Configuring a supply network in the presence of volume discounts

This paper considers a configuration problem for a manufacturer’s supply network in the presence of volume discounts. An integrated mixed-integer nonlinear programming (MINLP) model is developed to simultaneously determine the selection of suppliers and production levels under an uncertain demand environment. The suppliers provide discounts in terms of the total value of purchased items instead of the quantity of items. The objective of the model is to maximize the manufacturer’s expected profit, subject to both manufacturer and supplier capacities. An external function is developed to deal with integral parts of the model and is integrated with the general algebraic modeling system (GAMS) and its solvers to solve the complex problem. Two numerical examples are used to illustrate the effectiveness of the proposed model and solution approach. Management insights from the sensitivity analysis and comparisons of volume discount are also discussed.     

Scatter search algorithm for supplier selection and order lot sizing under multiple price discount environment

Supplier selection is one the most important elements of supply chain management. This function involves evaluation of many factors such as, cost of parts/materials, size of order, quality, and delivery performance. Therefore, this problem is categorized as multi-criteria decision making problems. Different approaches have been applied in order to assess and select the suppliers when suppliers offer discount on the unit price. In practical conditions, buyers may face a situation where different types of discount may be offered by candidate suppliers. None of the previous studies have considered different discount schemes simultaneously. In this article a mathematical model is introduced which consider different types of discount (all-unit cost, incremental discount, and total business volume discount) through multi-objective formulation for single item purchasing problem. In addition, constraints such as suppliers’ capacity and demand are taken into consideration in the model. Due to the complexity of the problem a proposed scatter search algorithm (SSA) is presented to solve this problem. Finally several sample problems have been solved by the proposed SSA and the exact (branch and bound) method. The results illustrate slight relative errors to compare with reasonable saving in computational times.     

Buyer–vendor coordination for fixed lifetime product with quantity discount under finite production rate

Buyer–vendor coordination has been widely addressed; however, the fixed lifetime of the product is seldom considered. In this paper, we study the coordination of an integrated production-inventory system with quantity discount for a fixed lifetime product under finite production rate and deterministic demand. We first derive the buyer’s ordering policy and the vendor’s production batch size in decentralised and centralised systems. We then compare the two systems and show the non-coordination of the ordering policies and the production batch sizes. To improve the supply chain efficiency, we propose quantity discount contract and prove that the contract can coordinate the buyer–vendor supply chain. Finally, we present analytically tractable solutions and give a numerical example to illustrate the benefits of the proposed quantity discount strategy.     

A mixed-integer non-linear program to model dynamic supplier selection problem

In a highly competitive scenario, suppliers play a vital role in making a business organization successful. Business of any organization is continuous process and therefore the supplier selection is also dynamic in nature. This is quite natural as the organization’s demand; supplier’s capacity, quality level, lead time, unit part cost and fixed transportation cost of supplier varies with time. Therefore, supplier identified for one period may not necessarily be same for the next period to supply the same set of parts. Hence, the supplier selection problem is highly dynamic in real practice. In this paper, a mixed-integer non-linear program (MINLP) is developed to address the dynamic supplier selection problem (DSSP). To validate the proposed MINLP data are generated randomly. A numerical illustration is also provided to demonstrate the proposed MINLP using LINGO.     

供应商驱动下异质零售商的量折扣协调机制

为解决多个异质零售商的订货策略及系统利润分配问题,提出了一种市场年总需求有限情形下的量折扣协调方法。理论分析表明,量折扣策略能保证零售商积极响应合作决策下的最优订货量,并有效提高渠道效益至一体化时的最优水平,供应链成员收益达到pareto改善,实现了供应链系统的完美协调。通过数值算例对上述结论作了进一步释义与分析。     

A multi-objective quantity discount and joint optimization model for coordination of a single-buyer multi-vendor supply chain

Supply chain management is concerned with the coordination of different parts of the production system. Companies have realized that they must closely collaborate with the suppliers of their strategic components or products. Recently, developing integrated inventory models for the supplier selection problem has attracted a significant amount of attention amongst researchers. In these models some incentives are required from the vendors to motivate the buyer to change his (her) policies to the policy which is optimal for the entire system. Quantity discount policies are used as common incentives in the literature. However, the literature on this problem does not incorporate quantity discount into the coordination model. This paper develops a multi-objective mixed integer nonlinear programming model to coordinate the system of a single buyer and multiple vendors under an all-unit quantity discount policy for the vendors. Due to the complexity of the problem two well known meta-heuristic algorithms are proposed to solve the problem. An illustrative example is given to show the behavior of the model. Results obtained from solving the sample problems show good performance of the proposed algorithms in finding the optimal solutions.     

A stochastic production planning problem with nonlinear cost

Most production planning models are deterministic and often assume a linear relation between production volume and production cost. In this paper, we investigate a production planning problem in a steel production process considering the energy consumption cost which is a nonlinear function of the production quantity. Due to the uncertain environment, the production demands are stochastic. Taking a scenario-based approach to express the stochastic demands according to the knowledge of planners on the demand distributions, we formulate the stochastic production planning problem as a mixed integer nonlinear programming (MINLP) model.Approximated with the piecewise linear functions, the MINLP model is transformed into a mixed integer linear programming model. The approximation error can be improved by adjusting the linearization ranges repeatedly. Based on the piecewise linearization, a stepwise Lagrangian relaxation (SLR) heuristic for the problem is proposed where variable splitting is introduced during Lagrangian relaxation (LR). After decomposition, one subproblem is solved by linear programming and the other is solved by an effective polynomial time algorithm. The SLR heuristic is tested on a large set of problem instances and the results show that the algorithm generates solutions very close to optimums in an acceptable time. The impact of demand uncertainty on the solution is studied by a computational discussion on scenario generation.     

Dynamic supplier selection model under two-echelon supply network

With the rise in competition levels and rapid changes in customer preferences, companies feel the pressure to create an efficient and effective base of suppliers in order to achieve the competitive advantage for them. The selection parameters of suppliers do not remain constant with respect to time and moreover; with highly fluctuating market demand, the suppliers are also expected to respond to it dynamically. This paper addresses a specific dynamic supplier selection problem (DSSP) under a two-echelon supply network (TESN) for the decision maker to allocate optimum order quantities to different levels of suppliers. The problem here considers a TESN with an integrated approach where the original equipment manufacturer (OEM) selects the first-tier suppliers and in turn with their opinion decides for the second-tier suppliers. Second-tier suppliers supply raw materials/parts/components to the first-tier suppliers, and then the first-tier suppliers supply the fabricated semi-finished product to OEM. In order to solve such a kind of problem, a mixed-integer non-linear programming (MINLP) is proposed to minimize the Total Cost (TC) of procurement for satisfying the OEM's demand. The problem incorporates parameters relevant to supplier's capacity, lead time, quality level of products, and transportation costs as a function of lead time. The model is validated through two cases with randomly generated data, and sensitivity analysis is conducted through Taguchi method using LINGO 15. This method not only helps to check the robustness of the parameters involved but also to set their optimum level. The analysis shows a significant reduction in the TC of procurement and the effect of each parameter on the TC are finally identified. The methodology adopted here can be extended to other organizations.     

Integrating the logistics network design with order quantity determination under uncertain customer demands

This paper aims to design an optimal logistics network including suppliers and retailers by taking into account the order quantity of products under uncertain consumer demand pattern. This research proposes a mixed-integer bi-level programming model and employs the iterative-optimization method. In the bi-level programming, the upper model is the logistics network design (LND) problem, which is designed for suppliers and consists of the hub locations, wholesale price of the products as well as the transportation flow of the commodity. The lower model is the order quantity determination (OQD) problem for retailers. It processes a special case of inventory problem in which the customer demand is stochastic and follows a series of assumed probability distributions. By applying the proposed methodology in a computational experiment, this research shows that if there were a large number of suppliers in the logistics system, retailers could order the product with relatively low price and the largest profit belongs to the retailer who could sell the commodity at the highest price.     

Supplier selection model with contingency planning for supplier failures

We consider the optimal allocation of demand across a set of suppliers given the risk of supplier failures. We assume items sourced are used in multiple facilities and can be purchased from multiple suppliers with different cost and reliability characteristics. Suppliers have production flexibility that allows them to deliver a contingency quantity in case other suppliers fail. Costs considered include supplier fixed costs and variable costs per unit, while failure to deliver to a demand point results in a particular financial loss. The model utilizes the decision tree approach to consider all the possible states of nature when one or more suppliers fail, as well as expand the traditional transportation problem. Unlike other supplier selection models, this model considers contingency planning in the decision process, minimizing the total network costs. This results in a base allocation to one or more of the available suppliers and a state of nature specific delivery contingency plan from the suppliers to each demand point. A numerical example, as well as sensitivity analysis, is presented to illustrate the model and provide insights.     

Sourcing with random yields and stochastic demand: A newsvendor approach

We studied a supplier selection problem, where a buyer, while facing random demand, is to decide ordering quantities from a set of suppliers with different yields and prices. We provided the mathematical formulation for the buyer's profit maximization problem and proposed a solution method based on a combination of the active set method and the Newton search procedure. Our computational study shows that the proposed method can solve the problem efficiently, and is able to generate interesting and insightful results that lead us to various managerial implications.

Scope and purpose

In today's globally competitive environment, decision makers in supply chains face numerous challenges particularly regarding the selection of suppliers or outsourcing partners. To assist in this endeavor, we examined a double-layered supply chain where a buyer facing the end users has the option of selecting among a cohort of suppliers. The available suppliers may have different yield rates and unit costs. The buyer has to decide, given the stochastic nature of the problem's governing parameters, whether or not to order from each supplier, and if so how much. We developed a ‘newsvendor-style’ model for the problem, and proposed a solution algorithm for it. Numerical studies were performed to provide some insights for supplier selection and order quantity decisions.     

Supplier selection and procurement decisions with uncertain demand,fixed selection costs and quantity discounts

In this paper, we study the supplier selection and procurement decision problem with uncertain demand, quantity discounts and fixed selection costs. In addition, a holding cost is incurred for the excess inventory if the buyer orders more than the realized demand and the shortage must be satisfied by an emergent purchase at a higher price otherwise. The objective is to select the suppliers and to allocate the ordering quantity among them to minimize the total cost (including selecting, procurement, holding and shortage costs, etc.). The problem is modeled as a Mixed Integer Programming (MIP) and is shown to be NP-hard. Some properties of the optimal policy are provided and an optimal algorithm is proposed based on the generalized Bender's decomposition. Numerical experiments are conducted to show the efficiency of the algorithm and to obtain some managerial insights.     

基于需求偏差的供应链协调问题

分析了单供应商和单零售商组成的供应链在需求预测偏差下的协调问题.当市场规模和价格敏感系数同时发生变化时,为使供应链收益最大,提出了调整生产计划和零售价格的协调机制.进一步证明了利用数量折扣机制可协调需求偏差下的分权供应链,而且该机制还实现了供应链收益在供应商和零售商间的任意分配.最后进行了实例分析.     

Inventory control model for a supply chain system with multiple types of items and minimum order size requirements

This paper considers a replenishment problem for a single buyer who orders multiple types of items from two or more heterogeneous suppliers in order to sell to end customers. The buyer periodically orders each type of item from the suppliers according to a select inventory control policy. Processing the order, each supplier enforces the policy that an order from the buyer must meet a predetermined minimum order quantity (MOQ). Therefore, the buyer must decide how much to order from each supplier considering the current inventory level, demand forecast, and MOQ requirement. The buyer's problem is formulated as an integer programming model and an efficient implementation strategy is suggested to apply the model to real problems. Numerical experiments are performed to test the validity of the proposed model as well as the efficiency of the implementation strategy. The experimental results show that this model combined with the implementation method yields a considerable cost reduction compared to the most efficient policy currently available.     

Improved optimization methods for refinery hydrogen network and their applications

Heavier market competition and tighter environmental legislation lead to the increasing demand for hydrogen in refinery. Hence it is necessary for refinery to seek effective hydrogen management strategies to satisfy the increasing hydrogen requirements. In this paper, two improved systematic mathematical methods are developed based on two-step approach and simultaneous optimization approach respectively to retrofit the hydrogen network in refinery. To make the proposed approaches more suitable for real systems, the flowrate and purity at the reactor inlet of hydrogen consumers and the hydrogen recovery of purification units are considered as variables, and the minimum pure hydrogen of hydrogen consumers must be satisfied. Then the corresponding optimization problem is mathematically transformed to a mixed-integer nonlinear programming (MINLP) problem. However, solving a MINLP model directly will result in inconsistency in solution quality and time. In this paper, the solving of the complex MINLP formulation is avoided by using a mixed-integer linear programming (MILP) linearization technique, resulting in better quality, stability, and efficiency than solving the MINLP model directly. The proposed approaches in this paper could make the best use of resources and consequently provide significant environmental and economic benefits for refinery. A real case study is presented to illustrate the applicability of the proposed approaches.     

零售商多种价格策略下的最优订购模型

同种产品以多种价格同时销售(多种价格策略)是零售商常用的折扣策略,为了找到多种价格策略下零售商实现期望利润最大化的订购量,对经典报童模型进行拓展,分别从无订购量约束和订购量约束两方面进行研究和讨论.在多种价格对应消费者需求不确定情况下建立一个新的报童模型,使用拉格朗日乘子法求解订购量约束问题,并设计近似最优总订购量的求解算法.数值分析结果表明:多种价格策略优于单一价格策略,订购量约束对零售商多种价格策略的选择会产生影响;在有订购量约束情况下,零售商的多种价格策略还会受到价格折扣系数、需求差异性的影响.     

基于异步时间段的原油混输调度连续时间建模研究

采用严格的数学规划方法对沿海炼厂的原油混输调度问题进行了建模和求解,提出了一个基于异步时间段的连续时间混合整数非线性规划(MINLP)数学模型。该模型具有整型变量少,求解快的优点。采用求解一系列MILP来近似求解非凸的MINLP,避免了原油混输MILP模型产生的浓度不一致问题。文中采用提出的新模型和算法对文献中的9个实例进行了计算,相同计算条件的4个实例比离散时间模型效率提高86％-1011％。     

Sole versus dual sourcing under order dependent lead times and prices

In this paper, we consider a dual-sourcing model with constant demand and stochastic lead times. Two suppliers may be different in terms of purchasing prices and lead-time parameters. The ordering takes place when the inventory level depletes to a reorder level, and the order is split among two suppliers. Unlike previous works in the order splitting literature, the supply lead time between vendor and buyer as well as unit purchasing prices is considered to be order quantity dependent. The proposed model finds out the optimal reorder point, order quantity and splitting proportion, using a solution procedure. Numerical results show that neglecting the relationship between ordering batch size and lead times is a shortcoming that hides one of order splitting advantages. Moreover, connecting unit prices to order quantity can decrease the percentage saving from dual sourcing compared to sole sourcing. Furthermore, sensitivity analysis shows some managerial insights.     

MINLP‐based hybrid strategy for operating mode selection of TES‐backed‐up refrigeration systems

This brief deals with the satisfaction of the daily cooling demand by a hybrid system that consists of a vapor‐compression refrigeration cycle and a thermal energy storage (TES) unit, based on phase change materials. The addition of the TES tank to the original refrigeration plant allows to schedule the cooling production regardless of the instantaneous demand, given that the TES tank can store cold energy and release it whenever deemed appropriate. The scheduling problem is posed as an optimization problem based on mixed‐integer nonlinear programming (MINLP) since it includes both discrete and continuous variables. The latter corresponds to the references on the main cooling powers involved in the problem (cooling production at the evaporator and TES charging/discharging), whereas the discrete variables define the operating mode scheduling. Therefore, in addition to the hybrid features of the physical plant, a hybrid optimal control strategy is also proposed. A receding horizon approach is applied, similar to model predictive control (MPC) strategies, while economic criteria are imposed in the objective function, as well as feasibility issues. The TES state estimation is also addressed since its instantaneous charge ratio is not measurable. The proposed strategy is applied in simulation to a challenging cooling demand profile, and the main advantages of the MINLP‐based strategy over a nonlinear MPC‐based scheduling strategy previously developed are highlighted, regarding operating cost, ease of tuning, and ability to adapt to cooling demand variations.     

Modeling and analysis of single item multi-period procurement lot-sizing problem considering rejections and late deliveries

Integer linear programming approach has been used to solve a multi-period procurement lot-sizing problem for a single product that is procured from a single supplier considering rejections and late deliveries under all-unit quantity discount environment. The intent of proposed model is two fold. First, we aim to establish tradeoffs among cost objectives and determine appropriate lot-size and its timing to minimize total cost over the decision horizon considering quantity discount, economies of scale in transactions and inventory management. Second, the optimization model has been used to analyze the effect of variations in problem parameters such as rejection rate, demand, storage capacity and inventory holding cost for a multi-period procurement lot-sizing problem. This analysis helps the decision maker to figure out opportunities to significantly reduce cost. An illustration is included to demonstrate the effectiveness of the proposed model. The proposed approach provides flexibility to decision maker in multi-period procurement lot-sizing decisions through tradeoff curves and sensitivity analysis.