A mathematical modelling approach for managing sudden disturbances in a three-tier manufacturing supply chain Abstract This paper aims to develop a recovery planning approach in a three-tier manufacturing supply chain, which has a single supplier, manufacturer, and retailer under an imperfect production environment, in which we consider three types of sudden disturbances: demand fluctuation, and disruptions to production and raw material supply, which are not known in advance. Firstly, a mathematical model is developed for generating an ideal plan under imperfect production for a finite planning horizon while maximizing total profit, and then we re-formulate the model to generate the recovery plan after happening of each sudden disturbance. Considering the high commercial cost and computational intensity and complexity of this problem, we propose an efficient heuristic, to obtain a recovery plan, for each disturbance type, for a finite future period, after the occurrence of a disturbance. The heuristic solutions are compared with a standard solution technique for a considerable number of random test instances, which demonstrates the trustworthy performance of the developed heuristics. We also develop another heuristic for managing the combined effects of multiple sudden disturbances in a period. Finally, a simulation approach is proposed to investigate the effects of different types of disturbance events generated randomly. We present several numerical examples and random experiments to explicate the benefits of our developed approaches. Results reveal that in the event of sudden disturbances, the proposed mathematical and heuristic approaches are capable of generating recovery plans accurately and consistently.

Computations of volumes and Ehrhart series in four candidates elections Abstract We describe several analytical results obtained in four candidates social choice elections under the assumption of the Impartial Anonymous Culture. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality voting with runoff. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes and counting functions encoded as Ehrhart series of polytopes.

Flow formulations for curriculum-based course timetabling Abstract In this paper we present two mixed-integer programming formulations for the curriculum based course timetabling problem (CTT). We show that the formulations contain underlying network structures by dividing the CTT into two separate models and then connect the two models using flow formulation techniques. The first mixed-integer programming formulation is based on an underlying minimum cost flow problem, which decreases the number of integer variables significantly and improves the performance compared to an intuitive mixed-integer programming formulation. The second formulation is based on a multi-commodity flow problem which in general is NP-hard, however, we prove that it suffices to solve the linear programming relaxation of the model. The formulations show competitiveness with other approaches based on mixed-integer programming from the literature and improve the currently best known lower bound on one data instance in the benchmark data set from the second international timetabling competition. Regarding upper bounds, the formulation based on the minimum cost flow problem performs better on average than other mixed integer programming approaches for the CTT.

Optimal credit term, order quantity and selling price for perishable products when demand depends on selling price, expiration date, and credit period Abstract The demand for perishable goods (e.g., baked goods, fruits, vegetables, meat, milk, and seafood) is influenced by product freshness which gradually declines over time and can be perceived by its expiration date. Also, selling price is an important factor on demand. Furthermore, most modern companies offer their products on various credit terms to increase sales. However, relatively little attention has been paid to the combined effects of selling price, expiration date and credit term affecting demand. In this paper, a three-echelon supplier-retailer-consumer supply chain for perishable goods is explored in which the retailer receives an upstream full trade credit from the supplier while granting a downstream partial trade credit to credit-risk customers, with demand as a multiplicative form of selling price, expiration date, and credit period. The proposed model includes numerous previous models as special cases. The optimal credit term, order size and selling price are derived simultaneously for the retailer to achieve maximum profit. Several numerical examples are conducted to gain managerial insights. For example, if the credit efficiency of demand increases, then the retailer shall offer a longer downstream credit period to raise sales volume, which in turn implies the retailer can raise a higher price, order a larger quantity, and gain a higher total profit. Conversely, an increase in portion of cash payment results in a lower demand rate. Hence, the retailer orders less quantity and earns less profit while decreasing price to stimulate sales. Finally, conclusions and future research directions are provided.

Necessary players, Myerson fairness and the equal treatment of equals Abstract This article addresses linear sharing rules on transferable utility games (TU-games) with various structures, namely communication structures and conference structures as defined by Myerson in two papers (Myerson in Mathematics of Operations Research 2:225–229, 1977; Myerson in International Journal of Game Theory 9:169–182, 1980). Here, using matrix expressions, we rewrite those sharing rules. With this presentation we identify the close relationship between the fairness property and an equal treatment of necessary players axiom. Moreover, we show that the latter is implied by the equal treatment of equals, linking the fairness property to the notion of equality.

Designing competitive loyalty programs: a stochastic game-theoretic model to guide the choice of reward structure Abstract We develop a game-theoretic model to guide the choice of the reward structure in customer loyalty programs. We model a duopoly market in which one firm adopts a loyalty program. Firms independently and simultaneously set the prices and rewards. Heterogeneous customers buy homogeneous products in a multi-period setting. Customers are segmented into three groups based on their level of strategic behavior, which is expressed in terms of their degree of forward-lookingness. We use two exogenous parameters to represent the size of each segment. A third parameter captures the point pressure effect, which refers to the increase in customer spending as they approach a reward threshold. In each period, customers choose the firm that maximizes their utility, which is a function of offered prices, rewards, and the distance to the next reward. We use the logit model to model the customer choice behavior. Customers’ accumulated purchases evolve as a Markov chain. We derive the limiting distribution of accumulated purchases, which is subsequently used to formulate the firm’s expected revenue functions. We develop two algorithms to find the Nash equilibrium for both the linear and nonlinear rewards in term of the three parameters. Using a thorough numerical analysis, we show that the choice of the structure becomes more critical as the size of the strategic segment increases. The nonlinear scheme is superior when the size of the highly-strategic segment is very small. The linear rewards is superior in markets where the size of the highly-strategic segment and the sensitivity to distance are simultaneously not small.

Modeling uncertainty of expert elicitation for use in risk-based optimization Abstract Capital budgeting optimization models, used in a broad number of fields, require certain and uncertain parameters. Often times, elicited subject matter expert (SME) opinion is used as a parameter estimate, which does not always yield perfect information or correspond to a single value. Because of the uncertainty of the elicitation, the unknown true value of a parameter can be modeled as a random variable from a to-be-determined distribution. We estimate a univariate distribution using four different approaches, the Beta and Gaussian distributions, a standard Gaussian Kernel estimate, and an exponential epi-spline. We also capture dependencies within the parameters through three multivariate approaches: the multivariate Gaussian distribution, the multivariate Kernel and the multivariate exponential epi-spline. This is the first three-dimensional application of the latter. Sampling from the densities, we generate scenarios and implement a superquantile risk-based, capital budgeting optimization model. Numerical experiments contrast the differences between estimators, as well as their effects on an optimal solution. Our findings demonstrate that naively averaging the SME observations for use in optimization, rather than incorporating uncertainty, results in an overly optimistic portfolio. The flexibility of the exponential epi-spline estimator to fuse soft information with observed data produces reasonable density functions for univariate and multivariate random variables. Including a decision-maker’s risk-averseness through risk-based optimization delivers conservative results while incorporating the uncertainty of unknown parameters. We demonstrate a 20% improvement for this specific case when using our approach as opposed to the naive method.

Managing premium wines using an $$(s - 1,s)$$ ( s - 1 , s ) inventory policy: a heuristic solution approach Abstract Operations research models are increasingly being used to support decision making in the wine industry. However, they have not yet been used to support inventory management decisions. In this paper, we develop a heuristic procedure for managing the stock of premium wines motivated by the operations of a small export-focused winery we worked with. Following an \((s-1,s)\) inventory policy, we assume that the decision maker aims to minimize the steady-state expected values of work in process, overage, and underage costs. The developed heuristic is as follows. First, we approximate the dynamics of the labeling process by a group scheduling policy to obtain the mean delays for each labeled product. Then, we address the problem of setting the inventory positions for the whole product portfolio by solving one newsvendor-type problem for each end-product. We provide some theoretical insights, a numerical example, and we analyze the accuracy of our procedure.

Sandwiches missing two ingredients of order four Abstract For a set \(\mathcal{F}\) of graphs, an instance of the \(\mathcal{F}\) -free Sandwich Problem is a pair \((G_1,G_2)\) consisting of two graphs \(G_1\) and \(G_2\) with the same vertex set such that \(G_1\) is a subgraph of \(G_2\) , and the task is to determine an \(\mathcal{F}\) -free graph G containing \(G_1\) and contained in \(G_2\) , or to decide that such a graph does not exist. Initially motivated by the graph sandwich problem for trivially perfect graphs, which are the \(\{ P_4,C_4\}\) -free graphs, we study the complexity of the \(\mathcal{F}\) -free Sandwich Problem for sets \(\mathcal{F}\) containing two non-isomorphic graphs of order four. We show that if \(\mathcal{F}\) is one of the sets \(\left\{ \mathrm{diamond},K_4\right\} \) , \(\left\{ \mathrm{diamond},C_4\right\} \), \(\left\{ \mathrm{diamond},\mathrm{paw}\right\} \) , \(\left\{ K_4,\overline{K_4}\right\} \) , \(\left\{ P_4,C_4\right\} \) , \(\left\{ P_4,\overline{\mathrm{claw}}\right\} \) , \(\left\{ P_4,\overline{\mathrm{paw}}\right\} \) , \(\left\{ P_4,\overline{\mathrm{diamond}}\right\} \) , \(\left\{ \mathrm{paw},C_4\right\} \) , \(\left\{ \mathrm{paw},\mathrm{claw}\right\} \) , \(\left\{ \mathrm{paw},\overline{\mathrm{claw}}\right\} \) , \(\left\{ \mathrm{paw},\overline{\mathrm{paw}}\right\} \) , \(\left\{ C_4,\overline{C_4}\right\} \) , \(\left\{ \mathrm{claw},\overline{\mathrm{claw}}\right\} \) , and \(\left\{ \mathrm{claw},\overline{C_4}\right\} \) , then the \(\mathcal{F}\) -free Sandwich Problem can be solved in polynomial time, and, if \(\mathcal{F}\) is one of the sets \(\left\{ C_4,K_4\right\} \) , \(\left\{ \mathrm{paw},K_4\right\} \) , \(\left\{ \mathrm{paw},\overline{K_4}\right\} \) , \(\left\{ \mathrm{paw},\overline{C_4}\right\} \) , \(\left\{ \mathrm{diamond},\overline{C_4}\right\} \) , \(\left\{ \mathrm{paw},\overline{\mathrm{diamond}}\right\} \) , and \(\left\{ \mathrm{diamond},\overline{\mathrm{diamond}}\right\} \) , then the decision version of the \(\mathcal{F}\) -free Sandwich Problem is NP-complete.

Enhanced indexing using weighted conditional value at risk Abstract We propose an enhanced indexing portfolio optimization model that not only seeks to maximize the excess returns over and above the benchmark index but simultaneously control the risk by introducing a constraint on the weighted conditional value at risk (WCVaR) of the portfolio. The constraint in the proposed model can be seen as hedging the risk described by WCVaR of the portfolio. To carry out a comparative analysis of the proposed model, we also suggest an enhanced indexing CVaR model. We analyze the performance of the proposed model at various risk levels on eight publicly available financial data sets from Beasley OR library, and S&P 500, S&P BSE 500, NASDAQ composite, FTSE 100 index, and their constituents, for average returns, Sharpe ratio, and upside potential ratio. Empirical analysis exhibits superior performance of the portfolios from the proposed WCVaR model over the respective benchmark indices and additionally the optimal portfolios obtained from various other enhanced indexing models that exist in the literature. Furthermore, we present evidence of better performance of WCVaR model over the CVaR model for long-term investment horizons.