Promotion Optimization for Multiple Items in Supermarkets

Sales promotions are important in the fast-moving consumer goods (FMCG) industry due to the significant spending on promotions and the fact that a large proportion of FMCG products are sold on promotion. This paper considers the problem of planning sales promotions for an FMCG product in a grocery retail setting. The category manager has to solve the promotion optimization problem (POP) for each product, i.e., how to select a posted price for each period in a finite horizon so as to maximize the retailer’s profit. Through our collaboration with Oracle Retail, we developed an optimization formulation for the POP that can be used by category managers in a grocery environment. Our formulation incorporates business rules that are relevant, in practice. We propose general classes of demand functions (including multiplicative and additive), which incorporate the post-promotion dip effect, and can be estimated from sales data. In general, the POP formulation has a nonlinear objective and is NP-hard. We then propose a linear integer programming (IP) approximation of the POP. We show that the IP has an integral feasible region, and hence can be solved efficiently as a linear program (LP). We develop performance guarantees for the profit of the LP solution relative to the optimal profit. Using sales data from a grocery retailer, we first show that our demand models can be estimated with high accuracy, and then demonstrate that using the LP promotion schedule could potentially increase the profit by 3%, with a potential profit increase of 5% if some business constraints were to be relaxed. The online appendix is available at https://doi.org/10.1287/opre.2016.1573

Introduction to distribution logistics

The location-allocation problem of manifolds, which is a part of subsea field layout optimization, directly affects the flowline cost. This problem has always been studied as a mixed-integer nonlinear programming (MINLP) problem, or an integer linear programming (ILP) problem when there are location options for the facilities. Making a MINLP model is surely convenient to interpret the optimization problem. However, finding the global optimum of the MINLP model is very hard. Hence, practically, engineers use approximation algorithms to search a good local optimum or give several good location options based on their experience and knowledge to reduce the MINLP model into an ILP model. Nevertheless, the global optimum of the original MINLP model is no longer guaranteed. In this study, enlightened by the graphic theories, we propose a new method in which we reduce the MINLP model into an ILP model---more precisely, a binary linear programming (BLP) model---without compromise of achieving global optimum, but also with extremely high efficiency. The breakthrough in both efficiency and accuracy of our method for the location-allocation problem of manifolds and wellheads is well demonstrated in various cases with comparison to the published methods and the commercial MINLP solver from LINDO. Besides, we also provide our results for larger-scale problems which were considered infeasible for the commercial MINLP solver. More generally, our method can be regarded as a specific MINLP/NIP (nonlinear integer programming) solver which can be used for many other applications. This work is the second of a series of papers which systematically introduce an efficient method for subsea field layout optimization to minimize the development cost.

Ranking in quadratic integer programming problems

Algorithms and computer-based tools for analyzing infeasible linear and nonlinear programs have been developed in recent years, but few such tools exist for infeasible mixed-integer or integer linear programs. One approach that has proven especially useful for infeasible linear programs is the isolation of an Irreducible Infeasible Set of constraints (IIS), a subset of the constraints defining the overall linear program that is itself infeasible, but for which any proper subset is feasible. Isolating an IIS from the larger model speeds the diagnosis and repair of the model by focussing the analytic effort. This paper describes and tests algorithms for finding small infeasible sets in infeasible mixed-integer and integer linear programs; where possible these small sets are IISs.

The use of an internal rate of return (IRR) measure is cammon in financial problems. Over the past decade the measure has been enjoying an increased usage in the new municipal debt issue market. Nauss (Management Science, July 1986) developed a procedure to minimize the IRR for competitive bids for new issues of municipal debt. While this problem was generally viewed as being a nonlinear integer program, it was shown that the problem could be linearized so that an integer linear program resulted. The linearization is possible because there is only one change in the sign of the cash flows over time. This assures that only one real root for the IRR exists. A special purpose branch and bound algorithm was developed to solve the problem in a matter of seconds so that it could be used in actual applications.

A system with several actors will increase the possibilities for collaboration and therefore, in many cases also increase the possibilities to affect the operation with economic and environmental benefits. In this paper a district heating system is studied with purpose to create economic and environmental operation guidelines in favour of the involved actors. The company LuleKraft owns a combined heat and power plant, while the four heating stations together with the DH network are owned by Luleå Energi. District heat is produced from the CHP plant and the four heating stations which are fired with process gases from an integrated steel plant, oil, wood pellets or electricity. The heat demand in the system is strongly depending on the outdoor temperature. In this paper, the system is modelled with Mixed Integer Non Linear Programming in order to optimise the profit and CO2 emission. A comparison between actual process data and modelled results is performed. A pareto front is derived to show the trade-off between economic benefits and CO2 emissions. It is found that the main suggestions, under conditions for optimised profit, are (1) to prioritize effective heat production instead of electricity production at cold outdoor temperatures and (2) redistribution of the accessible process gases between the CHP plant and one of the heating stations. This will lead to an improved operation strategy, resulting in increased profit and reduced energy consumption.

Communication among isolated networks (clusters) in delay tolerant networks (DTNs) can be supported by a message ferry, which collects bundles from clusters and delivers them to a sink node. When there are lots of distant static clusters, multiple message ferries and sink nodes will be required. In this paper, we aim to make groups each of which consists of physically close clusters, a sink node, and a message ferry. Our main objective is minimizing the overall mean delivery delay of bundles in consideration of both offered load of clusters and distance between clusters and their sink nodes. We first model this problem as a nonlinear integer programming, based on the knowledge obtained in our previous work. Because it might be hard to solve this problem directly, we take two-step optimization approach based on linear integer programming, which yields an approximate solution of the problem. Through numerical results, we show the two-step optimization approach works well.

The size of imperfect cartels, i.e. with participating (k-firms) and non-participating or fringe (j-firms), has been investigated by some researchers since early 1980s. Most papers, based on simple cost and demand functions, have proposed various conditions to find the optimal number of k-firms. Recently, other researchers have conducted numerical simulations or used an analytic approach to determine the size of stable cartels. In this paper, I investigate two standard models, with additional parameters in demand and cost functions. In the first model the cartel faces j competitive firms, while in the second model the cartel is the Stackelberg leader and the j-firms are Cournot followers. In the second model, the entry to and exit from cartel are formulated as integer non-linear programming, using some parameter bounds. Contrary to other studies who relied on simplified functions and found exact algebraic conditions, Mathematica fails to provide such conditions. It does provide though global numerical solutions. In the first model, irrespectively of the number of firms, the optimal number of k-firms is always three. In the second model, the number of k-firms is moderate higher, but lower than other studies find.

The cutting stock problem is that of finding a cutting of stock material to meet demands for small pieces of prescribed dimensions while minimising the amount of waste. Because changing over from one cutting pattern to another involves significant setups, an auxiliary problem is to minimise the number of different patterns that are used. The pattern minimisation problem is significantly more complex, but it is of great practical importance. In this paper, we propose an integer programming formulation for the problem that involves an exponential number of binary variables and associated columns, each of which corresponds to selecting a fixed number of copies of a specific cutting pattern. The integer program is solved using a column generation approach where the subproblem is a nonlinear integer program that can be decomposed into multiple bounded integer knapsack problems. At each node of the branch-and-bound tree, the linear programming relaxation of our formulation is made tighter by adding super-additive inequalities. Branching rules are presented that yield a balanced tree. Incumbent solutions are obtained using a rounding heuristic. The resulting branch-and-price-and-cut procedure is used to produce optimal or approximately optimal solutions for a set of real-life problems.

In this work, a supply chain with coordination mechanism consisting single vendor and buyer is considered. Further, instead of price sensitive linear or deterministic demand function, a price-sensitive non-linear demand function is introduced in this paper. To find the inventory cost, penalty cost and transportation cost, it is considered that the production and shipping functions of vendor are continuously harmonized and occur at the same rate. In this integrated supply chain, the Buyer’s Linear Program (LP), vendor’s Integer Program (IP) and coordinated Mixed Integer Program (MIP) models are formulated. Numerical examples presented in this research, which include the sensitivity of the key parameters to illustrate the models. The solution procedures demonstrate that the individual profit as well as joint profit could be increased by coordination mechanism even though the demand function is non-linear. In addition, the results illustrate that Buyer’s selling price with consumers purchasing price could be decreased which may increase the demand of the end market. Finally, a conclusion is drawn in favor of the coordinated supply chain with non-linear price sensitive demand function.

This paper presents a metaheuristic approach to solve the transmission network expansion planning (TNEP) problem considering non-conventional solution candidates. The TNEP consists on finding the set of new elements required in a power system to meet a given future demand at a minimum cost. The TNEP traditionally considers as candidate solutions the addition of new lines and transformers. The main contribution of this work is the inclusion of non-conventional solution candidates. Such non-conventional solution candidates are namely: repowering of existing circuits and reactive shunt compensation. Also, an AC modeling of the network that allows obtaining more realistic results than the traditional DC model has been considered. The TNEP is represented by means of a non linear mixed integer programming problem which is solved through a hybrid genetic algorithm (HGA). Several tests were performed on two benchmark power systems to show the applicability and effectiveness of the proposed approach.

Genetic algorithm for non-linear mixed integer programming problems and its applications

Parametric Solution Algorithms for Large-Scale Mixed-Integer Fractional Programming Problems and Applications in Process Systems Engineering

A Survey of Lagrangean Techniques for Discrete Optimization