Total Quantity Discounts Research Articles

Value- and Ambiguity-Based Approach for Solving Intuitionistic Fuzzy Transportation Problem with Total Quantity Discounts and Incremental Quantity Discounts

The cost of goods per unit transported from the source to the destination is considered to be fixed regardless of the number of units transported. But, in reality, the cost is often not fixed. Quantity discount is often allowed for large shipments. Furthermore, the transportation cost and the price break quantities are not deterministic. In this study, we introduce the concept of Value- and Ambiguity-based approach for solving the intuitionistic fuzzy transportation problem with total quantity discounts and incremental quantity discounts. Here, the cost and quantity price breakpoints are represented by trapezoidal intuitionistic fuzzy numbers. The Values and Ambiguities are defined as the degree of acceptance and rejection for trapezoidal intuitionistic fuzzy numbers. The trapezoidal intuitionistic fuzzy transportation problem is converted to a parametric transportation problem based on their Value indices and Ambiguity indices. Then, for different Values of the parameter, the transformed problem is reduced to the linear programming problem. Then, the linear programming problem is solved by using the classical methods. The proposed method is demonstrated with a numerical example. In conclusion, the intuitionistic fuzzy transportation problem with total quantity discounts is compared with the intuitionistic fuzzy transportation problem with incremental quantity discounts.

An integrated acquisition policy for supplier selection and lot sizing considering total quantity discounts and a quality constraint

We consider a two-stage supply chain where a buyer purchases a product from multiple capacitated suppliers to satisfy a constant demand rate over a finite planning horizon. Suppliers have different perfect rates and offer total quantity discounts. The buyer selects suppliers and allocates orders to them that satisfy a minimum average quality level. A mathematical model is proposed with the objective on minimizing the total cost per time unit. The model is solved by dualizing the quality constraint. The relaxed model is solved by an efficient dynamic programming algorithm. The subgradient method is used to solve the dual problem.

New solution approaches for the capacitated supplier selection problem with total quantity discount and activation costs under demand uncertainty

We study a multi-product multi-supplier procurement problem in the Automotive sector involving both supplier selection and ordering quantity decisions, and further complicated by the presence of total quantity discounts, business activation costs, and demand uncertainty. Recent works have shown the importance of explicitly incorporate demand uncertainty in this economic setting, along with the evidence about the computational burden of solving the relative Stochastic Programming models for a sufficiently large number of scenarios. In this work, we propose different solution strategies to efficiently cope with these models by taking advantage of the particular structure of the stochastic problem. More precisely, we propose and test several variants of a Progressive Hedging based heuristic approach as well as a Benders algorithm. The results obtained on benchmark instances show how the proposed methods outperform the existing ones and the state-of-the-art solvers in terms of efficiency and solution quality. In particular, thanks to the developed Progressive Hedging, we have been able to solve for the first time problem instances with up to 20 suppliers and 30 products.

The Capacitated Supplier Selection problem with Total Quantity Discount policy and Activation Costs under uncertainty

We study the Capacitated Supplier Selection problem with Total Quantity Discount policy and Activation Costs, a procurement problem where a company needs a certain quantity of different products from a set of potential suppliers, and introduce its variant under uncertainty. In its deterministic form, the problem aims at selecting a subset of the suppliers and the relative purchasing plan satisfying the demands at minimum cost, taking into account that the suppliers offer discounts based on the total quantity of products purchased and that the activation of a business activity with a supplier has a fixed cost. However, due to the long-term nature of the problem, several parameters may be affected by uncertainty. Thus, we propose a two-stage stochastic programming formulation with recourse, highlighting the strategic and the operational decisions involved, as well as the effect of the different sources of uncertainty. In particular, we focus on the cases in which only the products price or only the products demand are stochastic. The general model and the recourse actions are adapted for these special cases, and the resulting modeling approaches are validated on a large set of instances. The experiments show the convenience of having in place models considering uncertainty explicitly with respect to using expected values for approximating it, and give rise to interesting managerial insights. Due to the computational burden of solving the resulting stochastic models (for a sufficiently large number of scenarios), we also propose a branch-and-cut solution framework based on valid inequalities and other accelerating mechanisms.

GUB Flow Cover Inequalities and an Exact Algorithm for Transportation Service Procurement Problem with Total Quantity Discounts

In transportation service procurement, total quantity discounts offered by a strategic carrier are incentives to a shipper. While the shipper reduces total cost by activating discounts, the carrier can win a valuable contract as considerable freight volume is assigned. Moreover, a strategic shipper would achieve even more cost reduction by purchasing more freight volume than demand and securing more discounts. The total quantity discounts are thus important to the transportation service procurement problem, which optimizes to select carriers and allocate them freight volume with the aim of minimizing the total cost for the shipper. The transportation service procurement problem with total quantity discounts (TP-TQD) can be formulated into a multiple choice model. We introduce a new formulation based on Dantzig-Wolfe reformulation for the problem, whose lower bound from LP relaxation is proved to be tighter than that of the multiple choice model. Two sets of generalized upper bound (GUB) flow cover valid inequalities are proposed to strengthen the models and applicable to any problems with structure consisting of the more-than-demand constraints, capacities, and GUB constraints. Effective separation methods are developed. A branch-and-price-and-cut algorithm is developed to solve the TP-TQD. Extensive experiments on a set of test instances generated from practical data are carried out to evaluate the methods. The results indicate that the new valid inequalities are effective and the exact algorithm is competent to solve problem instances of practical sizes.

Optimization models and algorithms for problems in procurement logistics

Procurement problems study the operational setting of a company that needs to collect products or raw materials from suppliers. In general, the underlying decision processes aim to elaborate a purchasing plan that completely satisfies the products demand while minimizing the procurement costs. Since procurement is a capital intensive decision that typically accounts for a large portion of a firm total cost (regardless of the type of purchased goods), it is of crucial importance to have mathematical models and efficient solution algorithms in place. The present thesis proposes the study, under a quantitative perspective, of different optimization problems arising in the procurement logistics context, namely, the Capacitated Total Quantity Discount Problem (CTQDP), the Traveling Purchaser Problem (TPP), the Restricted Traveling Purchaser Problem with Total Quantity Discount (RTPP-TQD), the Traveling Purchaser Problem under Uncertainty (TPP-U), and the Multi-Vehicle Traveling Purchaser Problem with Pairwise Incompatibility Constraints (MVTPP-PIC). All these problems share some characteristics, as the presence of multiple suppliers and multiple products, as well as the possibility of selecting a subset of suppliers at an operating level. Moreover, in order to be more realistic, each singular problem takes into account one or several complicating features as quantity discount policies, transportation costs and routing optimization, multi-vehicle optimization, data uncertainty, and restrictions on the vehicle loading (capacity constraints or incompatibilities among products). For all these problems, optimization models and solution algorithms (exact and/or heuristic) are proposed, compared, and validated through consistent computational experiments. The solution algorithms generally rely on consolidated mathematical programming techniques and on the analysis of the specific problem.

An effective matheuristic for the capacitated total quantity discount problem

New generation algorithms focus on hybrid miscellaneous of exact and heuristic methods. Combining meta-heuristics and exact methods based on mathematical models appears to be a very promising alternative in solving many combinatorial optimization problems. In this paper we introduce a matheuristic method exploiting the optimal solution of mixed integer linear subproblems to solve the complex supplier selection problem of a company that needs to select a subset of suppliers so to minimize purchasing costs while satisfying products demand. Suppliers offer products at given prices and apply discounts according to the total quantity purchased. The problem, known as Total Quantity Discount Problem (TQDP), is strongly NP-hard thus motivating the study of effective and efficient heuristic methods. The proposed solution method is strengthened by the introduction of an Integer Linear Programming (ILP) refinement approach. An extensive computational analysis on a set of benchmark instances available in the literature has shown how the method is efficient and extremely effective allowing us to improve the existing best known solution values.

An exact algorithm for the Capacitated Total Quantity Discount Problem

In this paper we analyze the procurement problem of a company that needs to purchase a number of products from a set of suppliers to satisfy demand. The suppliers offer total quantity discounts and the company aims at selecting a set of suppliers so to satisfy product demand at minimum purchasing cost. The problem, known as Total Quantity Discount Problem (TQDP), is strongly NP-hard. We study different families of valid inequalities and provide a branch-and-cut approach to solve the capacitated variant of the problem (Capacitated TQDP) where the quantity available for a product from a supplier is limited. A hybrid algorithm, called HELP (Heuristic Enhancement from LP), is used to provide an initial feasible solution to the exact approach. HELP exploits information provided by the continuous relaxation problem to construct neighborhoods optimally searched through the solution of mixed integer subproblems. A streamlined version of the proposed exact method can optimally solve in a reasonable amount of time instances with up to 100 suppliers and 500 products, and largely outperforms an existing approach available in the literature and CPLEX 12.2 that frequently runs out of memory before completing the search.

The supplier selection problem with quantity discounts and truckload shipping

To minimize procurement expenditures both purchasing and transportation costs need to be considered. We study a procurement setting in which a company needs to purchase a number of products from a set of suppliers to satisfy customer demand. The suppliers offer total quantity discounts and transportation costs are based on truckload shipping rates. The goal is to select a set of suppliers so as to satisfy product demand at minimal total costs. The resulting optimization problem is strongly NP-hard. We develop integer programming based heuristics to solve the problem. Extensive computational experiments demonstrate the efficacy of the proposed heuristics and provide insight into the impact of instance characteristics on effective procurement strategies.

Exact algorithms for procurement problems under a total quantity discount structure

In this paper, we study the procurement problem faced by a buyer who needs to purchase a variety of goods from suppliers applying a so-called total quantity discount policy. This policy implies that every supplier announces a number of volume intervals and that the volume interval in which the total amount ordered lies determines the discount. Moreover, the discounted prices apply to all goods bought from the supplier, not only to those goods exceeding the volume threshold. We refer to this cost-minimization problem as the total quantity discount (TQD) problem. We give a mathematical formulation for this problem and argue that not only it is NP-hard, but also that there exists no polynomial-time approximation algorithm with a constant ratio (unless P = NP). Apart from the basic form of the TQD problem, we describe four variants. In a first variant, the market share that one or more suppliers can obtain is constrained. Another variant allows the buyer to procure more goods than strictly needed, in order to reach a lower total cost. We also consider a setting where the buyer needs to pay a disposal cost for the extra goods bought. In a third variant, the number of winning suppliers is limited, both in general and per product. Finally, we investigate a multi-period variant, where the buyer not only needs to decide what goods to buy from what supplier, but also when to do this, while considering the inventory costs. We show that the TQD problem and its variants can be solved by solving a series of min-cost flow problems. Finally, we investigate the performance of three exact algorithms (min-cost flow based branch-and-bound, linear programming based branch-and-bound, and branch-and-cut) on randomly generated instances involving 50 suppliers and 100 goods. It turns out that even the large instances of the basic problem are solved to optimality within a limited amount of time. However, we find that different algorithms perform best in terms of computation time for different variants.

Optimal procurement decisions in the presence of total quantity discounts and alternative product recipes

We describe the purchasing decisions faced by a multi-plant company. The suppliers of this company offer complex discount schedules based on the total quantity (rather than cost) of ingredients purchased. The schedules simultaneously account both for corporate purchases and for purchases at the individual plant level. The complexity of the purchasing decisions is further increased due to the existence of alternative production recipes for each final product. We formulate the corresponding cost-minimization problem as a nonlinear mixed 0–1 programming problem. We propose various ways to linearize this formulation, and we evaluate the quality of the resulting models on real-world data.