Marshall L. Fisher

The Lagrangian Relaxation Method for Solving Integer Programming Problems

(This article originally appeared in Management Science, January 1981, Volume 27, Number 1, pp. 1-18, published by The Institute of Management Sciences.) One of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints. Dualizing the side constraints produces a Lagrangian problem that is easy to solve and whose optimal value is a lower bound (for minimization problems) on the optimal value of the original problem. The Lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm. This approach has led to dramatically improved algorithms for a number of important problems in the areas of routing, location, scheduling, assignment and set covering. This paper is a review of Lagrangian relaxation based on what has been learned in the last decade.

An analysis of approximations for maximizing submodular set functions--I

LetN be a finite set andz be a real-valued function defined on the set of subsets ofN that satisfies z(S)+z(T)≥z(S⋃T)+z(S⋂T) for allS, T inN. Such a function is called submodular. We consider the problem maxS⊂N{a(S):|S|≤K,z(S) submodular}.Several hard combinatorial optimization problems can be posed in this framework. For example, the problem of finding a maximum weight independent set in a matroid, when the elements of the matroid are colored and the elements of the independent set can have no more thanK colors, is in this class. The uncapacitated location problem is a special case of this matroid optimization problem.We analyze greedy and local improvement heuristics and a linear programming relaxation for this problem. Our results are worst case bounds on the quality of the approximations. For example, whenz(S) is nondecreasing andz(0) = 0, we show that a “greedy” heuristic always produces a solution whose value is at least 1 −[(K − 1)/K]K times the optimal value. This bound can be achieved for eachK and has a limiting value of (e − 1)/e, where e is the base of the natural logarithm.

A generalized assignment heuristic for vehicle routing

Abstract : We consider a common variant of the vehicle routing problem in which a vehicle fleet delivers products stored at a central depot to satisfy customer orders. Each vehicle has a fixed capacity, and each order uses a fixed portion of vehicle capacity. The routing decision involves determining which of the demands will be satisfied by each vehicle and what route each vehicle will follow in servicing its assigned demand in order to minimize total delivery cost. We present a heuristic for this problem in which an assignment of customers to vehicles is obtained by solving a generalized assignment problem with an objective function that approximates delivery cost. This heuristic has many attractive features. It has outperformed the best existing heuristics on a sample of standard test problems. It will always find a feasible solution if one exists, something no other existing heuristic can guarantee. It can be easily adapted to accommodate many additional problem complexities. By parametrically varying the number of vehicles in the fleet, our method can be used to optimally solve the problem of finding the minimum size fleet that can feasibly service the specified demand.

Reducing the Cost of Demand Uncertainty Through Accurate Response to Early Sales

Traditionally, fashion products have incurred high losses due to stockouts and inventory obsolence because long lead times coupled with a concentrated selling season force all or at least most production to be committed before demand information is available. Under a Quick Response system, lead times are shortened sufficiently to allow a greater portion of production to be scheduled in response to initial demand. We model and analyze the decisions required under Quick Response and give a method for estimating the demand probability distributions needed in our model. We applied these procedures with a major fashion skiwear firm and found that cost relative to the current informal response system was reduced by enough to increase profits by 60%. Relative to the cost that would have been incurred if no response were used, optimized response reduces cost by enough to roughly quadruple profits.

Coordination of production and distribution planning

Abstract This paper is a computational study to investigate the value of coordinating production and distribution planning. The particular scenario we consider concerns a plant that produces a number of products over time and maintains an inventory of finished goods at the plant. The products are distributed by a fleet of trucks to a number of retail outlets at which the demand for each product is known for every period of a planning horizon. We compare two approaches to managing this operation, one in which the production scheduling and vehicle routing problems are solved separately, and another in which they are coordinated within a single model. The two approaches are applied to 132 distinct test cases with different values of the basic model parameters, which include the length of the planning horizon, the number of products and retail outlets, and the cost of setups, inventory holding and vehicle travel. The reduction in total operating cost from coordination ranged from 3% to 20%. These results indicate the conditions under which companies should consider the organizational changes necessary to support coordination of production and distribution.

Optimal Solution of Vehicle Routing Problems Using Minimum K-Trees

We consider the problem of optimally scheduling a fleet of K vehicles to make deliveries to n customers subject to vehicle capacity constraints. Given a graph with n + 1 nodes, a K-tree is defined to be a set of n + K edges that span the graph. We show that the vehicle routing problem can be modeled as the problem of finding a minimum cost K-tree with two K edges incident on the depot and subject to some side constraints that impose vehicle capacity and the requirement that each customer be visited exactly once. The side constraints are dualized to obtain a Lagrangian problem that provides lower bounds in a branch-and-bound algorithm. This algorithm has produced proven optimal solutions for a number of difficult problems, including a well-known problem with 100 customers and several real problems with 25–71 customers.

A dual algorithm for the one-machine scheduling problem

A branch and bound algorithm is presented for the problem of schedulingn jobs on a single machine to minimize tardiness. The algorithm uses a dual problem to obtain a good feasible solution and an extremely sharp lower bound on the optimal objective value. To derive the dual problem we regard the single machine as imposing a constraint for each time period. A dual variable is associated with each of these constraints and used to form a Lagrangian problem in which the dualized constraints appear in the objective function. A lower bound is obtained by solving the Lagrangian problem with fixed multiplier values. The major theoretical result of the paper is an algorithm which solves the Lagrangian problem in a number of steps proportional to the product ofn2 and the average job processing time. The search for multiplier values which maximize the lower bound leads to the formulation and optimization of the dual problem. The bounds obtained are so sharp that very little enumeration or computer time is required to solve even large problems. Computational experience with 20-, 30-, and 50-job problems is presented.

An Econometric Analysis of Inventory Turnover Performance in Retail Services

Inventory turnover varies widely across retailers and over time. This variation undermines the usefulness of inventory turnover in performance analysis, benchmarking, and working capital management. We develop an empirical model using financial data for 311 publicly listed retail firms for the years 1987-2000 to investigate the correlation of inventory turnover with gross margin, capital intensity, and sales surprise (the ratio of actual sales to expected sales for the year). The model explains 66.7% of the within-firm variation and 97.2% of the total variation (across and within firms) in inventory turnover. It yields an alternative metric of inventory productivity, adjusted inventory turnover, which empirically adjusts inventory turnover for changes in gross margin, capital intensity, and sales surprise, and can be applied in performance analysis and managerial decision making. We also compute time trends in inventory turnover and adjusted inventory turnover, and find that both have declined in retailing during the 1987-2000 period.

Demand Estimation and Assortment Optimization Under Substitution: Methodology and Application

Assortment planning at a retailer entails both selecting the set of products to be carried and setting inventory levels for each product. We study an assortment planning model in which consumers might accept substitutes when their favorite product is unavailable. We develop an algorithmic process to help retailers compute the best assortment for each store. First, we present a procedure for estimating the parameters of substitution behavior and demand for products in each store, including the products that have not been previously carried in that store. Second, we propose an iterative optimization heuristic for solving the assortment planning problem. In a computational study, we find that its solutions, on average, are within 0.5% of the optimal solution. Third, we establish new structural properties (based on the heuristic solution) that relate the products included in the assortment and their inventory levels to product characteristics such as gross margin, case-pack sizes, and demand variability. We applied our method at Albert Heijn, a supermarket chain in The Netherlands. Comparing the recommendations of our system with the existing assortments suggests a more than 50% increase in profits.

Assortment Planning: Review of Literature and Industry Practice

A retailer’s assortment is defined by the set of products carried in each store at each point in time. The goal of assortment planning is to specify an assortment that maximizes sales or gross margin subject to various constraints, such as a limited budget for purchase of products, limited shelf space for displaying products, and a variety of miscellaneous constraints such as a desire to have at least two vendors for each type of product.