The linear dynamic lot size problem with minimum order quantity

This paper continues the analysis of a special uncapacitated single item lot sizing problem where a minimum order quantity restriction, instead of the setup cost, guarantees a certain level of production lots. A detailed analysis of the model and an investigation of the particularities of the cumulative demand structure allow us to develop a solution algorithm based on the concept of atomic sub-problems. We present an optimal solution to an atomic sub-problem in an explicit form and prove that it serves as a construction block for the optimal solution of the original problem. Computational tests and a comparison with a published algorithm confirm the efficiency of the solution algorithm developed here.     

A local search algorithm for the optimization of the stochastic economic lot scheduling problem

This paper describes a model for the stochastic economic lot scheduling problem (SELSP) and a Local Search heuristic to find close to optimal solutions to this model. The SELSP considers multiple products, which have to be scheduled on a single facility with limited capacity and significant setup times and costs. The demand is modeled as a stationary compound renewal process. The objective is to find a schedule that minimizes the long-run average costs for setups and inventories while satisfying a given fill rate. We use a cyclic scheduling approach in which the individual cycle time of each product is a multiple of some basic period (fundamental cycle).For the deterministic version of the SELSP, efficient heuristics have been developed which guarantee the feasibility of the solution by adding an additional constraint to the problem. In our case this is not sufficient, because for the calculation of the average inventory levels and fill rates we need to develop a schedule with detailed timing of the lots. We present an efficient heuristic for this scheduling problem, which can also be used to check the feasibility of the solution. Thereby, the most time-consuming step (the calculation of average inventory levels and fill rates) is only performed for a limited set of candidates.The algorithm was tested on deterministic benchmark problems from literature and on a large set of stochastic instances. We report on the performance of the heuristic in both cases and try to identify the main factors influencing the objective.     

Optimal lot sizing in a non-cooperative material manager-controller game

It is important for manufacturing companies to optimise purchase order quantities. Inaccurate lot size planning raises costs and lowers profits, which top management of course attempts to avoid through controlling processes. The lot size decision becomes even more relevant in the case of just-in-time delivery within a supply chain.The interaction between lot sizing and auditing can be described in terms of a modified inspection game. This paper considers how probabilities, which are the basis for the mixed strategies at equilibrium in the inspection game, will change if the level of penalties accruing to the two players (material manager and controller) depends on the cost deviation caused by the material manager's poor lot-size planning. It is evident that the Nash equilibrium shifts to the strategy combination (methodically determined decision and low auditing level), if the penalties imposed on the controller and material manager increase.Penalties that depend on such deviations, and an accurate audit of the controller's report by top management, prove to be the best instruments for avoiding mismanagement by the material manager and poor controller work, both of which lead to high costs.     

Economic lot and delivery scheduling problem for multi-stage supply chains

The economic lot and delivery scheduling problem for a multi-stage supply chain comprising multiple items is studied in this paper. It is required to develop a synchronized replenishment strategy, and specify the sequence of production and the replenishment cycle time that achieves synchronization through the supply chain at minimum cost. The problem is presented in a novel formulation based on the quadratic assignment representation. The common cycle time and the integer multipliers policies are adopted to accomplish the desired synchronization. The two policies are represented by nonlinear models handled through a hybrid algorithm. The algorithm combines linearization, outer approximation and Benders decomposition techniques. Results of the two policies demonstrate that a cost reduction up to16.3% can be attained by employing the integer multipliers policy instead of the common cycle time. Computational experiments show the efficiency of the new formulation and solution algorithm by reaching the optimal solution for large problem instances in short time.     

Non-cooperative strategies for production and shipment lot sizing in one vendor-multi-buyer system

The supply chain structure examined in this paper consists of a single vendor (or manufacturer) with multiple heterogeneous buyers (or retailers). A continuous deterministic model is presented. To satisfy buyers demands, the vendor will deliver the product in JIT shipments to each buyer. The production rate is constant and sufficient to meet the buyers’ demands. The product is shipped in discrete batches from the vendor's stock to buyers’ stocks and all shipments are realized instantaneously. Special production-replenishment policies of the vendor and the buyers are analyzed. That is, the production batch is transferred to each buyer in several sub-batches in each production distribution cycle (PDC).This paper offers game model without prices, where agents minimize individual costs. It is a non-cooperative (1+N)-person game model with agents (a single vendor and N-buyers) choosing numbers and sizes of transferred batches. The model describes inventory patterns and cost structure of PDC. It is proved that there exist Nash equilibria in several types of sub-games of the considered game.     

Single-stage formulations for synchronised two-stage lot sizing and scheduling in soft drink production

This study deals with industrial processes that produce soft drink bottles in different flavours and sizes, carried out in two synchronised production stages: liquid preparation and bottling. Four single-stage formulations are proposed to solve the synchronised two-stage lot sizing and scheduling problem in soft drink production synchronising the first stage's syrup lots in tanks with the second stage's soft drink lots on bottling lines. The first two formulations are variants of the General Lot Sizing and Scheduling Problem (GLSP) with sequence-dependent setup times and costs, while the other two are based on the Asymmetric Travelling Salesman Problem (ATSP) with different subtour elimination constraints. All models are computationally tested and compared to the original two-stage formulation introduced in Ferreira et al. (2009), using data based on a real-world bottling plant. The results show not only the superiority of the single-stage models if compared to the two-stage formulation, but also the much faster solution times of the ATSP-based models.     

The deterministic EOQ and EPQ with partial backordering at a rate that is linearly dependent on the time to delivery

Our original models for the EOQ and EPQ with partial backordering assumed that the backordering rate, β, is a constant. In this paper we extend those models to allow β to increase linearly as the time until delivery decreases. We show how those previous models can be adapted to find the optimal decision variable values for this new assumption and develop, for each model type, a condition that the initial value of β must meet for partial backordering to be optimal.     

The maximum capture per unit cost location problem

This paper considers the problem of siting p new facilities of an entering firm to a competitive market so as to maximize the market share captured from competitors per unit cost. We first formulate the problem as a mixed 0-1 fractional programming model, in which we incorporate the fixed cost and transportation cost. The model can deal with the case where some demand nodes have two or more possible closest servers. We then re-formulate the problem as a 0-1 mixed integer linear program. We use a one-opt heuristic algorithm based on the Teitz-Bart method to obtain feasible solutions and compare them with the optimal solutions obtained by a branch-and-bound algorithm. We conduct computational experiments to evaluate the two algorithms. The results show that both algorithms can solve the model efficiently and the model is integer-friendly. We discuss other computational results and provide managerial insights.     

Incorporating uncertainty into a supplier selection problem

Supplier selection is an important strategic supply chain design decision. Incorporating uncertainty of demand and supplier capacity into the optimization model results in a robust selection of suppliers. A two-stage stochastic programming (SP) model and a chance-constrained programming (CCP) model are developed to determine a minimal set of suppliers and optimal order quantities with consideration of business volume discounts. Both models include several objectives and strive to balance a small number of suppliers with the risk of not being able to meet demand. The SP model is scenario-based and uses penalty coefficients whereas the CCP model assumes a probability distribution and constrains the probability of not meeting demand. Both formulations improve on a deterministic mixed integer linear program and give the decision maker a more complete picture of tradeoffs between cost, system reliability and other factors. We present Pareto-optimal solutions for a sample problem to demonstrate the benefits of the SP and CCP models. In order to describe the tradeoffs between costs and risks in an analytical form, we use multi-parametric programming techniques to more completely analyze the alternative Pareto-optimal supplier selection solutions in the CCP model. This analysis gives insights into the robustness of the solutions with respect to number of suppliers, costs and probability of not meeting demand.     

Ship routing problem with berthing time clash avoidance constraints

We consider a ship routing problem in which multiple vessels have to perform pickups and deliveries of cargoes at various locations. The loading and unloading time of cargoes at pickup and delivery locations is significant, and at each of these locations we need to assign a time slot to each vessel to perform the loading/unloading task so as to avoid time clashes. This problem is motivated by the operations of feeder vessels and company-owned cargo terminals, where the shipping company wishes to coordinate the routing and the berthing time of its vessels. We develop a heuristic algorithm for the problem using set partitioning formulation and column generation techniques. The effectiveness of the heuristic is tested via extensive computational experiments.     

Supply chain scheduling and coordination with dual delivery modes and inventory storage cost

We study a two-echelon supply chain scheduling problem in which a manufacturer acquires supplies from an upstream supplier and processes orders from the downstream retailers. The supply chain sells a single short-life product in a single season. We consider the scenario where the manufacturer can only accept some of the orders from the retailers due to its supplier's common production time window and its own two common production and delivery time windows. The upstream supplier processes materials and delivers the semi-finished products to the manufacturer within its time window. Then the manufacturer further processes these products to produce finished products and delivers them to the retailers within its two time windows, where one window is for production and normal delivery, and the other is for production and express delivery. Having to store the materials before processing them, the supplier incurs a storage cost, which depends on the order size and storage time. The manufacturer pays the transportation cost for delivering the finished products to the retailers. Due to double marginalization, the performance of the supply chain is sub-optimal. We model the supply chain problem as a flow shop scheduling problem with multiple common time windows. We derive some dominance properties and establish some theorems that help solve the sequencing problems for the orders and eliminate the idle time among the orders. Based on these results, we develop fast pseudo-polynomial dynamic algorithms to optimally solve the problem. We prove that the problem is NP-hard in the ordinary sense only. We develop two practically relevant and robust methods for the supply chain to achieve optimal profit-making performance through channel coordination.     

Vendor performance with supply risk: A chance-constrained DEA approach

The strategic importance of vendor evaluation is well established in the purchasing literature. Several evaluation methodologies that consider multiple performance attributes have been proposed for vendor evaluation purposes. While these techniques range from scoring models that utilize prior articulation of weights to derive composite scores for vendors to advanced mathematical models, methods that incorporate the inherent variability in vendor's performance attributes have been limited. The primary reason for the lack of development of such models is due to the complexities associated with stochastic approaches. In order to more accurately evaluate the performance of vendors, it is critical to consider variability in vendor attributes. This paper is an attempt to fill this void in vendor evaluation models by presenting a chance-constrained data envelopment analysis (CCDEA) approach in the presence of multiple performance measures that are uncertain. Our paper effectively demonstrates the first application of CCDEA in the area of purchasing, in general, and vendor evaluation, in particular. The model is demonstrated by applying it to a previously reported dataset from a pharmaceutical company.     

Myopic optimal policy for a multi-period, two-delivery-lead-times, stochastic inventory problem with minimum cumulative commitment and capacity

This paper studies a single-product, multi-period, stochastic inventory problem that imposes the lower and upper bounds on the cumulative order quantity during a planning horizon and allows two delivery lead times. This model includes three features. The first one is that a buyer purchases a fixed capacity from a supplier at the beginning of a planning horizon and the buyer’s total cumulative order quantity during the planning horizon is constrained with the capacity. The second one is that the buyer agrees to purchase the product at least a certain percentage of the purchased capacity during the planning horizon. The third one is that the supplier allows the buyer to order the product with two-delivery-lead-times. We identify conditions under which a myopic ordering policy is optimal. We also develop an algorithm to calculate the optimal capacity when the minimum cumulative order quantity commitment is a certain percentage of the capacity. We then use the algorithm to evaluate the effect of the various parameters on the buyer’s minimum expected total cost during the planning horizon. Our computation shows that the buyer would benefit from the commitments and two-delivery-lead-times.     

Active-set sequential quadratic programming method with compact neighbourhood algorithm for the multi-polygon mass production cutting-stock problem with rotatable polygons

The cutting-stock problem, which considers how to arrange the component profiles on the material without overlaps, can increase the utility rate of the sheet stock, and is thus a standard constrained optimisation problem. In some applications the components should be placed with specific orientations, but in others the components may be placed with any orientation. This study presents an overlap index and it is much more suitable for the active-set SQP method which can reduce the time spend for constraint consideration. Using this method, various object orientations can be considered easily and the number of object on the sheet stock can be improved by up to eight percent.     

Sheltering network planning and management with a case in the Gulf Coast region

This paper studies sheltering network planning and operations for natural disaster preparedness and responses with a two-stage stochastic program. The preparedness phase decides the locations, capacities and resources of new Permanent Shelters. Under each disaster scenario, both evacuees and resources are distributed to shelters in the response phase. To address the computational burden, the L-shaped algorithm is applied to decompose the problem into the scenario level with linear programs. A case study for hurricanes in the Gulf Coast region of the US is conducted to demonstrate the implementation of the proposed model.     

Budget allocation for permanent and contingent capacity under stochastic demand

We develop a model of budget allocation for permanent and contingent workforce under stochastic demand. The level of permanent capacity is determined at the beginning of the horizon and is kept constant throughout, whereas the number of temporary workers to be hired must be decided in each period. Compared to existing budgeting models, this paper explicitly considers a budget constraint. Under the assumption of a restricted budget, the objective is to minimize capacity shortages. When over-expenditures are allowed, both budget deviations and shortage costs are to be minimized. The capacity shortage cost function is assumed to be either linear or quadratic with the amount of shortage, which corresponds to different market structures or different types of services. We thus examine four variants of the problem that we model and solve either approximately or to optimality when possible. A comprehensive experimental design is designed to analyze the behavior of our models when several levels of demand variability and parameter values are considered. The parameters consist of the initial budget level, the unit cost of temporary workers and the budget deviation penalty/reward rates. Varying these parameters produce several trade-offs between permanent and temporary workforce levels, and between capacity shortages and budget deviations. Numerical results also show that the quadratic cost function leads to smooth and moderate capacity shortages over the time periods, whereas all shortages are either avoided or accepted when the cost function is linear.     

Analysis of a general Markovian two-stage continuous-flow production system with a finite buffer

Fluid flow models are used in the performance evaluation of production, computer, and telecommunication systems. In order to develop a methodology to analyze general Markovian continuous material flow production systems with two processing stages with an intermediate finite buffer, a general single-buffer fluid flow system is modelled as a continuous time, continuous-discrete state space stochastic process and the steady-state distribution is determined. Various performance measures such as the production rate and the expected buffer level are determined from the steady-state distributions. The flexibility of this methodology allows analysis of a wide range of models by specifying only the transition rates and the flow rates associated with the discrete states of each stage. Therefore, the method is proposed as a tool for performance evaluation of general Markovian continuous-flow systems with a finite buffer. The solution methodology is illustrated by analyzing a production system where each machine has multiple up and down states associated with their quality characteristics.     

Financial and operational decisions in the electricity sector: Contract portfolio optimization with the conditional value-at-risk criterion

The restructuring of electricity markets around the world have caused increased volatility and uncertainty of the price power. As a result, providers of power now face increased uncertainty and risk in the operational and financial decisions related to procurement. Providers must seek optimal ways to deliver the required volume of power to retailers and end users while managing risk. We consider a mixed-integer programming model for a power providing agent that jointly considers the problem of selecting custom electricity contracts and finding the optimal procurement strategy of meeting contract obligations under spot price uncertainty. A two-stage stochastic integer programming (SIP) model with a conditional value-at-risk (CVaR) constraint to incorporate risk aversion is developed. Computational results are presented that demonstrates the CVaR approach and the results are compared with a corresponding expected cost minimization approach. The SIP model with CVaR will allow acceptance of contracts at lower prices compared to an approach based on a corresponding risk-neutral model as a hedge against uncertainty and mis-specified arbitrage.     

An inventory model with non-resuming randomly interruptible lead time

We assume that an unreliable supplier in a single-item stochastic inventory system alternates randomly between two possible states (i.e., available and unavailable), following a two-state continuous-time homogeneous Markov chain. For a compound Poisson stream of demands and Erlang lead times, our model considers the scenario where the processing of the outstanding order (if any) is interrupted at every supplier's transition epoch from the available to the unavailable state, and is restarted from the outset upon the supplier's regaining its available state. We derive the stationary distribution of the on-hand inventory under a continuous-review policy and provide some numerical results.