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Solving a Real-world Supply-Chain ManagementProblem Using a Bilevel Evolutionary Approach

COIN Report Number 2017009

Zhichao Lua, Kalyanmoy Deba, Erik Goodmana, John Wassickb

aMichigan State UniversitybThe Dow Chemical Company

Abstract

Supply-chain management problems are common to most industries andthey involve a hierarchy of subtasks, which must be coordinated well toarrive at an overall optimal solution. Such problems also involve more thanone stake-holders, such as the supplier, the transport companies that supplygoods from source to destinations, and multiple management levels withinthe supplier. Thus, these problems involve a hierarchy of decision-makers,each having its own objectives and constraints, but importantly requiringa coordination of their actions to make the overall supply chain processoptimal from cost and quality considerations. In this paper, we considera specific supply-chain management problem from an industry, which in-volves two levels of coordination: (i) yearly strategic planning in whicha decision on establishing an association of every destination point witha supply point must be made so as to minimize the yearly transportationcost, and (ii) weekly operational planning in which, given the associationbetween a supply and a destination point, a decision on the preference ofavailable transport carriers must be made for multiple objectives: minimiza-tion of transport cost and maximization of service quality and satisfactionof demand at each destination point. Thus, the resulting problem is bilevel,for which the operational planning problem is nested within the strategicplanning problem. The problem is also multi-objective in nature. Moreover,the problem involves several practical challenges, such as uncertainty indemand at each destination, non-linearity and non-differentiability of thecost model, large dimensions, and others. We propose a customized multi-objective bilevel evolutionary algorithm, which is computationally tractable.We then present results on state-level and ZIP-level accuracy (involvingabout 40,000 upper level variables) of destination points over the mainland

USA. We compare our proposed method with current non-optimizationbased practices and report a considerable cost saving.

1. Introduction

Supply chain management (SCM) problems are common in most inte-grated industries, as a product is made from several raw or pre-finishedsub-products and the product itself, after being produced, must be trans-ported to several destination locations. SCM problems come in differentcomplexities and involve multiple goals, such as minimizing transportationcost, adhering to scheduled delivery time, maximizing service quality, etc.Importantly, different sub-tasks are hierarchical and interconnected. Thus,it is not surprising that SCM problems are often treated as optimizationproblems [1, 8, 15].

In this paper, we consider a particular routine SCM problem in which aproduct needs to be supplied from multiple source locations (manufacturingplants or warehouses) to several destinations across mainland US states andZIP locations. The task is considered routine, as the product needs to besupplied according to a given demand every week for an entire year at eachdestination location. The task is hierarchical having at least two levels. Thestrategic level task is to identify (i) an association of each destination locationwith a specific source location and also (ii) an assignment of a set of transportcarrier companies with each source location. The strategic task is done lessfrequently, may be once a year. For a specific combination of these twosub-tasks fixed for the entire year, every source location is then aware of thedestination locations where the product needs to be supplied to and alsothe carrier companies that will deliver the products. Then, at the operationallevel, a decision needs to be made to allocate an optimal proportion of dailydemand to be transported by a carrier company to every destination location.This task needs to be done more frequently, may be once every week ordaily. Thus, the operational (or lower) level task has as many optimizationproblems as the source locations. At the strategic (or upper) level, there isan overriding objective of minimizing the overall transportation cost fordelivering products from every source to every destination location. Atthe lower level, there can be a single or two objectives: (i) minimizing thetransport cost associated with every source location for completing thedelivery task to all its associated destination locations, and (ii) maximizinga service quality of all carrier companies chosen to deliver the product fromthe specific source location. A little thought will reveal that this problem is a

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bilevel optimization problem involving a single objective at the upper leveland at most two objectives at the lower level. The problem is more complexthan an usual bilevel optimization problem, as the lower level task involvesmultiple optimization problems in parallel.

Bilevel optimization is commonly characterized as a mathematical pro-gramming problem that involves two nested levels of optimization. Theouter problem is usually denoted as the upper level optimization task andthe inner optimization task is usually denoted as the lower level problem. Inthe are of game theory, bilevel optimization problems were first introducedby Stackelberg [12] and came to be known as Stackelberg games; ThenBracken and McGill [2] from the area of mathematical programming laterpresented bilevel problems as a constrained optimization task, in which thelower level optimization problem acts as a constraint to the upper level opti-mization problem. The interest in bilevel programming has been driven bya number of applications arising in the field of operation research, includingtransportation [10, 4, 3], management [13], facility location [7, 14, 13]. Overthe past few years, evolutionary computation (EC) has become a promis-ing mean in solving bilevel optimization problems because of its flexibleframework and population approach, offer viable means to address the com-plexities that are otherwise difficult to handle using classical optimizationtechniques.

In this paper, we develop bilevel evolutionary optimization methodsto solve single and multi-objective version of the above-mentioned SCMproblem. The study is also significant as after considering one state-leveldestination location, we have considered ZIP-level destination locations,involving about 40,000 nodes. This consideration increases the problem sizeby 1,000-fold and makes the overall bilevel programming task challenging.Moreover, one of the trade-mark challenges of SCM problems is that demandto each destination is usually uncertain and that demand changes weeklyand seasonally. We have considered demand uncertainties as well andinvestigated its effect in arriving at robust solutions of the problem. With allthese considerations, this study brings evolutionary computation close toreal-world application of a routine supply chain management problem.

In the remainder of this paper, the specific supply chain managementproblem is described in Section 2. Details of source and destination loca-tions and associated transport carriers are mentioned as well. Thereafter, inSection 3, we formulate the SCM problem as a bilevel optimization problem.Section 4 describes the evolutionary bilevel optimization procedure devel-oped for solving the SCM problem. Then, Section 5 presents the simulationresults for both single-objective and multi-objective bilevel versions of the

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SCM problem. Advantages of using the multi-objective version is also de-scribed in this section. Thereafter, in Section 6, the demand uncertainty isconsidered and results are shown. The difference between robust solutionsand deterministic solutions are then highlighted. Section 7 discusses themodified algorithm for handling ZIP-level destination locations, therebyaddressing a large-scale version of the SCM problem. Finally, Section 8concludes this extensive study.

2. Problem Description and Modeling

Figure 1: An example of the upper level problem showing destination (states) to sourcelocation associations.

The supply chain management problem with two levels of interactionsis described in Figure 1. We provide more details of the problem in thefollowing subsections.

2.1. Source Locations and Their CharacteristicsIn total, 10 source locations all over mainland USA are considered, as

shown in Table 1. Each source location is considered to have infinite capacity,so that any quantity of product can be supplied from each source. Here, we

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Table 1: Ten assumed source locations.

Location # City State Abbrev.1 Saint Louis MO StL2 Los Angeles CA LA3 Augusta GA AUG4 New Orleans LA NOR5 Detroit MI DET6 Minneapolis MN MIN7 Charlotte NC CHA8 Philadelphia PA PHI9 Houston TX HOU

10 Seattle WA SEA

also assume that a single product is supplied from all source locations to alldestination locations.

2.2. Destination Locations and Their Demand ModelingWe consider two scenarios related to destination locations. In the first

scenario, we consider one destination location at the capital city of everystate, thereby making 48 different destination locations all over mainlandUSA. In the second scenario (discussed in Section 7, we have consideredone location at every ZIP level, thereby making a total 40,000 destinationlocations. At each location, following parameters are chosen based onrealistic data obtained from a particular industry:

A yearly average demand is set to be 42,400 in shipments.

The demand at each destination location is proportional to its popula-tion statistics.

Two types of demands are considered, seasonal and non-se