Solving Fuzzy Chance-Constrained

Programming with Ant ColonyOptimization-Based Algorithms and Application

to Fuzzy Inventory Model

Yuanguo Zhu

Department of Applied Mathematics, Nanjing University of Scienceand Technology Nanjing, Jiangsu 210094, China

ygzhu@mail.njust.edu.cn

Abstract. An ant colony optimization algorithm is designed to solvecontinuous optimization models. Based on this algorithm, a hybrid intel-ligent algorithm combined with fuzzy simulation and neural network isintroduced for solving fuzzy chance constrained models. Several numer-ical examples are given to show the algorithms effective. As an applica-tion, a fuzzy inventory model is established and solved with the hybridintelligent algorithm.

1 Introduction

Dorigo [1] started the study and applications of ant colony optimization algo-rithm (ACOA) in the early nineties last century. In ACOA, the computationalresources are allocated to a set of relatively simple agents (artificial ants) that ex-ploit pheromone communication. ACOA and its improved versions have demon-strated considerable success in providing optimal solutions to many complexoptimization problems and received more and more attentions during the pastdecade such as [2] [3] [4].

As we have seen, heuristic approaches are effective in finding solutions ofuncertain programming. Liu and Iwamura [5] and Liu [6] integrated fuzzy sim-ulation, neural network(NN), and genetic algorithm(GA) to produce a hybridintelligent algorithm for solving many kinds of uncertain programming includingfuzzy chance constrained programming. In this paper, we design an ant colonyoptimization algorithm for solving continuous optimization problems. Based onthis algorithm, we present a hybrid intelligent algorithm combined fuzzy simu-lation and neural network for solving fuzzy chance constrained models.

2 Ant Colony Optimization Algorithm

Consider the following optimization problem⎧⎨

⎩

min f(x1, x2, · · · , xn)subject to:

g(x1, x2, · · · , xn) ≤ 0(1)

D.-S. Huang, K. Li, and G.W. Irwin (Eds.): ICIC 2006, LNAI 4114, pp. 977–983, 2006.c© Springer-Verlag Berlin Heidelberg 2006

978 Y. Zhu

where x1, x2, · · · , xn are decision variables which are in the feasible set of con-straints Ω = {(x1, x2, · · · , xn) ∈ IRn | g(x1, x2, · · · , xn) ≤ 0}. Without loss ofgenerality, we let Ω is included in a hypercube L = {(x1, x2, · · · , xn) ∈ IRn |ai ≤ bi, i = 1, 2, · · · , n}. Suppose that the intervals [ai, bi] are averagely di-vided into ri subintervals, where ri are positive integers, i = 1, 2, · · · , n. Thesesubintervals are associated to pheromone trails τ .

Given a fixed point a ∈ IR (called start point) not in L, set a0 = b0 = a,r0 = 1. Connect a with all subintervals of [a1, b1], and connect each subintervalof [ai, bi] with all subintervals of [ai+1, bi+1], i = 1, 2, · · · , n − 1. Artificial antsbuild candidate solutions by movement on these connections. The proceduresare described as follows.

(1) Initialization Process: Randomly generate a feasible solution s′ and sets = s′. For each ant, set τi,j(0) = τ0, j = 1, 2, · · · , ri, i = 1, 2, · · · , n, where τ0 isa parameter, and 0 < τmin ≤ τ0 < +∞ for some fixed parameter τmin.

(2) Ant Movement: At each step i after building the sequence 〈x1, x2, · · · , xi−1〉,select the next subinterval of [ai, bi] in probability following

pi,j =τi,j(t)

ri∑

k=1

τi,k(t). (2)

Randomly produce a point v from the chosen subinterval, and mutate this pointto xi so that it is still in the interval [ai, bi]. The mutation is made as follows:given a number M > 0, randomly produce a direction γ; choose M ′ < M suchthat xi = v + γM ′ ∈ [ai, bi]. If the new sequence is infeasible, then the ant isdropped and its movement is terminated.

(3) Pheromone Update: At each moment t, let s = (x1, x2, · · · , xn) be the bestfeasible solution found so far. Reinforce the pheromone trails on subintervals inwhich corresponding coordinates of s are, and evaporate the pheromone trailson others:

τij(t) =

⎧⎪⎨

⎪⎩

(1 − ρ)τi,j(t − 1) + ρg(s), if xi is in the jth subintervalof [ai, bi],

(1 − ρ)τi,j(t − 1), otherwise,

(3)

where ρ, 0 < ρ < 1 is the evaporation rate, and g(s), 0 < g(s) < +∞ is afunction with that g(s) ≥ g(s′) if f(s) < f(s′).

3 Fuzzy Chance-Constrained Programming

We first recall some concepts in fuzzy environment [6]. Let Θ be a nonempty set,and P(Θ) the power set of Θ. A function Pos : P(Θ) → IR is called a possibilitymeasure if (i) Pos{Θ} = 1, (ii) Pos{∅} = 0, (iii) Pos{∪iAi} = supi Pos{Ai}for any collection {Ai} in P(Θ). Then the triplet (Θ,P(Θ), Pos) is called apossibility space. The credibility measure (Liu and Liu [7]) of an element A in

Solving Fuzzy Chance-Constrained Programming 979

P(Θ) is defined by Cr{A} = 12 (Pos{A} + 1 − Pos{Ac}). A fuzzy variable ξ is

defined as a function from a possibility space (Θ,P(Θ), Pos) to the set of realnumbers IR. The membership function of a fuzzy variable ξ is derived from thepossibility measure Pos by μ(x) = Pos{θ ∈ Θ | ξ(θ) = x}, for x ∈ IR.

Fuzzy Chance-Constrained Programming(CCP) are studied such as in Liuand Iwamura [5] and Liu [6]. When the optimistic return is maximized, a generalsingle-objective fuzzy CCP may be formulated as

⎧⎪⎪⎨

⎪⎪⎩

max fsubject to:

Cr{f(x, ξ) ≥ f} ≥ βCr{gj(x, ξ) ≤ 0, j = 1, 2, · · · , p} ≥ α

(4)

where α and β are the predetermined confidence levels, x is a decision vector, ξis a fuzzy vector, f(x, ξ) is the return function, and gj(x, ξ) are fuzzy constraintfunctions for j = 1, 2, · · · , p. Liu and Iwamura [5] and Liu [6] integrated fuzzysimulation, neural network(NN), and genetic algorithm(GA) to produce a hybridintelligent algorithm for solving fuzzy CCP, which was illustrated to be effectivewith several numerical examples.

Now, we will present an ant colony optimization-based hybrid intelligent al-gorithm for solving fuzzy CCP, which are introduced as following section.

4 Hybrid Intelligent Algorithm

In order to solve the fuzzy CCP (4), we embed the fuzzy simulation and neuralnetwork(NN), which would be referred to [6] for detail, into ant colony optimiza-tion algorithm to form a hybrid intelligent algorithm.

Algorithm: (Hybrid Intelligent Algorithm)

Step 1. Generate training input-output data for uncertain functions like

U1 : x → max{f | Cr{f(x, ξ) ≥ f} ≥ β} (5)

U2 : x → Cr{gj(x, ξ) ≤ 0, j = 1, 2, · · · , p} (6)

by fuzzy simulations.Step 2. Train a neural network to approximate the uncertain functions by

the generated training data.Step 3. Partition the region which contains the optimal solution.Step 4. Initialize all pheromone trails with the same amount of pheromone

and randomly generate a solution whose feasibility may be checkedby the trained neural network.

Step 5. Ant movement according to the pheromone trails to produce so-lution whose feasibility may be checked by the trained neural net-work.

Step 6. Repeat the fifth step for a given number of ants.

980 Y. Zhu

Step 7. Pheromone update according to the best feasible solution in thecurrent algorithm iteration.

Step 8. Repeat the fifth to seventh steps for a given number of cycles ora terminate criterion.

Step 9. Report the best solution as the optimal solution.

5 Numerical Examples

In the following examples, the evaporation rate ρ and the probability of mutationpm are chosen by several experiments.

Example 1. Consider the following fuzzy CCP,⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

max f

subject to:

Cr{ln |x1 + ξ1| + ln |x2 + ξ2| + ln |x3 + ξ3| ≥ f} ≥ 0.95Cr{(x1 − ξ1)2 + (x2 − ξ2)2 + (x3 − ξ3)2 ≤ 25} ≥ 0.900 ≤ x1, x2, x3 ≤ 6

(7)

where ξ1, ξ2, and ξ3 are triangular fuzzy variables (−1, 0, 2), (0, 1.5, 2.5), and(1, 2, 3), respectively. Let us first generate input-output data for the uncertainfunctions

U1 : x → max{f | Cr{ln |x1 + ξ1| + ln |x2 + ξ2| + ln |x3 + ξ3| ≥ f} ≥ 0.95} (8)

U2 : x → Cr{(x1 − ξ1)2 + (x2 − ξ2)2 + (x3 − ξ3)2 ≤ 25} (9)

by fuzzy simulations. Then we train an NN (3 input neurons, 15 hidden neurons,2 output neurons) to approximate the functions U1(x), U2(x). We averagely di-vide the interval [0, 6] to 10 subintervals. Now the hybrid intelligent algorithm(6000 cycles in simulation, 2000 data in NN, 50 ants, 200 generations) is employedwith the evaporation rate ρ = 0.6 and the probability of mutation pm = 0.4 toshow that the optimal solution is x∗ = (3.4634, 2.6106, 3.1366), whose objectivevalue is 3.6542.

Table 1 shows that parameters ρ and pm less effect the objective value butgeneration numbers.

Example 2. Consider the following fuzzy chance-constrained goal programming,⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

lexmin{d−1 , d−2 , d−3 }subject to:

Cr{0.7 − | sin(ξ1 − x1)| ≤ d−1 } ≥ 0.90Cr{0.6 − | sin(ξ1 + ξ2 − x1 − x2)| ≤ d−2 } ≥ 0.85Cr{0.75 − | sin(ξ2 + ξ3 − x2 − x3)| ≤ d−3 } ≥ 0.80x2

1 + x22 + x2

3 ≤ 25x1, x2, x3, d

−1 , d−2 , d−3 ≥ 0

(10)

Solving Fuzzy Chance-Constrained Programming 981

Table 1. Several experiment results

ρ pm generation numbers objective values

0.2 0.8 1308 3.6517

0.2 0.7 1287 3.6535

0.3 0.6 959 3.6535

0.5 0.6 959 3.6535

0.5 0.5 703 3.6533

0.6 0.4 182 3.6542

0.8 0.2 299 3.6417

0.2 0.3 602 3.6534

0.5 0.3 602 3.6534

0.5 0.2 1324 3.6534

where ξ1, ξ2 and ξ3 are triangular fuzzy variables (0, 1, 2), (−2,−1, 0), and(1, 2, 3), respectively. The d−i is the optimistic negative deviation from the targetof goal i, defined as

d−1 = min{d ∨ 0 | Cr{0.7 − | sin(ξ1 − x1)| ≤ d} ≥ 0.90}, (11)

d−2 = min{d ∨ 0 | Cr{0.6 − | sin(ξ1 + ξ2 − x1 − x2)| ≤ d} ≥ 0.85}, (12)

d−3 = min{d ∨ 0 | Cr{0.75 − | sin(ξ2 + ξ3 − x2 − x3)| ≤ d} ≥ 0.80} . (13)

It is easy to know that the feasible set is contained in the following hypercubeL = {(x1, x2, x3) | 0 ≤ x1 ≤ 5, 0 ≤ x2 ≤ 5, 0 ≤ x3 ≤ 5}. Let us first generateinput-output data for the uncertain functions

U1 : x → max{d | Cr{| sin(ξ1 − x1)| ≥ d} ≥ 0.90}, (14)

U2 : x → max{d | Cr{| sin(ξ1 + ξ2 − x1 − x2)| ≥ d} ≥ 0.85}, (15)

U3 : x → max{d | Cr{| sin(ξ2 + ξ3 − x2 − x3)| ≥ d} ≥ 0.80} (16)

by fuzzy simulations. Then we train an NN (3 input neurons, 15 hidden neurons,3 output neurons) to approximate the functions U1(x), U2(x), U3(x). Note that

d−1 = [0.7 − U1(x)] ∨ 0, d−2 = [0.6 − U2(x)] ∨ 0, d−3 = [0.75 − U3(x)] ∨ 0 . (17)

We averagely divide the interval [0, 5] to 10 subintervals. Now the hybrid intel-ligent algorithm (6000 cycles in simulation, 2000 data in NN, 50 ants, 400 genera-tions) is employed with the evaporation rate ρ = 0.2 and the probability of muta-tion pm = 0.2 to show that the optimal solution is x∗ = (0.0088, 2.3504, 0.7164),which satisfies the first two goals, but the third objective value is 0.0447.

982 Y. Zhu

6 Inventory Model with Fuzzy Demands

Inventory problem is classic in management. Due to the complexity of socialsociety, many factors in management behave uncertainty such as fuzzy variables.Researchers then presented some fuzzy inventory models, for example, inventorymodels with fuzzy demands [8], fuzzy production quantity [9], fuzzy holdingcost [10], and so on.

We in this section establish a new fuzzy inventory model. First present someparameters as follows: x represents order quantity per block; y represents reorderpoint; ξ represents fuzzy demands per year; η represents fuzzy demands per dayat a cycle; ζ represents fuzzy time between order and delivery (unite: day); c1

represents order cost per block; c2 represents holding cost per item at a year;c3 represents penalty cost per item out of stock per day; T represents totalinventory cost per year.

Total inventory cost per year is the sum of yearly order cost, yearly holdingcost, and yearly penalty cost. Then T = c1

ξx + c2Q1 + c3

ξxQ2, where Q1 =

y + x2 −ηζ, and Q2 = ηζ−y if y < ηζ, 0 if y ≥ ηζ. We will consider the following

fuzzy CCP model if we want to minimize the α-pessimistic cost⎧⎪⎪⎪⎨

⎪⎪⎪⎩

min T

subject to:

Cr{T ≤ T} ≥ α

x > 0, y > 0

(18)

where T is the α-pessimistic cost.Suppose that c1 = 100, c2 = 150, c3 = 30, α = 0.90 and ξ, η, ζ are trapezoidal

fuzzy variable (440, 450, 470, 480), trapezoidal fuzzy variable (1, 1.5, 2, 3), trianglefuzzy variable (9, 10, 11), respectively.

We can let the feasible set be contained in the hypercube L = {(x, y) | 0 <x ≤ 100, 0 < y ≤ 50}. Let us first approximate the uncertain function U :(x, y) → min{T | Cr{T ≤ T} ≥ 0.90} by fuzzy simulation. Then the solution ofproblem (18) can be approximated by the following problem

⎧⎪⎨

⎪⎩

min U(x, y)subject to:

0 < x ≤ 100, 0 < y ≤ 50 .

(19)

Divide averagely the interval [0, 100] to 10 subintervals, and [0, 50] to 5 subin-tervals. Now the ant colony optimization algorithm (50 ants, 500 generations)is employed with the evaporation rate ρ = 0.2 and the probability of muta-tion pm = 0.3 to show that the optimal solution is x = 31.4673, y = 23.4291,whose objective value is 5792.43. This means that when the amount of inventoryitems is 23.4291, it is required to order items 31.4673 so that yearly total cost isminimum.

Solving Fuzzy Chance-Constrained Programming 983

7 Conclusions

Based on ant colony optimization algorithm, we designed a hybrid intelligentalgorithm mixed fuzzy simulation and neural network for solving fuzzy chanceconstrained programming. Then we gave some numerical examples to illustratethe efficiency of the algorithms. Several experiments showed that the algorithm isstable with regard to main parameters. As an application, finally we establisheda fuzzy inventory model and solved it with the hybrid intelligent algorithm.

References

1. Dorigo, M.: Optimization, Learning and Natural Algorithm. Ph.D. dissertain (inItalian). Department of Electronics, Politecnico di Milano, Italy(1992)

2. Bullnheimer, B., Hartl, R.F., Strauss, C.: A New Rank-based Version of the AntSystem: A Computational Study. Central European Journal for Operations Re-search and Economics. 7 (1999) 25-38

3. Parpinelli, R.S., Lopes, H.S., Freitas, A.A.: Data Mining with an Ant Colony Op-timization Algorithm. IEEE Transactions on Evolutionary Computing. 6 (2002)321-332

4. Stutzle, T., Dorigo, M.: A Short Convergence Proof for a Class of Ant Colony Op-timization Algorithm. IEEE Transactions on Evolutionary Computation. 6 (2002)358-365

5. Liu, B., Iwamura, K.: Chance Constrained Programming with Fuzzy Parameters.Fuzzy Sets and Systems. 94 (1998) 227-237

6. Liu, B.: Theory and Practice of Uncertain Programming. Physica-Verlag, Heidel-berg(2002)

7. Liu, B., Liu, Y.-K.: Expeccted Value of Fuzzy Variable and Fuzzy Expected ValueModels. IEEE Transactions on Fuzzy Systems. 10 (2002) 445-450

8. Chen, S., Hsieh, C.H.: Optimization of Fuzzy Simple Inventory Models. IEEE In-ternational Fuzzy System Conference Proceedings. 1 (1999) 240-244

9. Hsieh, C.H.: Optimization of Fuzzy Production Inventory Models. Information Sci-ence. 146(2002) 29-40

10. Park, K.S.: Fuzzy Set Theoretic Interpetation of Economic Order Quantity. IEEETransactions on Systems, Man, and Cybernetics SMC. 17 (1987) 1082-1084

11. Liu, B.: Uncertainty Theory: An Introduction to its Axiomatic Foundations.Springer-Verlag, Berlin(2004)