A Mixed Integer Programming Approach for Supply ChainManagement Using Particle Swarm Optimization AlgorithmGAO Bo WU YuanyuanCollege of Mechanical & Electric Engineering, Agricultural University of Hebei, Hebei, P.R.China,071001gaobo@hebau.edu.cn

Abstract As a novel optimization technique, particle swarm optimization (PSO) algorithm works as anevolutionary computation technique and has been applied to solve non-linear problems in many domains.But this algorithm always works as problem-depended so that the application method analysis alwaysattracts researchers interests. In this paper we address the research on solving a kind of programmingproblems- mixed integer programming (MIP) that usually occur in operation or management regions.Following the way of standard PSO algorithm, we give a general formula for MIP and emphasize onconstructing a feasible pattern which can be used in PSO so that the constrained condition which mostMIP problems happen to are transformed into unconstrained condition by request of PSO algorithm.Finally we give a practical case to prove such methodology and the testing result verifies its validity.Key words Mixed integer programming; particle swarm optimization; penalty function; supply chainoptimization

1 IntroductionAlthough have emerged for about more than 20 years, Supply chain Management (SCM) still hasnot a universal definition. But we can find in literatures of some definitions on Supply Chain (SC). Forexample, SC is defined as a system of suppliers, manufacturers, distributors, retailers and customerswhere material, financial and information flows connect participants in both directions [P.fiala, 2005].Asusually an important and strategic operation management problem in SC, SC design is to provide anoptimal platform for effective and efficient SCM[Yan, et al, 2003].The significance of modeling the interruptions of a production/inventory system in the SC processshould be attributed to the severity of their potential negative impacts on operating costs and customerservice measures in todays modern manufacturing and business environment [Mohebbi, 2003]. Thereexist many papers on quantitative techniques for the improvement and optimization of SCs withoutglobal considerations, and mixed integer programming models are among the most widely usedtechniques. Most models address the problem in a regional, local or single-country environment, whereinternational factors do not have a significant impact on the design of the SC [Goetschalckx, et al, 2002].Literatures can be categorized into conceptual, empirical and quantitative frameworks for designing andoperating SC networks. In such frameworks conceptual models play an important role to provide astrong foundation for developing rigorous quantitative frameworks[Talluri & Baker, 2002].

2

An quantitative model of two-phased SC

Supply chain management has an inherent relativity with retailer-provider management and therehave been proposed many mathematical models for such t subjects. In this paper we choose a typicaltwo-tier SC model[Huang&Lu,2003] to illustrate its structure and solution.m

m n

q

minZ = f i zi + cijk xijki =1

(1)

i =1 j =1k =1

m

s.t. xijk = 1

j , k

i =1

969

(2)

m

zi p

(3)

i =1

i, j , k

zi + xijk 0

s j d jk xijk wi

i

j k

xijk 0 j = j1 , j2 , L, jr

zi = {0,1}0 x ijk 1

(4)(5)

k = k , k , L, k

ii, j , k

1

2

r

(6)(7)(8)

Where i represents the facility of inventory and retailers, j represents the number of production itemand k represents the end retailers or consumers. There are two variants influence the effect of thedecision that is zi and xijk . zi is a binary variant that its value should be 1which means i th facilityhas been chosen or 0 not. The other variant xijk represents the market share of i th facility at j thproduct for the k th retailer or consumer. For the objective function (Eq. 1) there are two parts ofvariants: f i means the annual cost of i th facility and cijk represents theannual cost of xijk .Parameter p in equation (3) denote the maximum of total number of facilities. si in Eq.(5) denotes thespace of j th product. d jk denotes the k th retailers or consumers demand of j th product at a fixedinterval. wi represents the maximum capability of i th facility.The above functional optimization problem is a typical mixed integer programming (MIP) issue. Inpractice some optimization problems usually require one or more decision variants or optimal solutionsshould be integer. The traditional programming approaches are: processing the non-integer solution intointeger by rounding off which may not obtain an available solution or an optimization solution; anothercommon method is integer programming algorithms such as branch and bound algorithm. But suchalgorithm demands the objective function continuous and smoothly as well as strict restraint conditions.When the model contains non-linear function the solution may reach a local optimum depending on itsparameters or initial values. Along with the enlargement of extent and complexity especial for non-linearmixed integer model, the computing time would increase extremely and the online optimization couldnot be implemented by this algorithm[Feng, et al,2004]. So some researchers introduced more advancedalgorithms to solve such large scale non-linear dynamic systems optimization issues.For such issues one can use classical step-wise regression, procedure as well as some moresophisticated methodologies such as simulated annealing, genetic algorithms and evolution algorithm.Among them GAs and EAs are optimization techniques simulating biological systems which areclassified as a category of the research of so-called artificial life. Particle swarm optimization (PSO)algorithm as a relatively new optimization technique in this category can also be used as an excellentoptimizer which originated as a simulation of simplified social system.

3 Particle swarm optimization algorithmPSO is proposed by Kennedy and Eberhart in 1995 and Particle swarm optimization (PSO) isregarded as one of the swarm intelligent algorithms. PSO is a population-based algorithm that exploits apopulation of individuals, to synchronously probe promising regions of a given research space. In thiscontext, the population is called a swarm, and the individuals are called particles. For a definite searchspace the job of each particle is to move and retain in memory the best position it ever encountered.There are two types of variant in PSO: global variant and local variant. Exploration is the ability to testvarious regions in the problem space in order to locate a good optimum, hopefully the global one;Exploitation is the ability to concentrate the search around a promising candidate solution in order tolocate the optimum precisely.PSO is a new branch in evolutionary algorithms, which were inspired in group dynamics and its970

synergy and were originated from computer simulations of the coordinated motion in flocks of birds orschools of fish. As these animals wander through a three-dimensional space, searching for food orevading predators, these algorithms make use of particles moving in an n-dimensional space to searchfor solutions for an n-variable function optimization problem.PSO has proved to be competitive with GA in several tasks, mainly in optimization areas. Thereemerged some variants of PSO including Discrete Particle Swarm Optimizer (DPSO), LinearDecreasing Weight Particle Swarm Optimizer (LDWPSO) and Constricted Particle Swarm Optimizer(CPSO).Assume an n-dimensional search space, S R n , and a swarm consisting of NP particles. The ithparticle is in effect an n-dimensional vector X i = ( xi1 , xi 2 , L , xin )Many real optimization problems can be formulated as the following functional optimizationproblemrMin f (x), x xrs.t. x [ai , bi ]rwhere f (x), is the objective function , and x x is the decision vector consisting of n variables.3.1 Basic standard PSO algorithmWe adopt a vector notation style to recommend the basic PSO algorithm as follows:r rrrr rr rrrrv k +1 = a v k + b1 r1 ( p1 x k ) + b2 r2 ( p 2 x k )r rrr rx k +1 = c x k + d v k +1

(9)

(10)rWhere the signal v k called the velocity for particle k which represents the distance to be traveled bythis particle from its current position , The symbol denotes element-by-element vectorrmultiplication. At iteration k , the velocity v k is updated on its current value affected by a momentumrrrfactor( a ); p1 represents the best previous position of particle k known as local-best position and p2represents the best position among all particles in the population known as global-bestrrrposition; xk represents the position of particle k ; b1 and b2 are positive constant parameters calledrracceleration coefficients which control the maximum step size; r1 and r2 are usually selected asuniform random numbers in the range of [0,1] .For such algorithm the work flow can be listed as follows:Step 1: Initialize for each particle k in the population;rStep 1.1: Initialize v k randomly;rStep 1.2: Initialize x k randomly;Step 1.3 Evaluate f kvStep 1.4 Initialize p2 with the index of the particle with the best function value among thepopulation.rrStep 1.5 Initialize p1 with the copy of. v k , k NStep 2: Repeat until a stopping criterion is satisfiedvStep 2.1: find p2 such that f [ p2 ] f k , k NrrStep 2.2: for each particle k , p1 = x k , if f k < f best [k ], k NrvStep 2.3: for each particle k , update v k and x k according to equation 1 and 2.Step2.4: evaluate f k for all particles.The search work flow PSO is shown in figure 1.

3.2 Variants of PSOThere are several variants of PSO, typically differing in the representation: Discrete or Continuous971

PSO, Linear Decreasing Weight or Constricted PSO; Predator Prey or Collision Avoiding Swarms andChaotic PSO. In this paper the last one is adopted to solve the optimization problemsIn PSO, proper control of global exploration and local exploitation is a core in finding the optimumsolution efficiently. The performance of PSO depends on its parameters and the parameter- inertiarweight a is the moduluswhich controls the impact of previous velocity on the current one. So the balance between explorationrand exploitation in PSO is dictated by the value of a . In order to achieve trade-off between explorationrand exploitation, we set a varying adaptively in response to the objective values of the particles. Sowe construct a function to adapt inertia weight factor.r

r

r

v

Initialization ( x k , v k , p 2 , p1 )Evaluating f k

Next k

v

If f k > f pv2 and If p 2 [a i , bi ]then

rvp2 = xk

v

r

If f k > f pv1 and If p1 [ai , bi ]then

r

Update x kUpdate v k

rvp1 = xk

Satisfying criterionOutput the solutionFigure 1 Work flow of PSO Algorithm

3.3 Tset ExampleWe construct a vegetable supply chain example to test the serach result of PSO algotithm.Theactual meaning of the model is ignored and just a formular model is given as following:m n

min f ( xi , y ij ) = bi xi + cij y ij

(11)

i =i j =1

s.t. xi = {0,1}

(12)

m n

y ij = 1

(13)

i =1 j =1

m

xi E

(14)

i =1

y ij xi

(15)

0 y ij 1

(16)

o

y ij Q =

l

k

qk

(17)

k =1

m

y

ij

Gj

i =1

The values of the main parameters are given in table 1.

972

(18)

3.4 Numerical simulationvvThe population size is 20, fixed the total number of function evaluation as 2000, b1 and b2 are setrto 2.0, and v max is clamped to be 15% of the research space, IWF a varying from 1.2 at the beginning ofthe research to 0.2 at the end. We introduce two indexes SR and AVEN to evaluate the algorithmsperformance as follows:SR = N v / RT 100%(19)(20)AVEN = ni / N vWhere RT (run time) denotes the total number of run; N v denotes the number of success runamong RT ; ni denotes the number of function evaluation of the i th success run.Table 2 Testing Result

Table 1 The Main Parameters of SC ModelParameter

bi

m n E Gj

Population1020304050

cij[0.711,0.77,0713]

0.400.300.38Value

[0.771,0.76,0.843]4

3

4 0.45

0.39

SR (%)100100100100100

AVEN104388582774270066752

[0.944,0.92,0.915]0.25

0.35

[0.87,0.863,0.881]

4 ConclusionsObjective function can be directly used as fitness function to guide the search in PSO that make iteasy to handle non-linear and non-differentiable optimization problem. Using discrete numbers forparticles and velocity, PSO can beapplied to discrete problems or mixed integer problems easily. The results of numerical simulation proveits effectiveness and efficiency.

References

[1]P.fiala. Information sharing in supply chains.Omega, 33(2005),p419 423[2]Hong Yan, Zhenxin Yu, T.C.Edwin Cheng. A strategic model for supply chain design with logicalconstraints: formulation and solution. Computers & Operations Research, 30(2003),p2135 2155.[3]Esmail Mohebbi. Supply interruptions in a lost-sales inventory system with random lead time.Computer& Operations Research, 30(2003),p411 426.[4]Mare Goetschalckx, Carlos J. Vidal, Koray Dogan. Modeling and design of global logistics systems: Areview of integrated strategic and tactical models and design algorithms. European Journal of OperationalResearch, 143(2002),p 1 18.[5]Srinivas Tallori, R.C.Baker. A multi-phase mathematical programming approach foe effective supplychain design. European Journal of Operational Research, 141(2002),p544 558[6]Huang Xiao-yuan, Lu Zhen. Application of two-stages by supply chain model in server and distributionsystem. Systems engineering-theory methodology applications,2003,12(3),p228 231[7] Feng Jian-rong, Liu Zhi-he, Liu Zheng-he. Programming problems and simulation implementing.Journal of system simulation,2004,16(4),p845 848

973