Proactive data envelopment analysis: Effective production and capacity expansion in stochastic environments

Demand fluctuations that cause variations in output levels will affect a firm’s technical inefficiency. To assess this demand effect, a demand-truncated production function is developed and an “effectiveness” measure is proposed. Often a firm can adjust some input resources influencing the output level in an attempt to match demand. We propose a short-run capacity planning method, termed proactive data envelopment analysis, which quantifies the effectiveness of a firm’s production system under demand uncertainty. Using a stochastic programming DEA approach, we improve upon short-run capacity expansion planning models by accounting for the decreasing marginal benefit of inputs and estimating the expected value of effectiveness, given demand. The law of diminishing marginal returns is an important property of production function; however, constant marginal productivity is usually assumed for capacity expansion problems resulting in biased capacity estimates. Applying the proposed model in an empirical study of convenience stores in Japan demonstrates the actionable advice the model provides about the levels of variable inputs in uncertain demand environments. We conclude that the method is most suitable for characterizing production systems with perishable goods or service systems that cannot store inventories.     

Integrating effective flexibility measures into a strategic supply chain planning model

This paper develops models for capacity, product mix, distribution and input supply flexibility and integrates them in a strategic level, mixed integer supply chain (SC) planning model as a way of addressing demand and supply uncertainty, as well as improving market responsiveness. Capacity flexibility is modeled via the SC’s production capacity planning to address budgeted demand and ensure the fulfillment of prospective demand increases when considering various market scenarios. This model selects an optimal number of products from fast moving and extended product range options—based on the product mix flexibility. The model confirms a quick response to a changing marketplace by considering elements like transportation and supply lead time along with the probabilities of stock out options when addressing input supply and distribution flexibility. This paper proposes a solution procedure to solve the model for real world problems, and investigates the sensitivity of the model outputs with respect to changes in flexibility measures.     

Flexible capacity strategy with multiple market periods under demand uncertainty and investment constraint

We establish a flexible capacity strategy model with multiple market periods under demand uncertainty and investment constraints. In the model, a firm makes its capacity decision under a financial budget constraint at the beginning of the planning horizon which embraces n market periods. In each market period, the firm goes through three decision-making stages: the safety production stage, the additional production stage and the optimal sales stage. We formulate the problem and obtain the optimal capacity, the optimal safety production, the optimal additional production and the optimal sales of each market period under different situations. We find that there are two thresholds for the unit capacity cost. When the capacity cost is very low, the optimal capacity is determined by its financial budget; when the capacity cost is very high, the firm keeps its optimal capacity at its safety production level; and when the cost is in between of the two thresholds, the optimal capacity is determined by the capacity cost, the number of market periods and the unit cost of additional production. Further, we explore the endogenous safety production level. We verify the conditions under which the firm has different optimal safety production levels. Finally, we prove that the firm can benefit from the investment only when the designed planning horizon is longer than a threshold. Moreover, we also derive the formulae for the above three thresholds.     

The valuation of options on capacity with cost and demand uncertainty

Options contracts can provide trading partners with enhanced flexibility to respond to uncertain market conditions and allow for superior capacity planning thanks to early information on future demand. We develop an analytical framework to value options on capacity for production of non-storable goods or dated services. The market consists of a sequence of contract and spot market. Reservations are made during the contract market session in period 0, where the buyer’s future demand, the seller’s future marginal costs as well as the future spot price are uncertain, the latter being impacted neither by the buyer nor the seller. During the spot market session in period 1, the buyer may execute his options or satisfy his entire or additional demand from a competing seller in the spot market. The seller allocates reserved capacity now being called and attempts to sell remaining capacity into the spot market. Analytical expressions for the buyer’s optimal reservation quantity and the seller’s tariff are derived, making explicit the risk-sharing benefits of options contracts. The combination of an options contract and a spot market is demonstrated to be Pareto improving as compared to alternative market schemes. An analysis of the determinants of the efficiency gain characterizes industries particularly suitable to the options approach.     

Planning production and working time within an annualised hours scheme framework

Production flexibility is essential for industrial companies that have to deal with seasonal demand. Human resources are one of the main sources of flexibility. Annualising working hours (i.e., the possibility of irregularly distributing the total number of working hours over the course of a year) is a tool that provides organisations with flexibility; it enables a firm to adapt production capacity to fluctuations in demand. However, it can involve a worsening of the staff working conditions. To take this into account, the planning and scheduling of working time should comply with constraints derived from the law or from a collective bargaining agreement. Thus, new and more difficult working-time and production planning and scheduling problems are arising. This paper proposes two mixed-integer linear program models for solving the problem of planning the production, the working hours and the holiday weeks of the members of a human team operating in a multi-product process in which products are perishable, demand can be deferred and temporary workers are hired to stand in for employees. The results of a computational experiment are presented. Supported by the Spanish MCyT projects DPI2001-2176 and DPI2004-05797, co-financed by FEDER.     

Managing capacity flexibility in make-to-order production environments

This paper addresses the problem of managing flexible production capacity in a make-to-order (MTO) manufacturing environment. We present a multi-period capacity management model where we distinguish between process flexibility (the ability to produce multiple products on multiple production lines) and operational flexibility (the ability to dynamically change capacity allocations among different product families over time). For operational flexibility, we consider two polices: a fixed allocation policy where the capacity allocations are fixed throughout the planning horizon and a dynamic allocation policy where the capacity allocations change from period to period. The former approach is modeled as a single-stage stochastic program and solved using a cutting-plane method. The latter approach is modeled as a multi-stage stochastic program and a sampling-based decomposition method is presented to identify a feasible policy and assess the quality of that policy. A computational experiment quantifies the benefits of operational flexibility and demonstrates that it is most beneficial when the demand and capacity are well-balanced and the demand variability is high. Additionally, our results reveal that myopic operating policies may lead a firm to adopt more process flexibility and form denser flexibility configuration chains. That is, process flexibility may be over-valued in the literature since it is assumed that a firm will operate optimally after the process flexibility decision. We also show that the value of process flexibility increases with the number of periods in the planning horizon if an optimal operating policy is employed. This result is reversed if a myopic allocation policy is adopted instead.     

Optimal inventory policies with two modes of freight transportation

Theoretical inventory models with constant demand rate and two transportation modes are analyzed in this paper. The transportation options are truckloads with fixed costs, a package delivery carrier with a constant cost per unit, or using a combination of both modes simultaneously. Exact algorithms for computing the optimal policies are derived for single stage models over both an infinite and a finite planning horizon.     

Production planning via scenario modelling

Several Linear Programming (LP) and Mixed Integer Programming (MIP) models for the production and capacity planning problems with uncertainty in demand are proposed. In contrast to traditional mathematical programming approaches, we use scenarios to characterize the uncertainty in demand. Solutions are obtained for each scenario and then these individual scenario solutions are aggregated to yield a nonanticipative or implementable policy. Such an approach makes it possible to model nonstationarity in demand as well as a variety of recourse decision types. Two scenario-based models for formalizing implementable policies are presented. The first model is a LP model for multi-product, multi-period, single-level production planning to determine the production volume and product inventory for each period, such that the expected cost of holding inventory and lost demand is minimized. The second model is a MIP model for multi-product, multi-period, single-level production planning to help in sourcing decisions for raw materials supply. Although these formulations lead to very large scale mathematical programming problems, our computational experience with LP models for real-life instances is very encouraging.     

Joint management of capacity and inventory in make-to-stock production systems with multi-class demand

We evaluate the benefits of coordinating capacity and inventory decisions in a make-to-stock production environment. We consider a firm that faces multi-class demand and has additional capacity options that are temporary and randomly available. We formulate the model as a Markov decision process (MDP) and prove that a solution to the optimal joint control problem exists. For several special cases we characterize the structure of the optimal policy. For the general case, however, we show that the optimal policy is state-dependent, and in many instances non-monotone and difficult to implement. Therefore, we consider three pragmatic heuristic policies and assess their performance. We show that the majority of the savings originate from the ability to dynamically adjust capacity, and that a simple heuristic that can adjust production capacity (based on workload fluctuation) but uses a static production/rationing policy can result in significant savings.     

Impact of equity and equality on stability and collusion in a decentralized network

We consider a firm that procures a product from a regular supplier whose production is subject to both supply disruption and random yield risks and a backup supplier whose production capacity requires reservation in advance. Under both deterministic and stochastic demand, we study the impact of the two types of supply risks on the firm’s optimal procurement decisions and the importance of correctly identifying the source of supply risks. We find that if the overall supply risk is unchanged but its main source shifts from random yield to supply disruption, the firm should order more from the regular supplier and reserve less capacity from the backup supplier. Ignoring the existence of supply disruption leads to under-utilization of the regular supplier and over-utilization of the backup supplier. Moreover, we examine the option value of the reserved capacity that is affected by the uncertainty of customer demand. We find that the option value increases/decreases in demand uncertainty if the reservation capacity is exercised after/before demand is realized.     

Multi-stage stochastic programming models for provisioning cloud computing resources

We focus on the resource provisioning problem of a cloud consumer from an Infrastructure-as-a-Service type of cloud. The cloud provider offers two deployment options, which can be mixed and matched as appropriate. Cloud instances may be reserved for a fixed time period in advance at a smaller usage cost per hour but require a full commitment and payment for the entire contract duration. In contrast, on-demand instances reflect a pay-as-you-go policy at a premium. The trade-off between these two options is rooted in the inherent uncertainty in demand and price and makes it attractive to complement a base reserved capacity with on-demand capacity to hedge against the spikes in demand. This paper provides several novel multi-stage stochastic programming formulations to enable a cloud consumer to handle the cloud resource provisioning problem at a tactical level. We first formulate the cloud resource provisioning problem as a risk-neutral multi-stage stochastic program, which serves as the base model for further modeling variants. In our second set of models, we also incorporate a certain concept of system reliability. In particular, chance constraints integrated into the base formulation require a minimum service level met from reserved capacity, provide more visibility into the future available capacity, and smooth out expensive on-demand usage by hedging against possible demand fluctuations. An extensive computational study demonstrates the value of the proposed models by discussing computational performance, gleaning practical managerial insights from the analysis of the solutions of the proposed models, and quantifying the value of the stochastic solutions.     

A large-scale multilocation capacity planning model

In this paper, a large-scale multilocation capacity planning model is described. The model chooses a multiperiod schedule of openings, expansions, and closings of facilities, and assigns demand locations to these facilities. Although generic in nature, this model was developed to plan the evolution of material logistics systems over time. In order to have a truly practical tool, numerous features are considered including existing configuration, arbitrary demand patterns, concave operating costs, single-source assignments, demand location reassignment costs, and others.Such capacity planning models are highly combinatorial in nature, and are solved, in general, by heuristics. Our solution method has three major modules. First, an initial solution is generated by solving successively single-period problems using network optimization techniques complemented by other heuristics. Next, opening and closing decisions are adjusted and improved. Finally, demand location assignment decisions throughout the planning horizon are modified. The heuristic was tested on many problems of various sizes; computational experience is described.     

An Approximation of a Markov Decision Process for Resource Planning

This study presents an approximation of a Markovian decision process to calculate resource planning policies for environments with probabilistic resource demand. These policies provide a means of periodic determination of the quantity of resources required to be available. Managers may also use these approximation models to perform the sensitivity analysis of resource demand and the cost/reward parameters. The decision policy can be applied to many resource planning situations including manufacturing or construction equipment purchasing or leasing, airline capacity, professional services staffing, and computer/management information systems capacity.     

Models for planning capacity expansion of convenience stores under uncertain demand and the value of information

Several stochastic optimization models for planning capacity expansion for convenience store chains (or other similar businesses) are developed that incorporate uncertainty in future demand. All of these models generate schedules for capacity expansion, specifying the size, location, and timing of these expansions in order to maximize the expected profit to the company and to remain within a budget constraint on available resources. The models differ in how uncertainty is incorporated, specifically they differ in the point in the decision-making process that the uncertainty in the demand is resolved. Several measures of the value of information are defined by comparing the results from the different models. Two sample problems are given and their solutions for the various approaches compared.     

Competitive production scheduling: A two-firm,noncooperative finite dynamic game

In this paper, we study the production scheduling problem in a competitive environment. Two firms produce the same product and compete in a market. The demand is random and so is the production capacity of each firm, due to random breakdowns. We consider a finite planning horizon. The scheduling problem is formulated as a finite dynamic game. Algorithms are developed to determine the security, hazard, and Nash policies. Numerical examples are discussed. A single-firm optimization model is also analyzed and it is observed that the production control policy from the single-firm optimization model may not perform well in a competitive environment.     

Designing a two-echelon distribution network under demand uncertainty

This paper proposes a comprehensive methodology for the stochastic multi-period two-echelon distribution network design problem (2E-DDP) where product flows to ship-to-points are directed from an upper layer of primary warehouses to distribution platforms (DPs) before being transported to the ship-to-points. A temporal hierarchy characterizes the design level dealing with DP location and capacity decisions, as well as the operational level involving transportation decisions as origin-destination flows. These design decisions must be calibrated to minimize the expected distribution cost associated with the two-echelon transportation schema on this network under stochastic demand. We consider a multi-period planning horizon where demand varies dynamically from one planning period to the next. Thus, the design of the two-echelon distribution network under uncertain customer demand gives rise to a complex multi-stage decisional problem. Given the strategic structure of the problem, we introduce alternative modeling approaches based on two-stage stochastic programming with recourse. We solve the resulting models using a Benders decomposition approach. The size of the scenario set is tuned using the sample average approximation (SAA) approach. Then, a scenario-based evaluation procedure is introduced to post-evaluate the design solutions obtained. We conduct extensive computational experiments based on several types of instances to validate the proposed models and assess the efficiency of the solution approaches. The evaluation of the quality of the stochastic solution underlines the impact of uncertainty in the two-echelon distribution network design problem (2E-DDP).     

Pricing policy in dynamic markets and a generalization of the Ramsey rule

Pricing policy in a regulated monopoly industry is usually based on maximizing welfare or some other measure of utility level of return on investment. Previously, the Ramsey pricing policy which states that the percentage deviation of quasi-optimal price from marginal cost for each product must be inversely proportional to its price elasticity of demand, has been developed for a static market. The Ramsey framework assumes instantaneous demand response to price changes; empirical evidence suggests demand changes occur dynamically through time.In this paper an optimum pricing rule for a profit maximizing firm based on a general time varying demand model in a dynamic market is obtained assuming a single price change at the beginning of the planning period. A dynamic market equivalent of the well known inverse elasticity law of the static market is developed. Defining the concept of average price elasticity for dynamic markets we show that the inverse elasticity law of static markets takes an inequality form in dynamic markets. For demand functions which decrease, increase or are constant with time the optimum price markups are greater than, less than, or equal to the inverse of the average price elasticity, respectively.The results are then generalized to the case of a constrained welfare maximizing firm. This leads to the development of a dynamic market generalization of the well known Ramsey pricing rule. A simple rule for making quantitative arguments about the relative size of the optimum price in static and dynamic markets is also derived.This work was completed when the author was with Bell Laboratories, USA.     

On optimal bidding and inventory control in sequential procurement auctions: the multi period case

We consider the problem of a firm that in each cycle of a planning horizon builds inventory of identical items that it acquires by participating in auctions in order to satisfy its own market demand. The firm’s objective is to have a procurement strategy that maximizes the expected present value of the profit for an infinite planning horizon of identical cycles. We formulate this problem as a Markov decision process. We establish monotonicity properties of the value function and of the optimal bidding rule.     

Ensuring responsive capacity: How to contract with backup suppliers

Firms that source from offshore plants frequently perceive the lack of reliability and flexibility to be among the major drawbacks of their strategy. To mitigate against imminent mismatches of uncertain supply and demand, establishing capacity hedges in the form of responsive backup suppliers is a way out that many firms follow. This article analyzes how firms should contract with backup suppliers, inducing the latter to install responsive capacity. We show that supply options are appropriate to achieve sourcing channel coordination under forced compliance, whereas any firm commitment contract imposes a deadweight loss on the system. Whereas price-only contracts are unable to coordinate the sourcing channel under voluntary compliance, utilization-dependent price-only contracts are. Under the former contract, a price-focused strategy on the part of the manufacturer turns out to diminish the system’s service level and possibly has negative implications on installed backup capacity, and not least on the manufacturer’s profit.     

An analytical approach to the facility location and capacity acquisition problem under demand uncertainty

This article presents an analysis of facility location and capacity acquisition under demand uncertainty. A novel methodology is proposed, in which the focus is shifted from the precise representation of facility locations to the market areas they serve. This is an extension of the optimal market area approach in which market area size and facility capacity are determined to minimize the total cost associated with fixed facility opening, variable capacity acquisition, transportation, and shortage. The problem has two variants depending on whether the firm satisfies shortages by outsourcing or shortages become lost sales. The analytical approach simplifies the problem considerably and leads to intuitive and insightful models. Among several other results, it is shown that fewer facilities are set up under lost sales than under outsourcing. It is also shown that the total cost in both models is relatively insensitive to small deviations in optimal capacity choices and parameter estimations.