A solution procedure for mixed-integer nonlinear programming formulation of supply chain planning with quantity discounts under demand uncertainty

Quantity discount policy is decision-making for trade-off prices between suppliers and manufacturers while production is changeable due to demand fluctuations in a real market. In this paper, quantity discount models which consider selection of contract suppliers, production quantity and inventory simultaneously are addressed. The supply chain planning problem with quantity discounts under demand uncertainty is formulated as a mixed-integer nonlinear programming problem (MINLP) with integral terms. We apply an outer-approximation method to solve MINLP problems. In order to improve the efficiency of the proposed method, the problem is reformulated as a stochastic model replacing the integral terms by using a normalisation technique. We present numerical examples to demonstrate the efficiency of the proposed method.     

The joint replenishment problem involving multiple suppliers offering quantity discounts

In this article, we deal with the problem of determining the economic operating policy when a number of items are to be procured from a number of suppliers offering different quantity discounts schedules. In such inventory problems, a fixed cost is incurred with each replenishment order, independent of the suppliers as well as the items involved in the order. Further, the item involves a minor fixed cost. In such a system, it includes the supplier selection problem when considering the quantity discounts as well as the general joint replenishment problem. We develop a hybrid genetic algorithm for this NP-hard decision problem and extend it to systems with resource restrictions.     

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.     

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.     

A supplier-selection model with classification and joint replenishment of inventory items

Since inventory costs are closely related to suppliers, many models in the literature have selected the suppliers and also allocated orders, simultaneously. Such models usually consider either a single inventory item or multiple inventory items which have independent holding and ordering costs. However, in practice, ordering multiple items from the same supplier leads to a reduction in ordering costs. This paper presents a model in capacity-constrained supplier-selection and order-allocation problem, which considers the joint replenishment of inventory items with a direct grouping approach. In such supplier-selection problems, the following items are considered: a fixed major ordering cost to each supplier, which is independent from the items in the order; a minor ordering cost for each item ordered to each supplier; and the inventory holding and purchasing costs. To solve the developed NP-hard problem, a simulated annealing algorithm was proposed and then compared to a modified genetic algorithm of the literature. The numerical example represented that the number of groups and selected suppliers were reduced when the major ordering cost increased in comparison to other costs. There were also more savings when the number of groups was determined by the model in comparison to predetermined number of groups or no grouping scenarios.     

A fuzzy model for supplier selection in quantity discount environments

Traditionally, supplier selection should simultaneously take into account numerous heterogeneous criteria, and then is a tedious task for the purchasing decision makers. It becomes especially complicated when quantity discounts are considered at the same time. Under such manner, most studies often formulate such a problem as a Multi-Objective Linear Programming (MOLP) problem, and then scale it down to a Mixed Integer Programming (MIP) problem to handle the inherited multi-objectives simultaneously. However, this approach often neglects to consider scaling and subjective weighting issues. In order to ease the problem mentioned above and to obtain a more reasonable compromise solution for allocating order quantities among suppliers with their quantity discount rate offered, the Analytical Hierarchy Process (AHP) and fuzzy compromise programming are introduced in this study. An illustrated example is presented to demonstrate the proposed model and to illuminate two kinds of attitudes for decision makers. The information from the experiments can be utilized further to explain the suppliers’ possible improvement and to help create win–win policies.     

The optimization of warehouse location and resources distribution for emergency rescue under uncertainty

China is one of the countries that suffer the most natural disasters in the world. The situation of emergency response and rescue is extremely tough. Establishing the emergency warehouse is one of the important ways to cope with rapid-onset disasters. In this paper, a mixed integer programming (MIP) model based on time cost under uncertainty is proposed, which help solve the emergency warehouse location and distribution problem. Comprehensive consideration of factors such as time cost, penalty cost for lack of resources, alternative origins of resources from both suppliers and emergency warehouses, different means of transportation and multiple resources types are involved in our study. We also introduce uncertain scenarios to describe the severity of the disaster. Particle swarm optimization (PSO) and variable neighborhood search (VNS) are designed to solve the MIP model of different scales of instances. Numerous examples have been tested to compare two heuristics with commercial solver (CPLEX). Both of two algorithms can obtain the exact solution same as CPLEX in small-scale instances while show great performance on larger instances with 10 candidate warehouses, 25 disasters and 50 scenarios.     

Interval valued demand related inventory model under all units discount facility and deterioration via parametric approach

In the yard of business organization, due to a major number of competitors and uncertainty of customers’ demand, everyone faces some disturbances for smoothly running of their business. To survive in the fierce competition some modern business policies are required and accordingly the formulation of an advanced inventory model is indispensable by handling the uncertainty of customers’ demand. In this model formulation, discount facility is considered as a business policy whereas interval valued demand is proposed to tackle the uncertainty of customers’ demand. The purpose of this work is to analyze a model with interval valued parameters for deteriorating items considering two situations (with shortages and without shortage) in the discount environment. Moreover, deterioration rate is taken into account as interval valued and unit carrying cost is supposed to be a function of the length of storage time as well as interval valued purchase cost. Also, the interval valued purchase cost is assumed to be a step function (decreasing) of lot size under discount business policy. Considering without shortage and with shortages separately, using parametric approach of interval differential equations, the proposed model is formulated in two different cases. After formulating the proposed model properly, the corresponding profit functions are obtained for two different cases (without shortage and with shortages). In order to optimize the profit, various types of meta-heuristic algorithms (weighted quantum behaved particle swarm optimization and Gaussian quantum behaved particle swarm optimization) are used and then the obtained results are compared. Finally, to justify the reality of the model, numerical illustrations are presented and also post optimality analyses are performed with the changes of known parameters.

     

A hybrid intelligent approach for supply chain management system

The process of supplier selection is one of the key processes in the industry since the suppliers play a key role in regards to the competitive advantage. It is therefore important to identify and select good suppliers. Supplier selection can be considered a decision-making problem with many criteria, which include both qualitative and quantitative elements, such as; purchase cost, quality level, supplier risk etc. The best process to be designed for selecting the suppliers may require a compromise in terms of these elements. A general framework, which combines Analytic Hierarchy Process (AHP), Fuzzy AHP and Genetic Algorithm (GA) in order to determine the best set of suppliers, is proposed in this study.     

The effect of advance payment with discount facility on supply decisions of deteriorating products whose demand is both price and stock dependent

Generally, in the business world, it is observed that suppliers give different kinds of benefits to retailers due to advance payment. One of the popular benefits is instant cash discount due to advance payment. If a retailer pays off his total purchase cost before receiving the products, then he receives a certain percentage of cash discount instantly. However, if the retailer pays off a certain fraction of the total purchasing cost, then price discount is given only at the time of receiving the products while paying the remaining amount of the total purchasing cost. Using this concept, this paper formulates, under both cases of advance payment (full or partial), an inventory model for deteriorating products where shortages are allowed and demand function is considered as price and stock dependent. The closed‐form solutions for each case are presented and two numerical examples are solved. In addition, a sensitivity analysis is also performed to show the effects of advance payment with discount facility.     

A fuzzy TOPSIS based approach for e-sourcing

This study presents a methodology for solving the sealed bid, multi-attribute reverse auction problem of e-sourcing in which the sales item is defined by several attributes, the buyer is auctioneer, and the suppliers are the bidders. There is only one buyer and a number of suppliers. Both qualitative and quantitative attributes of benefit and cost types are considered for solving the winner determination (WD) problem of reverse auction. Here, the WD problem is considered as multi-criteria decision making problem (MCDM). In order to address the imprecision of suppliers or decision makers in formulating the preference value of various attributes in MCDM, a fuzzy TOPSIS based methodology along with a mechanism for determination of fuzzy linguistic value of each attribute is proposed in this article. Entropy method is utilised to enumerate the weights of various attributes automatically without involvement of decision makers. An illustrative example is presented to demonstrate the applicability of the proposed methodology.     

A genetic algorithm for a bicriteria supplier selection problem

In this paper, we discuss the problem of selecting suppliers for an organisation, where a number of suppliers have made price offers for supply of items, but have limited capacity. Selecting the cheapest combination of suppliers is a straightforward matter, but purchasers often have a dual goal of lowering the number of suppliers they deal with. This second goal makes this issue a bicriteria problem – minimisation of cost and minimisation of the number of suppliers. We present a mixed integer programming (MIP) model for this scenario. Quality and delivery performance are modelled as constraints. Smaller instances of this model may be solved using an MIP solver, but large instances will require a heuristic. We present a multi-population genetic algorithm for generating Pareto-optimal solutions of the problem. The performance of this algorithm is compared against MIP solutions and Monte Carlo solutions.     

MRP lot sizing with multiple purchase discounts

Five MRP lot sizing models are modified to accommodate multiple purchase discounts and tested in an experiment involving different levels of requirements variability, discount attractiveness, and order cost. The testing of multiple purchase discounts is an extension of earlier research that compared MRP lot sizing models under a simplified discount structure involving a single price break [3, 4, 7]. The models tested here have either been implemented in some actual MRP systems or have been proposed as effective models by other researchers. Five hundred test problems were solved by each model, generating 2500 cost figures for comparison. The results indicate that an incremental version of part-period balancing and, to a lesser extent, the least unit cost method are the best models of those tested.     

Supplier selection and supply quantity allocation of common and non-common parts with multiple criteria under multiple products

With limited capacity of suppliers, how to reduce the total operating cost of the enterprise by determining the most suitable production capacity allocation has become the major issue faced by various enterprises in producing multiple types of products. In addition, when manufacturing multiple types of products, due to the high demand of common and non-common parts, which is applicable to various products, enterprises will place special emphasis on the procurement of common and non-common parts, to select most suitable suppliers of parts with the highest quality and minimum time and costs, in order to cut down on operating costs of enterprises. This research first lists parts of various products through bill of material (BOM), and constructs an optimal mathematical model suitable for multi-phase products’ parts, in order to assess the assembling relationship of various parts; it makes use of the linkage among those to select the supplier of common and non-common parts when assessing multiple products. Then considering the limited production capacity of suppliers, it selects the best combination of suppliers of special common and non-common parts. To solve the optimal mathematical model, a genetic algorithm (GA) is proposed to find the acceptable results of the supply selection and quantity allocation problem. It then provides a benchmark for enterprise in current diversified market to purchase and assess common and non-common parts, and makes such benchmark a normal standard for selection of suppliers in the future.     

Modeling and solving a multi-objective supplier quota allocation problem considering transaction costs

Suppliers’ evaluation is a subject, which has attracted the attention of many researchers. The performance of potential suppliers is evaluated against multiple criteria rather than considering a single factor such as cost or quality. One of the major objectives of suppliers’ evaluation is to determine the optimal quota assigned to each supplier while needing to replenish an order. This problem has been studied by many researchers as a multi-objective problem. The usual objectives are minimizing the purchasing cost, rejected units, and late delivered units. However, in a few researches maximizing the evaluation scores of the selected suppliers is considered as fourth objective. In this paper, we present a model with five objectives including minimizing the transaction costs of purchasing from suppliers as well as the four addressed objectives. We convert the model to a single objective one using the well-known weighting method, solve it utilizing two meta-heuristic algorithms, and analyze the efficiency of the heuristics. The reason why we utilize the meta-heuristic algorithms is that the problem is proved to be an NP-hard one.     

A solid transportation problem for an item with fixed charge, vechicle cost and price discounted varying charge using genetic algorithm

In this paper, discount in transportation cost on the basis of transportated amount is extended to a solid transportation problem. In a transportation model, the available discount is normally offered on items/criteria, etc., in the form AUD (all unit discount) or IQD (incremental quantity discount) or combination of these two. Here transportation model is considered with fixed charges and vechicle costs where AUD, IQD or combination of AUD and IQD on the price depending upon the amount is offered and varies on the choice of origin, destination and conveyance. To solve the problem, genetic algorithm (GA) based on Roulette wheel selection, arithmetic crossover and uniform mutation has been suitably developed and applied. To illustrate the models, numerical examples have been presented. Here, different types of constraints are introduced and the corresponding results are obtained. To have better customer service, the entropy function is considered and it is displayed by a numerical example. To exhibit the efficiency of GA, another method—weighted average method for multi-objective is presented, executed on a multi-objective problem and the results of these two methods are compared.     

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.     

现货市场补充作用下基于总量折扣的运输服务采购问题研究

产品需求的不确定性使得企业经常产生临时性或突发性的物流需求,物流服务现货市场采购在应对上述物流需求时发挥了重要的作用.在物流服务采购决策过程中,长期合同采购所提供的价格折扣受采购量的影响,因此对托运企业而言,如何平衡长期合同采购以及现货市场采购,从而在满足其物流需求的基础上使得总体采购成本最低,成为一个需要解决的问题.基于托运人的角度,建立以最小化运输服务采购成本及最大化整体满意度为目标,综合考虑承托双方偏好、承运方总量折扣及承运方运力接受范围等因素的运输服务混合采购决策模型,并设计多目标粒子群算法对上述模型进行求解.通过算例分析验证所提模型及算法的有效性,算例结果表明,所建立的模型可显著降低运输服务总采购成本,并有效提升整体的满意度水平,研究成果对于降低运输服务采购成本、促进承托双方的稳定合作具有一定的理论意义.     

Effects of uncertainty on a tire closed-loop supply chain network

In a closed-loop supply chain (CLSC) network, there are both forward and reverse supply chains. In this research, a tire remanufacturing CLSC network is designed and optimized based on tire recovery options. The objective of the optimization model is to maximize the total profit. The optimization model includes multiple products, suppliers, plants, retailers, demand markets, and drop-off depots. The application of the model is discussed based on a realistic network in Toronto, Canada using map. In addition, a new decision tree-based methodology is provided to calculate the net present value of the problem in multiple periods under different sources of uncertainty such as demand and returns. Furthermore, the discount cash flow is considered in the methodology as a novel innovative approach. This methodology can be applied in comparing the profitability of different design options for CLSCs.