process. Because of the problem complexity, a multicriteria model is required to bring a comprehensive solution. Several Pareto-optimal solutions are obtained through application of the ε-constraint technique. The original formulation is a nonconvex one that is then transformed to obtain a disjunctive linear model that guarantees a global result.
A multinational company's purchases go well beyond basic raw materials; they include catalysts, indirect materials, additives, etc. Strategic sourcing contracts offer several advantages and are common practice in many industries, especially the chemical industry. However, contracts come in various shades of price, commitments, duration, terms, flexibility, lead time, quality, discounts, product bundling, etc. Selecting the best contracts and suppliers for a company's globally distributed sites in an integrated and global business environment can be nontrivial. In this paper, we propose a relatively comprehensive classification of material supply contracts and propose a multiperiod mathematical programming model that selects optimal contracts for the minimum total procurement cost in the face of several practical considerations such as different contract types, multitier prices and discounts, logistics and inventory costs, quantity/dollar purchase commitments, spot market, product bundling, etc. The model also identifies the optimal distribution of materials from various suppliers to plant sites. Our examples demonstrate substantial savings over ad hoc or heuristic methods.
In this article, the design and retrofit problem of a supply chain (SC) consisting of several production plants, warehouses and markets, and the associated distribution systems, is considered. The first problem formulation modifies and extends other previously presented models, in order to include several essential characteristics for realistically representing the consequences of design decisions on the SC performance. Then, in order to take into account the effects of the uncertainty in the production scenario, a two-stage stochastic model is constructed. The problem objective, i.e., SC performance, is assessed by taking into account not only the profit over the time horizon, but also the resulting demand satisfaction. This approach can be used to obtain different kinds of solutions, that may be valuable at different levels. On one hand, the SC configurations obtained by means of deterministic mathematical programming can be compared with those determined by different stochastic scenarios representing different approaches to face uncertainty. Additionally, this approach enables to consider and manage the financial risk associated to the different design options, resulting in a set of Pareto optimal solutions that can be used for decision-making.
This work presents a novel approach for modeling of different types of contracts that a company may sign with its suppliers and customers. The main objective is to expand the scope of current planning and supply chain optimization models by including the selection of the types of contracts as an additional decision. The solution approach relies on representing the decision of choosing different contracts using disjunctive programming for both short-term and long-term production planning models. The resulting formulation is converted into a mixed-integer linear programming (MILP) problem. The advantages of the proposed models are highlighted in two case studies of increasing complexity.
This paper addresses the optimal design and planning of sustainable chemical supply chains (SCs) in the presence of uncertainty in the damage model used to evaluate their environmental performance. The environmental damage is assessed through the Eco-indicator 99, which includes recent advances made in life cycle assessment (LCA). The overall problem is formulated as a bi-criterion stochastic non-convex mixed-integer nonlinear program (MINLP). The deterministic equivalent of such a model is obtained by reformulating the joint chance-constraint employed to calculate the environmental performance of the SC in the space of uncertain parameters. The resulting bi-criterion non-convex MINLP is solved by applying the epsilon constraint method. To guarantee the global optimality of the Pareto solutions found, we propose a novel spatial branch and bound method that exploits the specific structure of the problem. The capabilities of our modeling framework and the performance of the proposed solution strategy are illustrated through a case study.
This paper addresses the optimization of supply chain design and planning under responsive criterion and economic criterion with the presence of demand uncertainty. The supply chain consists of multi-site processing facilities and corresponds to a multi-echelon production network with both dedicated and multiproduct plants. The economic criterion is measured in terms of net present value, while the criterion for responsiveness accounts for transportation times, residence times, cyclic schedules in multiproduct plants and inventory management. By using a probabilistic model for stock-out, the expected lead time is proposed as the quantitative measure of supply chain responsiveness. The probabilistic model can also predict the safety stock levels by integrating stock-out probability with demand uncertainty. These are all incorporated into a multi-period mixed-integer nonlinear programming (MINLP) model, which takes into account the selection of manufacturing sites and distribution centers, process technology, production levels, scheduling and inventory levels. The problem is formulated as a bi-criterion optimization model that maximizes the net present value and minimizes the expected lead time. The model is solved with the ɛ-constraint method and produces a Pareto-optimal curve that reveals how the optimal net present value, supply chain network structure and safety stock levels change with different values of the expected lead time. A hierarchical algorithm is also proposed based on the decoupling of different decision-making levels (strategic and operational) in the problem. The application of this model and the proposed algorithm are illustrated with two examples of polystyrene supply chains.
Discrete-continuous non-linear optimization models are frequently used to formulate problems in process system engineering. Major modeling alternatives and solution algorithms include generalized disjunctive programming and mixed integer non-linear programming (MINLP). Both have advantages and drawbacks depending on the problem they are dealing with. In this work, we describe the theory behind logmip, a new computer code for disjunctive programming and MINLP. We discuss a hybrid modeling framework that combines both approaches, allowing binary variables and disjunctions for expressing discrete choices. An extension of the logic-based outer approximation (OA) algorithm has been implemented to solve the proposed hybrid model. Computational experience is reported on several examples, which are solved using disjunctive, MINLP and hybrid formulations.
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