Optimizing of nonlinear and NP-hard logistic problems
concerning uncertainty, time dependency and prospect theory
Péter Földesi CSc.
Széchenyi István University
Győr
Hungary
2011.
CONTENT
1. Introduction
2. The objective of the research
3. Results of the research
4. Summary and future research
References
2
1. Introduction
The logistic (transportation) system is a set of complex human, social, economic, and political
interactions, and traditionally probability theory has been used for handling the obvious uncertainty. On
the basis of recent research possibility theory offers a more useful way of handling the uncertain
situations that often arise in logistic (transportation) analysis [20,27,39]. The two classes of uncertainty
in transportation are vagueness, which is associated with the lack of clarity of the definition, such as the
loss caused by a delivery delay, and ambiguity, which is associated with the lack of clarity in
information, e.g., the predicted cost elements are often ambiguous.
Considering real road transport, logistic networks, especially in city logistics, the actual circumstances
and condition of the transit process are subject to not only the topography of the given network but to
timing as well. Referring to the phenomenon of cyclic peak-hours and also to the weekly (monthly,
yearly) periodicity of traffic on road, it can be stated that the unit cost of traveling is also a variable, and
it can be described as a time series rather than a constant [10].
Obviously the physical distances can be considered as constant values in a given relation, but transit
times are subject to external factors [3]. Furthermore the actual costs are rarely constant and predictable,
so fuzzy cost coefficient can be applied in order to represent the uncertainty. In modern logistics systems
uncertainty and inaccuracy are not tolerated due to the widespread just-in-time approach, service time
windows are considered [21]. Multiobjective models have been presented [11,23] concerning a set of
alternative solutions.
Time become an important factor of logistics service level. The satisfaction of customers has – in
general terms - nonlinear characteristics, moreover, when time is included in the optimization, subjective
elements have a significant affect on evaluation, so prospect theory must be involved in the analysis.
The typical logistic optimization problem are very often discrete and NP-hard tasks, so the efficiency of
the applied heuristics, (e.g., evolutionary algorithms) is a critical issue of the problem.
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2. The objective of the research
Considering the special features of the logistic processes dedicated optimising principles, strategies,
techniques and tools must be set in order to explore methods that can be provide appropriate means for
practical use.
The main focus of the research is to identify the principal difference between the general optimization
theory and logistic optimization, such as the nature and uncertainty of input data, the nonlinear
behaviour of the objective functions, and also the side-effects of the decision-makers’ subjective
assessment, the problems that have to be solved when NP-hard problems describes the real life tasks.
There are many artificial intelligence methods proposed in the literature, and each of them has its own
advantage (and also disadvantage), so after classifying and evaluating them it is vital to determine their
possible usability for logistic tasks. Obviously the tools must be improved either by extending the
original model (e.g., fuzzy extension), or redefining the restriction and premises (e.g., „road transport
TSP2). In some cases the restructured model requires new solutions in technical terms (e.g., the training
of fuzzy neural networks, reconsidering gene transfer strategy, introducing fuzzy fitness even in case of
crisp problems), thus adequate methodology must be developed and tested in order to improve the
efficiency.
3. Results of the research
Thesis 1.
Considering the uncertainty of relevant data of logistic processes it can be stated, that one’s estimated
travel time by automobile between two points has possibilistic features due to the imprecision and
perception of measurement. It can be seen that the circumstances and conditions are significantly
changing in time that is the actual value of a given cost matrix element cij should be subject to the timing
of transit between nodei and nodej and an appropriate representation of imprecision can be done with
use of fuzzy numbers. In this sense geographical optimization alone is not appropriate, and the road
transport operation has to be scheduled in time as well, since the actual cost of the trip from nodei to
nodej is determined not only by the selection of edges [3,10].
For solving the above-mentioned fuzzy road-transport traveling salesman problem (FRTTSP) I
suggested an eugenic bacterial memetic algorithm (EBMA) since that algorithm is suitable for global
optimization of even non-linear, high-dimensional, multi-modal and discontinuous problems. As
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numerical example a modified TSP (FRTTSP) instance is considered, in which the elements of cost
matrix are dependent on the time elapsed from the beginning of the given operation, and loss aversion of
the decision maker is taken into consideration as well.
Theses 2.
Background and reasoning
In case of optimizing transport processes (e.g., TSP), when the distances between the cities are described
by fuzzy numbers, it must be discussed how these fuzzy numbers are summed up in a tour in order to
calculate the total distance. The arithmetic of fuzzy numbers is based on the extension principle [20]. I
use triangular shaped fuzzy numbers, which can be characterized by three values to represent the
boundaries of the support and the core value. When the total distance of a tour is being calculated, then
instead of adding fuzzy numbers by the extension principle, we can do an easier calculation based on the
defuzzified values of the fuzzy numbers. Some defuzzification methods have invariance properties
meaning that the result is invariant under linear transformations, thus there is no need to determine the
whole outcome using the extension principle but only to compute the sum of the defuzzified values of
each fuzzy number. If we are using triangular shaped fuzzy numbers then the Averaging Level Cuts
(ALC) type of defuzzification method used in [33] gives the same result as the Center of Gravity (COG)
method. So, in the first step the fuzzy numbers are defuzzified by the COG method (which is simply the
arithmetic mean of the three characteristic points of the fuzzy number) and then these crisp numbers are
summed up providing the total distance of the tour.
The uncertainty of logistic optimizing problems cannot be handled properly if only defuzzified values
are considered for evaluating the cost of the operation. I propose two other approaches for the
evaluation of the costs based not only on the geographical length described by the defuzzified values but
the uncertainty involved in the fuzzy numbers as well. Since now we need not only the defuzzified values
but – e.g., in case of TSP- the length of the tour as a fuzzy number, the fuzzy distances along the tour
have to be summed up. As we have triangular fuzzy numbers described by their three breakpoints, the
addition can be easily computed by summing the corresponding breakpoints [3,10].
Thesis 3. :
Background and reasoning
The special features of road transportation and supply chains encourage the modification of the classical
TSP, eliminating most of the original constraints, but making the problem more complicated in some
sense. Solutions in the literature are devoted to the classical problem, so after redefining the TSP and
transforming it to FRTTSP a novel approach is proposed. In the FRTTSP the costs between the nodes
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may depend on time and they have imprecise values involving uncertainties modeling the real life
processes.
These uncertainties can be represented by fuzzy numbers, which are capable of expressing the loss
aversion as well, since the symmetric and asymmetric features of a fuzzy number and the peak value support length rate can give a base for parametric quantification of the expected subjective loss
[4,5,8,10].
For the solution of the FRTTSP the eugenic bacterial memetic algorithm was proposed. This approach
combines the bacterial evolutionary algorithm performing a global search with local search techniques
and improving the candidate solutions in order to speed up the evolutionary process. The simulation
results confirm the effectiveness of the proposed technique. On the other hand the experiments
emphasize the importance of fine-tuning the model parameters responsible for converting the nature of
human thinking into numerical representation.
Thesis 4.:
Background and reasoning
For designing and developing logistic processes and outputs it is vital to know the relevancy of the
performance generated by each technical attribute and how they can increase customer satisfaction.
Improving the parameters of technical attributes requires financial resources, and the budgets are
generally limited. Thus the optimum target can be the achievement of the minimum overall cost for a
given satisfaction level. Kano’s quality model classifies the relationships between customer satisfaction
and attribute-level performance and indicates that some of the attributes have a non-linear relationship to
satisfaction, rather power-function should be used. For the customers’ subjective evaluation these
relationships are not deterministic and are uncertain. Also the cost function is uncertain, where the loss
aversion of decision makers should be considered as well [12,28].
Customer satisfaction is the key element of profitability and in the first step of achieving this satisfaction
is based in the designing and resource allocating process. The customers’ assessment of technical
attributes is very uncertain especially at the beginning of product life cycle so in Kano’s model the
exponents of satisfaction functions cannot be considered as deterministic values. I proposed the fuzzy
extension of the model in order to explore the possible alternative sets of technical attributes [4,8].
In the numerical example the fuzzy solution is significantly different from the crisp (deterministic)
version, not only in terms of total cost but – what is more important – in terms of technical attribute
levels. The benefit of fuzzy extension and the interpretation of loss aversion can be measured by the
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advantage we obtain analyzing the outputs. Difference between overall cost values is to be considered,
but what is more important the structure of technical attribution has to be examined.
Thesis 5. :
Background and reasoning
Time has limits; consumers have become time-sensitive [29, 30]. The time necessary to obtain a
product/service (access time) is involved in product utility to an increasing extent, the assurance of
which is the task of logistics [18,32]. There are more reasons for the shortening of this access time; one
of the most important is the change in customer expectations. Increasing rapidity is also encouraged by
the sellers in the competition against each other based on time, because of the pressure to reduce costs
and inventory, and to increase the efficiency and customer satisfaction [13]. Customer satisfaction is
determined by human factors as well, based on the loss aversion and impatient features of human
thinking: “future utility is less important” [22, 26].
The affects of time sensitivity have objective and subjective features. The utility of possessing goods in
time, and the accuracy of delivery times can be described by univariate functions, also the cost of
performing that given lead-time can be considered as a hyperbolic function. When maximizing the
utility-cost ratio, another, subjective element can be embedded in the model, by extending the meaning
of the power functions used, and time dependent exponents are used. The overall effects of that kind of
impatience can be detected by using simulations. Particle swarm optimization is an efficient method to
explore the side effects of loss aversion that are turned out to be digressive [4, 5].
Thesis 6.
Background and reasoning
In a non-linear search space the application of analytical approaches is limited; a heuristic search can be
used. The fuzzy extensions for heuristic based optimizing algorithms often face the problem of an
increased number of calculations required to find the solutions, since even in the case of the simplest
fuzzy number (the triangular one) the necessary calculation is three times larger when considering the
core value and supporting interval as minimum information. Moreover when the fuzzy power function
has to be computed the use of the extension principle requires a large number of calculations that makes
the run of a heuristic search - such as bacterial evolutionary algorithm - very slow. In the literature it is
shown that in some cases extension principle is hard to calculate, thus for practical reasons a simple
parametric representation of fuzzy power function must be used, in order to keep the required
computation time and resources at a reasonable level.
The approximation of fuzzy arithmetic operators are proposed in the literature [20] as the application of
the exact extension principle. These solutions require relatively large computational effort and
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resources, which is a key issue especially in evolutionary algorithms. Soft computing applications set
different requirements regarding the representation of uncertain, fuzzy values. These requirements are
based on the nature of uncertainty and fuzziness, and the characteristics and features of applied
algorithms must be assessed as well.
There are several methods that can be a theoretic foundation of parametric representation, but in the
case of the power function and fuzzy exponents the asymmetric features of function values must be
handled as well, since for practical reasons (computation time and resources) the continuous formulas
cannot be used efficiently. I proposed a simple solution for parametric representation of fuzzy power
function was presented that enables the decision-maker to fit the model to any defuzzification technique
by calculating and selecting the appropriate λ parameter. Also the appropriate selection of the number
and position of α-cuts can fine-tune the model and can give a proper representation of a real life
decision situation [1, 8].
Thesis 7.
Background and reasoning
Effective management of a supply chain has been increasingly recognized as a key factor in
differentiating product and service offerings and gaining competitive advantage for firms. It demands
close integration of internal functions within a firm and effective linkages with the external operations of
channel members in the chain. Coordination between different firms in the supply chain is key to its
effective implementation. It is necessary for firms to understand their supply chain activities and the
associated performance impact on the other member firms in the supply chain. Goal for firms in
achieving a competitive edge is to manage their supply chain performance (SCP) to gain advantages in
cost and service differentiation [15, 16, 19].
There are three levels of performance measurement at the company: strategic, tactical and operational.
First, at strategic level, there is a Balanced Scorecard (BSC) performance measurement tool used by the
management of the companies. The main area of use should be the prediction of important BSC
indicators based on inputs that can be forecasted or guessed. With the altering of those inputs an optimal
solution for the fulfilment of the BSC indicator can be chosen.
The selected tool should be able to predict the strategic level result based on operational level inputs.
The operational level inputs can be “manipulated” in short term (for example not allowing overtime for
the workers) but strategic level outcomes must be accepted and altering them in short term is not
possible.
Based on the complex and inter-related connection between the inputs and outputs in logistic systems
my proposal is to use artificial fuzzy neural networks as a computational model for establishing
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connection between strategic and operational level indicators. I proposed a novel computation of fuzzy
exponent in the sigmoid functions of fuzzy neural networks. The proposed fuzzy neural network applies
fuzzy input signals and crisp connection weights in the network’s hidden and output layers. The applied
calculation of fuzzy exponent is based on a parametric representation of the fuzzy exponent that is able
to provide a crisp output instead of the extension principle’s fuzzy output and requires less
computational effort than the learning based on  -cuts. [1, 2]
For the training of the network [34] the bacterial memetic algorithm was applied. The method was tested
on a benchmark problem and on two real datasets. The practical use of trained FNN is that the decisionmakers are able to check different scenarios concerning the future operation circumstances. If some of
the input parameters that cannot be affected by the operator are forecasted then the rest of the parameters
can be set at an appropriate level so that the required output can be achieved.
Thesis 8.
Background and reasoning
Evolutionary methods and in particular Bacterial Memetic Algorithms (BMA) are widely adopted
means of population based metaheuristics, which have the ability to perform robust search on a discrete
problem space, which very often arise in logistic optimisation [24]. These methods are categorized as
black-box search heuristics and tend to be quite good at finding generally good approximate solutions on
certain problem domains such as the Traveling Salesman Problem and Bin-packing Problem. The
effectiveness of the BMA can be improved by selecting appropriate gene transfer strategy and fitness
function [35]. The main goal was to evaluate different infection strategies on a certain NP-complete
combinatorial optimization task and perform simulations on multiple problem instances. Another goal
was to select the most promising infection strategy [36, 38]. Concerning the fitness function, the main
focus is on the performance differences when it comes to the fuzzy and crisp versions of fitness
calculations.
Bacterial memetic algorithms are widely used on discrete combinatorial problems, which are essential
in the field of logistics and forwarding, such as the well known Traveling Salesman Problem. The
original Bacterial Evolutionary Algorithm proposed by Nawa and Furuhashi [31, 37] has a predefined
set of operators such as bacterial mutation and gene transfer also known as infection. The traditional
bacterial infection operator is proven to be far from optimal. An altemative gene transfer operator that
is applied on the metric Traveling Salesman Problem can be more effective. This altemative infection
algorithm has superior rate of convergence while reducing the risk of getting stuck in a local optima [6,
7]. In NP-hard crisp optimizing processes if fuzzy fitness function is applied, the evolutionary algorithm
can result faster convergence even in the case of crisp problems, the mutation can be more effective by
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keeping “almost feasible” solutions [9].
Computational results show that the fuzzy additions to the fitness calculation have a positive effect on
the produced results. It was clearly shown that keeping “almost feasible” mutations by using fuzzy
fitness function can accelerate the convergence in terms of generations, and a trade-off must be set
finding the way of reducing the necessary computational effort.
4. Summary and future research
The brief summary of the theses is:
Considering the uncertainty of relevant data of logistic processes it can be stated that geographical
optimization alone is not appropriate, and the road transport operation has to be scheduled in time as
well, since the actual cost of the trip from nodei to nodej is determined not only by the selection of edges.
(Th.1)
The uncertainty of logistic optimizing problems cannot be handled properly if only defuzzified values
are considered for evaluating the cost of the operation. As we have triangular fuzzy numbers described
by their three breakpoints, the addition can be easily computed by summing the corresponding
breakpoints. (Th.2)
The uncertainties can be represented by fuzzy numbers, which are capable of expressing the loss
aversion as well, since the symmetric and asymmetric features of a fuzzy number and the peak value support length rate can give a base for parametric quantification of the expected subjective loss. (Th.3)
The customers’ assessment of technical attributes is very uncertain especially at the beginning of
product life cycle so in Kano’s model the exponents of satisfaction functions cannot be considered as
deterministic values. Fuzzy extension of the model is able to explore the possible alternative sets of
technical attributes. (Th.4)
The affects of time sensitivity have objective and subjective features. The utility of possessing goods in
time and the accuracy of delivery times can be described by univariate functions, also the cost of
performing that given lead-time can be considered as a hyperbolic function. Subjective element can be
embedded in the model, by extending the meaning of the power functions used, and time dependent
exponents are used. The overall effects of that kind of impatience can be detected by using simulations.
(Th.5)
10
In the case of the power function and fuzzy exponents the asymmetric features of function values must
be handled. A simple solution for parametric representation of fuzzy power function enables the
decision-maker to fit the model to any defuzzification technique by calculating and selecting the
appropriate λ parameter. Also the appropriate selection of the number and position of α-cuts can finetune the model and can give a proper representation of a real life decision situation. (Th.6)
Based on the complex and inter-related connection between the inputs and outputs in logistic systems
my proposal is to use artificial fuzzy neural networks as a computational model for establishing
connection between strategic and operational level indicators. I proposed a novel computation of fuzzy
exponent in the sigmoid functions of fuzzy neural networks. The applied calculation of fuzzy exponent
is based on a parametric representation of the fuzzy exponent. (Th.7)
Bacterial memetic algorithms are widely used on discrete combinatorial problems, which are essential in
the field of logistics and forwarding. The traditional bacterial infection operator is proven to be far from
optimal. An altemative gene transfer operator that is applied on the metric Traveling Salesman Problem
can be more effective. This altemative infection algorithm has superior rate of convergence while
reducing the risk of getting stuck in a local optima. (Th.8)
In NP-hard crisp optimizing processes if fuzzy fitness function is applied, the evolutionary algorithm
can result faster convergence even in the case of crisp problems, the mutation can be more effective by
keeping “almost feasible” solutions. (Th.8)
The main focus of the future research:
Future research is concerned with the parameter setting of the heuristics. The target is to minimize the
required level of apriori knowledge. Evolutionary algorithms have several setting parameters, and the
appropriate initial set can make the optimizing process more efficient, that is the required computational
effort (running time, memory) can be reduced. By embedding the algorithm parameters into the
algorithm itself an evolutionary procedure can be launch for finding the best parameter settings. This
task is not merely question of heuristic techniques, thorough investigation must be done in order to
explore and identify the prospective solutions. Another crucial point is to explore the affect of prospect
theory and loss aversion on the efficiency of human beings’ decision. The question is, that in complex
human-machine systems whether the presence of prospect theory is beneficial or not. Comparative
investigations and tests must be carried out to answer the question.
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