A dyadic age-replacement policy for a periodically inspected equipment item subject to random deterioration

A periodic review replacement system is considered. The amount of deterioration over successive periods forms a sequence of i.i.d. random variables. A replacement policy of the dyadic type is in effect whereby the used equipment item is discarded and immediately replaced by a new identical equipment item if at the end of a period the old equipment has service aged by an amount in excess of S or has been in operation for exactly N periods whichever comes first. Using a theorem on renewal reward processes, an expression for the total steady-state expected cost per period is derived, consisting of a fixed replacement cost and a linear cost of operation. Optimal values of S and N that minimize this steady state cost are computed for a few numerical examples, when the service aging per period has a gamma distribution.     

An efficient approach for solving the lot-sizing problem with time-varying storage capacities

We address the dynamic lot size problem assuming time-varying storage capacities. The planning horizon is divided into T periods and stockouts are not allowed. Moreover, for each period, we consider a setup cost, a holding unit cost and a production/ordering unit cost, which can vary through the planning horizon. Although this model can be solved using O(T³) algorithms already introduced in the specialized literature, we show that under this cost structure an optimal solution can be obtained in O(T log T) time. In addition, we show that when production/ordering unit costs are assumed to be constant (i.e., the Wagner–Whitin case), there exists an optimal plan satisfying the Zero Inventory Ordering (ZIO) property.     

Optimization of a Multiperiod Resource Allocation Model

In this paper we examine multiperiod resource allocation problems, such as allocating a given marketing budget among T periods. The return functions of each period are assumed to be concave functions of the effective effort variable, which is composed of the expenditures in all previous periods and the present one. Assuming that the effect of an amount spent in period t is decreasing by a fixed rate in successive periods, necessary and sufficient conditions for a non-boundary optimal policy are derived. Under these conditions the optimal policy which maximizes total returns is obtained.     

Managing stochastic inventory systems with free shipping option

In many industries, customers are offered free shipping whenever an order placed exceeds a minimum quantity specified by suppliers. This allows the suppliers to achieve economies of scale in terms of production and distribution by encouraging customers to place large orders. In this paper, we consider the optimal policy of a retailer who operates a single-product inventory system under periodic review. The ordering cost of the retailer is a linear function of the ordering quantity, and the shipping cost is a fixed constant K whenever the order size is less than a given quantity – the free shipping quantity (FSQ), and it is zero whenever the order size is at least as much as the FSQ. Demands in different time periods are i.i.d. random variables. We provide the optimal inventory control policy and characterize its structural properties for the single-period model. For multi-period inventory systems, we propose and analyze a heuristic policy that has a simple structure, the (s, t, S) policy. Optimal parameters of the proposed heuristic policy are then computed. Through an extensive numerical study, we demonstrate that the heuristic policy is sufficiently accurate and close to optimal.     

Joint pricing and inventory control with a Markovian demand model

We consider the joint pricing and inventory control problem for a single product over a finite horizon and with periodic review. The demand distribution in each period is determined by an exogenous Markov chain. Pricing and ordering decisions are made at the beginning of each period and all shortages are backlogged. The surplus costs as well as fixed and variable costs are state dependent. We show the existence of an optimal (s, S, p)-type feedback policy for the additive demand model. We extend the model to the case of emergency orders. We compute the optimal policy for a class of Markovian demand and illustrate the benefits of dynamic pricing over fixed pricing through numerical examples. The results indicate that it is more beneficial to implement dynamic pricing in a Markovian demand environment with a high fixed ordering cost or with high demand variability.     

Revenue deficiency under second-price auctions in a supply-chain setting

Consider a firm, called the buyer, that satisfies its demand over two periods by assigning both demands to a supplier via a second-price procurement auction; call this the Standard auction. In the hope of lowering its purchase cost, the firm is considering an alternative procedure in which it will also allow bids on each period individually, where there can be either one or two winners covering the two demands; call this the Multiple Winner auction. Choosing the Multiple Winner auction over the Standard auction can in fact result in a higher cost to the buyer. We provide a bound on how much greater the buyer’s cost can be in the Multiple Winner auction and show that this bound is tight. We then sharpen this bound for two scenarios that can arise when the buyer announces his demands close to the beginning of the demand horizon. Under a monotonicity condition, we achieve a further sharpening of the bound in one of the scenarios. Finally, this monotonicity condition allows us to generalize this bound to the T-period case in which bids are allowed on any subset of period demands.     

A mixed integer linear programming formulation of the optimal mean/Value-at-Risk portfolio problem

In this paper, we consider an extension of the Markovitz model, in which the variance has been replaced with the Value-at-Risk. So a new portfolio optimization problem is formulated. We showed that the model leads to an NP-hard problem, but if the number of past observation T or the number of assets K are low, e.g. fixed to a constant, polynomial time algorithms exist. Furthermore, we showed that the problem can be formulated as an integer programming instance. When K and T are large and α^VaR is small—as common in financial practice—the computational results show that the problem can be solved in a reasonable amount of time.     

Non-stationary inventory position processes

This paper investigates single-product non-stationary inventory problems associated with non-stationary demand processes, backorders allowed, stochastically independent inter-review-period demands, and review periods not necessarily of equal length. Sufficient conditions are obtained for the convergence in distribution of inventory positions in 〈R, r, T〉 inventory systems; in particular, the uniform long-run distribution is found robust in 〈nQ, r, T〉 systems for a large class of demand processes. It is also shown that in 〈R, r, T〉 systems the subsequences of cyclic non-stationary inventory position process associated with cyclic behavior of demand patterns converge in distribution.     

Production-inventory control policy under warm/cold state-dependent fixed costs and stochastic demand: partial characterization and heuristics

The motivation for our study comes from some production and inventory systems in which ordering/producing quantities that exceed certain thresholds in a given period might eliminate some setup activities in the next period. Many examples of such systems have been discussed in prior research but the analysis has been limited to production settings under deterministic demand. In this paper, we consider a periodic-review production-inventory model under stochastic demand and incorporate the following fixed-cost structure into our analysis. When the order quantity in a given period exceeds a specified threshold value, the system is assumed to be in a “warm” state and no fixed cost is incurred in the next period regardless of the order quantity; otherwise the system state is considered “cold” and a positive fixed cost is required to place an order. Assuming that the unsatisfied demand is lost, we develop a dynamic programming formulation of the problem and utilize the concepts of quasi-K-convexity and non-K-decreasing to show some structural results on the optimal cost-to-go functions. This analysis enables us to derive a partial characterization of the optimal policy under the assumption that the demands follow a Pólya or uniform distribution. The optimal policy is defined over multiple decision regions for each system state. We develop heuristic policies that are aimed to address the partially characterized decisions, simplify the ordering policy, and save computational efforts in implementation. The numerical experiments conducted on a large set of test instances including uniform, normal and Poisson demand distributions show that a heuristic policy that is inspired by the optimal policy is able to find the optimal solution in almost all instances, and that a so-called generalized base-stock policy provides quite satisfactory results under reasonable computational efforts. We use our numerical examples to generate insights on the impact of problem parameters. Finally, we extend our analysis into the infinite horizon setting and show that the structure of the optimal policy remains similar.     

Optimization and sensitivity analysis of controlling arrivals in the queueing system with single working vacation

This paper analyzes the F-policy M/M/1/K queueing system with working vacation and an exponential startup time. The F-policy deals with the issue of controlling arrivals to a queueing system, and the server requires a startup time before allowing customers to enter the system. For the queueing systems with working vacation, the server can still provide service to customers rather than completely stop the service during a vacation period. The matrix-analytic method is applied to develop the steady-state probabilities, and then obtain several system characteristics. We construct the expected cost function and formulate an optimization problem to find the minimum cost. The direct search method and Quasi-Newton method are implemented to determine the optimal system capacity K, the optimal threshold F and the optimal service rates (μ_B,μ_V) at the minimum cost. A sensitivity analysis is conducted to investigate the effect of changes in the system parameters on the expected cost function. Finally, numerical examples are provided for illustration purpose.     

The single item uncapacitated lot-sizing problem with time-dependent batch sizes: NP-hard and polynomial cases

This paper considers the uncapacitated lot sizing problem with batch delivery, focusing on the general case of time-dependent batch sizes. We study the complexity of the problem, depending on the other cost parameters, namely the setup cost, the fixed cost per batch, the unit procurement cost and the unit holding cost. We establish that if any one of the cost parameters is allowed to be time-dependent, the problem is NP-hard. On the contrary, if all the cost parameters are stationary, and assuming no unit holding cost, we show that the problem is polynomially solvable in time O(T³), where T denotes the number of periods of the horizon. We also show that, in the case of divisible batch sizes, the problem with time varying setup costs, a stationary fixed cost per batch and no unit procurement nor holding cost can be solved in time O(T³ logT).     

Triangular norms and TNF-sigma-algebras

Following the suggestion made by Klement [8], an axiomatic theory of TNF-σ-algebras is given, T being any measurable triangular norm and N any negation. Most of the results about T-fuzzy σ-algebbrs obtained in [8] are extended to the case of TNF-σ-algebras. Some other properties of TNF-σ-algebras are also discussed. Particularly, we point out: (1) there exists a large family of triangular norms, which contains the whole Yager family and almost the whole Sugeno family as subfamilies, such that for any negation N each TNF-σ-algebra is generated, and (2) given a set U with |U|?2 and a measurable triangular norm T, in order that for every negation N each TNF-σ-algebra on U is generated it is necessary that T is Archimedean.     

Fixed point iteration processes for asymptotic pointwise nonexpansive mappings in Banach spaces

Let X be a uniformly convex Banach space with the Opial property. Let T:C→C be an asymptotic pointwise nonexpansive mapping, where C is bounded, closed and convex subset of X. In this paper, we prove that the generalized Mann and Ishikawa processes converge weakly to a fixed point of T. In addition, we prove that for compact asymptotic pointwise nonexpansive mappings acting in uniformly convex Banach spaces, both processes converge strongly to a fixed point.     

A single-period inventory placement problem for a supply chain with the expected profit objective

Consider the expected profit maximizing inventory placement problem in an N-stage, supply chain facing a stochastic demand for a single planning period for a specialty item with a very short selling season. Each stage is a stocking point holding some form of inventory (e.g., raw materials, subassemblies, product returns or finished products) that after a suitable transformation can satisfy customer demand. Stocking decisions are made before demand occurs. Because of delays, only a known fraction of demand at a stage will wait for shipments. Unsatisfied demand is lost. The revenue, salvage value, ordering, shipping, processing, and lost sales costs are proportional. There are fixed costs for utilizing stages for stock storage. After characterizing an optimal solution, we propose an algorithm for its computation. For the zero fixed cost case, the computations can be done on a spreadsheet given normal demands. For the nonnegative fixed cost case, we develop an effective branch and bound algorithm.     

The price of anarchy for non-atomic congestion games with symmetric cost maps and elastic demands

By showing that there is an upper bound for the price of anarchyρ(Γ) for a non-atomic congestion game Γ with only separable cost maps and fixed demands, Roughgarden and Tardos show that the cost of forgoing centralized control is mild. This letter shows that there is an upper bound for ρ(Γ) in Γ for fixed demands with symmetric cost maps. It also shows that there is a weaker bound for ρ(Γ) in Γ with elastic demands.     

Combined pricing and inventory control with multiple unreliable suppliers

We study a general finite horizon, periodic review combined inventory and pricing model with N suppliers and T periods, where both the demands and the supply mechanisms are random. The random supply mechanisms are of a general type that includes most structures encountered in practice. Demands are price dependent according to general, stochastic demand functions. We characterize the optimal combined pricing and ordering policies to all N suppliers. The general results pertain to general independent supply mechanisms. Under random capacities—one of the special random supply mechanisms—they also extend to suppliers that are positively dependent on each other.     

A fuzzy algorithm for continuous capacitated location allocation model with risk consideration

This paper presents a continuous capacitated location-allocation model with fixed cost as a risk management model. In the presented model, the fixed cost consists of production and installation costs. The model considers risk as percent of unsatisfied demands. The fixed cost is assigned to a zone with a predetermined radius from its center. Because of uncertain environment, demand in each zone is investigated as a fuzzy number. The model is solved by a fuzzy algorithm based on α-cut method. After solving the model based on different α-values, the zones with the largest possibilities are determined for locating new facilities and the best locations are calculated based on the obtained possibilities. Then, the model is solved based on different α-values to determine best allocation values. Also, this paper proposes a Cross Entropy (CE) algorithm considering multivariate normal and multinomial density functions for solving large scale instances and is compared with GAMS. Finally, a numerical example is expressed to illustrate the proposed model.     

Scheduling policies in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1 make-to-stock queue

In this paper, we analyse a production/inventory system modelled as an M/G/1 make-to-stock queue producing different products requiring different and general production times. We study different scheduling policies including the static first-come-first-served, preemptive and non-preemptive priority disciplines. For each static policy, we exploit the distributional Little's law to obtain the steady-state distribution of the number of customers in the system and then find the optimal inventory control policy and the cost. We additionally provide the conditions under which it is optimal to produce a product according to a make-to-order policy. We further extend the application area of a well-known dynamic scheduling heuristic, Myopic(T), for systems with non-exponential service times by permitting preemption. We compare the performance of the preemptive-Myopic(T) heuristic alongside that of the static preemptive-bμ rule against the optimal solution. The numerical study we have conducted demonstrates that the preemptive-Myopic(T) policy is superior between the two and yields costs very close to the optimal.     

Dynamic lot-sizing model for major and minor demands

This paper deals with a lot-sizing model for major and minor demands in which major demands are specified by time windows while minor demands are given by periods. For major demands, the agreeable time window structure is assumed where each time window is not strictly nested in any other time windows. To incorporate the economy of scale of large production quantity, especially from major demands, concave cost structure in production must be considered. Investigating the optimality properties, we propose optimal solution procedures based on dynamic program. For a simple case when only major demands exist, we propose an optimal procedure with running time of O(n²T)$\mathrm{O}(n^{2}T)$ where n is the number of demands and T   is the length of the planning horizon. Extending the algorithm to the model with major and minor demands, we propose an algorithm with complexity O(n²T²)$\mathrm{O}(n^2{T}^2)$.