The stochastic lot-sizing problem with quantity discounts

This paper addresses the stochastic lot-sizing problem with quantity discounts. In particular, we examine the uncapacitated finite-period economic lot-sizing problem in which the parameters in each period are random and discrete. When an order is placed, a fixed cost is incurred and an all-unit quantity discount is awarded based on the quantity ordered. The lead time is zero and the order is delivered immediately. First we study the case with overstocks by which the excess inventory incurs a holding cost. The objective in this case is to minimize the expected total cost including ordering and holding costs. The stochastic dynamics is modeled with a scenario tree. We characterize properties of the optimal policy and propose a polynomial time algorithm with complexity O ( n 3 ) for single discount level, where n is the number of nodes in the scenario tree. We extend the results to cases allowing stockout and multi-discount levels. Numerical experiments are conducted to evaluate the performance of the algorithm and to gain the man- agement insights

Modeling Industrial Lot Sizing Problems: A Review

In this paper we give an overview of recent developments in the field of modeling single-level dynamic lot sizing problems. The focus of this paper is on the modeling various industrial extensions and not on the solution approaches. The timeliness of such a review stems from the growing industry need to solve more realistic and comprehensive production planning problems. First, several different basic lot sizing problems are defined. Many extensions of these problems have been proposed and the research basically expands in two opposite directions. The first line of research focuses on modeling the operational aspects in more detail. The discussion is organized around five aspects: the set ups, the characteristics of the production process, the inventory, demand side and rolling horizon. The second direction is towards more tactical and strategic models in which the lot sizing problem is a core substructure, such as integrated production-distribution planning or supplier selection. Recent advances in both directions are discussed. Finally, we give some concluding remarks and point out interesting areas for future research

Dynamic Programming and Heuristic for Stochastic Uncapacitated Lot-Sizing Problems with Incremental Quantity Discount

The stochastic uncapacitated lot-sizing problems with incremental quantity discount have been studied in this paper. First, a multistage stochastic mixed integer model is established by the scenario analysis approach and an equivalent reformulation is obtained through proper relaxation under the decreasing unit order price assumption. The proposed reformulation allows us to extend the production-path property to this framework, and furthermore we provide a more accurate characterization of the optimal solution. Then, a backward dynamic programming algorithm is developed to obtain the optimal solution and considering its exponential computation complexity in term of time stages, we design a new rolling horizon heuristic based on the proposed property. Comparisons with the commercial solver CPLEX and other heuristics indicate better performance of our proposed algorithms in both quality of solution and run time

Dynamic lot sizing with multiple suppliers, backlogging and quantity discounts

This paper studies the dynamic lot sizing problem with supplier selection, backlogging and quantity discounts. Two known discount types are considered separately, incremental and all-units quantity discounts. Mixed integer linear programming (MILP) formulations are presented for each case and solved using a commercial optimization software. In order to timely solve the problem, a recursive formulation and its efficient implementation are introduced for each case which result in an optimal and a near optimal solution for incremental and all-units quantity discount cases, respectively. Finally, the execution times of the MILP models and forward dynamic programming models obtained from the recursive formulations are presented and compared. The results demonstrate the efficiency of the dynamic programming models, as they can solve even large-sized instances quite timely. © 201

Single Item Supplier Selection and Order Allocation Problem with a Quantity Discount and Transportation Costs

In this paper, we address a single item supplier selection, economic lot-sizing, and order assignment problem under quantity discount environment and transportation costs. A mixed-integer nonlinear program (MINP) model is developed with minimization of cost as its objective, while lead-time, the capacity of the supplier and demand of the product are incorporated as constraints. The total cost considered includes annual inventory holding cost, ordering cost, transportation cost and purchase cost. An efficient and effective genetic algorithm (GA) with problem-specific operators is developed and used to solve the proposed MINP model.  The  model is illustrated through a numerical example and the results show that the GA can solve the model in less than a minute. Moreover, the results of the numerical illustration show that the item cost and transportation cost are the deciding factors in selecting suppliers and allocating orders. Keywords: Supplier selection, Economic Order Quantity, Order allocation, Mixed-integer nonlinear programming

Telecommunication carrier selection under volume discounts: a case study.

During 2001 many of the European mobile phone markets have reached saturation, and hence mobile phone operators have shifted their attention from growth and market share to cutting costs. One way of doing so is to reduce spending on international calls which are routed via network operating companies (carriers). These carriers charge per call-minute for each destination and may use a joint business volume discount to price their services. We developed a software system that supports the decision of allocating destinations to carriers. The core of this system is a min-cost flow routine that is embedded in a branch-and-bound framework. Our system not only solves the operational problem to optimality, it is also capable of performing what-if analyses and sensitivity analysis. It has been implemented at a major telecommunication services provider. The main benefits realized are twofold: the business process of allocating carriers to destinations has been structured and the costs arising from routing international calls have significantly decreased.Selection; Case studies; Studies; Markets; Market; Costs; International; Companies; Software; Decision; Framework; Sensitivity; Processes;

Truckload Shipment Planning and Procurement

This dissertation presents three issues encountered by a shipper in the context of truckload transportation. In all of the studies, we utilize optimization techniques to model and solve the problems. Each study is inspired from the real world and much of the data used in the experiments is real data or representative of real data. The first topic is about the freight consolidation in truckload transportation. We integrate it with a purchase incentive program to increase truckload utilization and maximize profit. The second topic is about supporting decision making collaboration among departments of a manufacturer. It is a bi-objective optimization model. The third topic is about procurement in an adverse market. We study a modification of the existing procurement process to consider the market stochastic into marking decisions. In all three studies, our target is to develop effectively methodologies to seek optimal answers within a reasonable amount of time

Selecting cost-minimal delivery profiles and assessing the impact on cost and delivery schedule stability in area forwarding inbound logistics networks in the automotive industry

Automobilhersteller in Europa nutzen Gebietsspeditionsnetzwerke, um Frachtkosten im Stückgutverkehr einzusparen. Die Materialflüsse im Logistiknetzwerk werden durch vom Automobilhersteller erzeuge Lieferabrufe gesteuert. Verschiedene Ansätze zur Erzeugung von Lieferabrufen können eingesetzt werden, um die Ziele Kostenreduktion und Lieferabrufstabilität auszubalancieren. Ein vielversprechender Ansatz zur Optimierung der Lieferabrufe, der in der Literatur diskutiert und in der Praxis angewandt wird, sind Anlieferprofile. Geschickt ausgewählte Anlieferprofile können sowohl die Logistikkosten reduzieren als auch die Stabilität der Lieferabrufe erhöhen. In dieser Arbeit wird eine Methode zur Auswahl kostenoptimaler Anlieferprofile zur Steuerung von Gebietsspeditionsnetzwerken in der Automobilindustrie vorgestellt und hinsichtlich des Einflusses auf die Logistikkosten und die Stabilität der Lieferabrufe bei einem Einsatz im rollierenden Planungshorizont im Rahmen einer Fallstudie untersucht. Die wesentlichen Aspekte der Problemstellung werden im Kontext der Planungsprozesse in der Automobilindustrie durchleuchtet, wobei ein besonderer Fokus auf der operativen Bestellmengenplanung liegt. Ein auf einem Dekompositionsverfahren basierender Lösungsalgorithmus für das Planungsproblem wird vorgestellt. Neben einem gemischt-ganzzahligen Modell werden zwei heuristische Lösungsansätze für das Problem vorgestellt. Eine Erweiterung von Modell und Lösungsalgorithmen für den Einsatz in einem zweistufigen stochastischen Verfahren werden präsentiert. Eine Fallstudie mit praxisnahen Daten wird genutzt, um den Einfluss auf die Logistikkosten und die Stabilität der Lieferabrufe darzustellen und einen Vergleich zu neuesten algorithmischen Ansätzen zu ziehen.Automotive manufacturers in Europe use area forwarding based inbound logistic networks to obtain cost advantages in the inbound logistics section. Delivery schedules that are frequently generated by the automotive manufacturers are used to control the material flow in the area forwarding networks. Different delivery schedule generation approaches can be used to balance between the objectives of cost reduction and delivery schedule stability. A promising approach discussed in literature and successfully applied in retailer business are delivery profiles. When chosen wisely, this control rule is said to reduce both logistic cost and schedule instability. In this thesis, a method to select cost-minimal delivery profiles under the consideration of area forwarding networks in the automotive industry is presented and its impact on both cost and delivery schedule stability in a rolling horizon environment is assessed in a case study. To identify the aspects of the problem setting that have to be considered, a description of the planning processes in the automotive industry and the operational order lot sizing in particular is given. An appropriate solution algorithm which uses a decomposition technique to overcome runtime issues is developed. A mixed integer formulation and heuristic algorithms, a sequential algorithm and a genetic algorithm that can be used in the solution algorithm are presented. The model and the solution algorithms are then extended to a two-stage stochastic program in order to consider demand uncertainties in the solution process. A large scale industry case study is then used to assess the impact on both cost and delivery schedules. A comparison with state-of-the-art algorithmic delivery schedule generation approaches is conducted to enlighten the pros and cons of both approaches.Tag der Verteidigung: 13.06.2013Paderborn, Univ., Diss., 201

Lot-Sizing Problem for a Multi-Item Multi-level Capacitated Batch Production System with Setup Carryover, Emission Control and Backlogging using a Dynamic Program and Decomposition Heuristic

Wagner and Whitin (1958) develop an algorithm to solve the dynamic Economic Lot-Sizing Problem (ELSP), which is widely applied in inventory control, production planning, and capacity planning. The original algorithm runs in O(T^2) time, where T is the number of periods of the problem instance. Afterward few linear-time algorithms have been developed to solve the Wagner-Whitin (WW) lot-sizing problem; examples include the ELSP and equivalent Single Machine Batch-Sizing Problem (SMBSP). This dissertation revisits the algorithms for ELSPs and SMBSPs under WW cost structure, presents a new efficient linear-time algorithm, and compares the developed algorithm against comparable ones in the literature. The developed algorithm employs both lists and stacks data structure, which is completely a different approach than the rest of the algorithms for ELSPs and SMBSPs. Analysis of the developed algorithm shows that it executes fewer number of basic actions throughout the algorithm and hence it improves the CPU time by a maximum of 51.40% for ELSPs and 29.03% for SMBSPs. It can be concluded that the new algorithm is faster than existing algorithms for both ELSPs and SMBSPs. Lot-sizing decisions are crucial because these decisions help the manufacturer determine the quantity and time to produce an item with a minimum cost. The efficiency and productivity of a system is completely dependent upon the right choice of lot-sizes. Therefore, developing and improving solution procedures for lot-sizing problems is key. This dissertation addresses the classical Multi-Level Capacitated Lot-Sizing Problem (MLCLSP) and an extension of the MLCLSP with a Setup Carryover, Backlogging and Emission control. An item Dantzig Wolfe (DW) decomposition technique with an embedded Column Generation (CG) procedure is used to solve the problem. The original problem is decomposed into a master problem and a number of subproblems, which are solved using dynamic programming approach. Since the subproblems are solved independently, the solution of the subproblems often becomes infeasible for the master problem. A multi-step iterative Capacity Allocation (CA) heuristic is used to tackle this infeasibility. A Linear Programming (LP) based improvement procedure is used to refine the solutions obtained from the heuristic method. A comparative study of the proposed heuristic for the first problem (MLCLSP) is conducted and the results demonstrate that the proposed heuristic provide less optimality gap in comparison with that obtained in the literature. The Setup Carryover Assignment Problem (SCAP), which consists of determining the setup carryover plan of multiple items for a given lot-size over a finite planning horizon is modelled as a problem of finding Maximum Weighted Independent Set (MWIS) in a chain of cliques. The SCAP is formulated using a clique constraint and it is proved that the incidence matrix of the SCAP has totally unimodular structure and the LP relaxation of the proposed SCAP formulation always provides integer optimum solution. Moreover, an alternative proof that the relaxed ILP guarantees integer solution is presented in this dissertation. Thus, the SCAP and the special case of the MWIS in a chain of cliques are solvable in polynomial time

Comprehensive quantity discount model for dynamic green supplier selection and order allocation

We model and solve a deterministic multi-period single-product green supplier selection and order allocation problem in which the considered suppliers’ availability, cost, and green performance change from one period to another in the planning horizon. Moreover, the available suppliers may offer an all-unit or an incremental quantity discount (QD) scheme, resulting in three problem configurations. In one configuration, all suppliers offer all-unit QD. In the second, all suppliers offer incremental QD. In the third, some suppliers offer all-unit QD, and others offer incremental QD. The problem is modelled using a bi-objective integer linear programming formulation that maximizes the total green value of the purchased items from all the suppliers and minimizes their total corresponding cost, including the fixed cost, variable cost, inventory holding cost, and shortage cost. The proposed bi-objective model is scalarized and solved using the branch-and-cut algorithm and a population-based heuristic. A numerical analysis is conducted, which allows first to validate the heuristic approach using small-size instances by comparing its results with those of the exact approach. Moreover, an extensive comparison between the exact and heuristic solution approaches is carried out. The results reveal different findings. First, the economic and environmental solutions of an instance are different, and the environmental solution is independent of the suppliers’ pricing schemes. Second, the maximum difference between the heuristic approach and the exact approach in terms of the bi-objective function value is 4.72%, which makes the proposed heuristic recommended for large-size instances due to its short computation time and good accuracy. Third, there is no difference in terms of the heuristic performance between the combined model and the models with a single type of discount. Fourth, the all-unit discount scheme seems to be generally better in terms of the trade-off between the green value of purchasing and cost