A Genetic Algorithm for Integrated Decisions on Spare Part ... · OutlineIntroductionThe ProblemMathematical ModelSolution AlgorithmComputational StudySummary A Genetic Algorithm

Outline Introduction The Problem Mathematical Model Solution Algorithm Computational Study Summary

A Genetic Algorithm for Integrated Decisions onSpare Part Inventories and Cross-Training Policies

in Repairable Inventory Systems

Hasan Huseyin Turan

Capability Systems CentreSchool of Engineering and Information Technology

University of New South Wales, Canberra

2nd Annual IEEE Systems Modelling ConferenceOctober 4, 2018

1 / 24


Outline

1 Introduction

2 Problem definition

3 Mathematical Model

4 Solution Algorithm

5 Computational Study

6 Summary

2 / 24


IntroductionImportance of Maintenance and Costs of Downtimes

Oil platforms∼ $600,000 per day

Aircrafts∼ $10,000 per hour

Lithographic machines(semiconductors)

∼ $150,000 per hour3 / 24


IntroductionDowntimes and Spare Parts Supply

There are multiple ways to decrease occurring downtimes.

Two of them are:Increase the availability (inventory) of the necessary spare parts.Optimize the replenishment/repair process.

Local warehouse

Local warehouse

Local warehouse

Central Depot

Repair Shop

Installed Base

Emergency Shipment

4 / 24


Problem definitionSingle Facility with Multi-Skill Servers and Inventories

The analyzed system consists of:A repair shop with several multi-skill servers.Multiple stocking points for each type of repairable spare parts(a.k.a. stock keeping units, SKUs).

1

1

2 3

3

4 5

Failed

Parts

Installed Base

Spare PartInventories

Repair Shop

Random Failures of Parts

(a) Pooled Design with Two Clusters

1

1

2 3

3

4 5

Failed

Parts

Random Failures of Parts

(b) Not Pooled (‘N’ and ‘W’ Structures) Design with Two Clusters

5 / 24


Problem definitionThe objective of research and different repair shop designs

The objectives of this research are:

(a) How many should spare parts be kept in stock ?

(b) What is the optimal repair shop design?

(b1) What should be the capacity of repair facility?(b2) How should servers be trained to repair different types of failed

spares? That is, what is the appropriate cross-training scheme foreach server?

λ1

λ2

λ3

λ4

(a) No cross-training (dedicated)

Failed Spares Servers

λ1

λ2

λ3

λ4

(b) Pooling (partial cross-training)


λ1

λ2

λ3

λ4

(c) Chaining (partial cross-training)


λ1

λ2

λ3

λ4

(d) Mixed (dedicated andpartial cross-training)


λ1

λ2

λ3

λ4

(e) Full cross-training


6 / 24


Problem definitionKey assumptions

A model with an infinite time planning horizon is chosen andsteady-state analysis is utilized in continues time [0,∞) in order todetermine inventory levels.

It is assumed that failures occur according to a Poisson process with aconstant rate.

All repair times are exponentially distributed and mutually independent.

First come first served(FCFS) queuing discipline is adopted and nopriority exists among failed spares.

It is assumed that holding costs for every SKU are linear in the initialstock level and are paid per SKU per time unit.

It is assumed that a positive cross-training or cost of flexibility occurswhenever an additional skill is assigned to a server.

Each cluster inside the repair shop is modeled as a multi-class multi-serverM/M/k queuing system with dedicated queues.

The clusters inside the repair shop are mutually exclusive (disjoint) andcollectively exhaustive.

7 / 24


Mathematical ModelParameters and Decision Variables

Problem parameters:

N: Number of distinct type of repairables (SKUs)λi Failure rate of SKU type i ( i = 1, . . . ,N)µi : Service rate of SKU type i ( i = 1, . . . ,N)hi : Inventory holding cost of SKU type i per unit time per part ( i = 1, . . . ,N)b: Penalty cost for each backordered demand per unit time, which is equivalent

to the cost of downtime of the systemf : Operation cost of a server per unit time (e.g., annual wage)ci : Cost of having a skill to repair SKU type i per unit time per server

(e.g., annual qualification bonus)ε: Very small positive real number

Decision variables:

Si : Initial inventory quantity (basestock level) kept on stock for SKU typewhere S = (S1, . . . , SN)

zk : Number of the operational servers in the cluster k (k = 1, . . . , y)where Z = (z1, . . . , zy )

xik : Binary variable indicating that whether the cluster k has a skill to repair SKU

type i or not, where Xk = (x1k , . . . , xNk)T and X = [X1| . . . |Xy ]y : Number of clusters in the repair shop 8 / 24


Mathematical ModelThe Objective function

The Mixed Integer Nonlinear Objective Function

minS, X, Z

y∑k=1

fzk︸︷︷︸Cost of Servers

+

y∑k=1

zk

(N∑i=1

cixik

)︸︷︷︸

Cost of Training

+N∑i=1

hiSi︸︷︷︸Holding Cost

+ bN∑i=1

EBOi [Si ,X,Z]︸︷︷︸Expected Backorder Cost

(1)

The penalty (backorder) cost term is calculated using the penaltycost b and the expected total number of backordered partsEBOi [Si ,X,Z] for each SKU type i in the steady-state; under thegiven initial inventory level Si , pooling scheme of the repair shop Xand the server assignment policy ZThe variable X represents a (N × y)–matrix of the binary decisionvariables xik denoting how SKUs are pooled in the repair shop.The variable Z represents a 1 × y– row matrix of integer decisionvariables zk denoting the number of servers in each cluster of therepair shop.

9 / 24


Mathematical ModelConstraints

y∑k=1

xik = 1 i = 1, . . . ,N (2)

N∑i=1

xikλi

µi≤ (1− ε) zk k = 1, . . . , y (3)

xik ∈ {0, 1} i = 1, . . . ,N k = 1, . . . , y (4)

zk ∈ Z+ k = 1, . . . , y (5)

Si ∈ N0 i = 1, . . . ,N (6)

y ∈ {1, . . . ,N} (7)

constraint set (2) ensures that pooling scheme X satisfies mutuallyexclusive and total exhaustive condition for each cluster, i.e., anySKU type being repaired by exactly one cluster,constraint set (3) guarantees the system stability by assigningsufficient number of servers to each cluster,constraint sets between Eqs.(4)-(6) are required for non-negativityand integrality of the variables

10 / 24


Solution AlgorithmA two-stage iterative heuristic algorithm

The number of possible pooling schemes X increases exponentially for increasingnumber of SKUs in the system.

Genetic Algorithm (GA) generates a set of feasible pooled repair shop designpolicies, and solutions are passed through fitness evaluation function to findoptimal values of server assignment policy Z and inventory levels of spares S.

Start

Input:GA parametersFailure/Service Rates

Cost Parameters

Initial Population Generationgeneration:=0

Fitness Evaluation

generation:=+1

generation> genmax?

Yes

No

Stop

Crossover

Mutation

Local Search for CapacityOptimization

Multi-Class Multi-ServerSolver

Z SX

Total Cost

11 / 24


Genetic AlgorithmThe general idea

The GA methods are population-based. Hence, they can escape fromlocal extrema, and easily combinedwith simulations for evaluating com-plex objective functions.

GA steps:

Step 1: Initial population generation

Step 2: Evaluation of individuals

Step 3: Parents selection

Step 4: Reproduction by crossoverand/or mutation

Step 5: Repeating steps 2–4 till thestop condition is reached.

Source: Wikimedia Commons

2 2 26 93 3 34 4

2 2 26 93 3 24 4

2 2 26 93 3 34 4

2 2 36 93 3 24 4 12 / 24


Evaluation of FitnessDecomposition of problem

For given X, the problem is reduced to a capacity and inventory optimizationproblem CIO

(X)

as follows:

CIO(X)

= minS, Z

y∑k=1

zk(f + Ck

(X) )

+N∑i=1

hiSi + bN∑i=1

EBOi

[Si ,Z,X

]From independence of clusters, we can separate and rewrite the CIO

(X)

foreach cluster k with the constraints as follows:

CIOk

(X)

= minSi , zki∈Nk

zk(f + Ck

(X) )

+∑i∈Nk

hiSi + b∑i∈Nk

EBOi

[Si , zk ,X

](8)

Subject to: zk ≥

⌈∑i∈Nk

λi

µi

⌉(9)

zk ∈ Z+ (10)

Si ∈ N0 i ∈ Nk (11)

where Nk is the index set of the SKUs in the cluster k13 / 24


Evaluation of FitnessDecomposition of problem II

We performs local greedy search to find the optimal zk by fixing server numbers ateach cluster and solving the following optimization subproblems:

CIOk

(X, zk

)=

{minSi

( ∑i∈Nk

hiSi +bEBOi

[Si , zk ,X

] )|Si ∈ N0

}k = 1, . . . , y

Objective function of model 12 has separable structure and it can be rewritten asfollows:

CIOk

(X, zk

)=

{ ∑i∈Nk

minSi

(hiSi +bEBOi

[Si , zk ,X

] )|Si ∈ N0

}k = 1, . . . , y

Require: X, y

Ensure: zk ≥⌈∑

i∈Nkx ikλi

µi

⌉1: for k ∈ {1, . . . , y} do2: Calculate: Γk (zk )3: while Γk (zk ) < 0 do

4: Optimize: CIOk

(X, zk

)5: zk := +16: end while7: end for

14 / 24


Evaluation of FitnessQueuing Approximation

Each cluster k in the repair shop for a given number of servers can be viewed as amulti-class multi-server M/M/zk queuing system. The probability distribution of thenumber of failed SKU type i at the steady-state

limt→∞

Pi

[Qi (t) = q,X,Z

]= pi (q) , i = 1, . . . ,N q = 0, 1, . . .

is required to evaluate EBOi

[Si ,Z,X

].

In this approximation, marginal probability distribution (and several performancecharacteristics) of the SKU type i in the cluster k is derived by aggregating all otherSKUs in the cluster k into a single SKU type (class). The procedure is repeated toobtain the remaining distributions for other SKUs in the cluster.

λN

N -class M/M/zk system

λ1

⇒

Original System

ΛA′

3-class M/M/zk system

λ1

ΛA + +

λN

3-class M/M/zk system

ΛA

ΛA′

Decomposed Approximated System

15 / 24


Evaluation of FitnessFinal Remarks

We can approximate the value of EBOi

[Si ,Z,X

]as:

EBOi

[Si ,Z,X

]u∞∑q=0

max {0, q − Si} pi (q)

u∞∑

q=Si+1

(q − Si ) pi (q)

where pi (q) is the approximation of the marginal probability distribution of thenumber of failed SKU type i , i.e., pi (q). The objective function (12) is the

summation of holding and backorder costs EHBi

[Si ,X,Z

]= hiSi + bEBOi

[Si ,Z,X

]and is optimized by at:

EHBi [Si ,X,Z] is convex on its whole domain Si ∈ N0 (:= N ∪ {0}) for given poolingpolicy X and capacity levels Z, and it is minimized at the smallest Si ∈ N0 for which

Si∑q=0

pi (q) ≥b − hi

bi = 1, . . . ,N

16 / 24


Evaluation of FitnessFinal Remarks

We can approximate the value of EBOi

[Si ,Z,X

]as:

EBOi

[Si ,Z,X

]u∞∑q=0

max {0, q − Si} pi (q)

u∞∑

q=Si+1

(q − Si ) pi (q)

where pi (q) is the approximation of the marginal probability distribution of thenumber of failed SKU type i , i.e., pi (q). The objective function (12) is the

summation of holding and backorder costs EHBi

[Si ,X,Z

]= hiSi + bEBOi

[Si ,Z,X

]and is optimized by at:

EHBi [Si ,X,Z] is convex on its whole domain Si ∈ N0 (:= N ∪ {0}) for given poolingpolicy X and capacity levels Z, and it is minimized at the smallest Si ∈ N0 for which

Si∑q=0

pi (q) ≥b − hi

bi = 1, . . . ,N

17 / 24


Numerical ExperimentsTestbed

A full factorial design of experiment (DoE) with seven factors and two levels per factoris used to generate a total of 128 instances.

Factors Levels

No. of SKUs (N) [10, 20]No. of Initial servers (M) [5, 10 ]Utilization Rate (ρ) [ 0.65, 0.80 ]Minimum Holding Cost (hmin) [ 1, 100 ]Maximum Holding Cost (hmax ) 1000Holding cost/Workload relation [ IND, HPB ]Server Cost (f ) [ 10hmax , 100hmax ]Cross-Training Cost (ci ) [ 0.01f , 0.10f ]

Penalty Cost (b) 50∑N

i=1 λi hi∑Ni=1

λi,

The holding costs, hi , are generated using two different patterns: (i) IND: completelyrandomly (independent) within a range [hmin, hmax ], and (ii) HPB: hyperbolicallyrelated to the workloads wi = λi/µi = δiρM:

hi =hmax − hmin + 10

9 wi−wminwmax−wmin

+ 1− 10 + hmin + ξi

where

ξi ∈ U[−hmax − hmin

20,hmax − hmin

20], wmin = min

i=1,...,Nwi and wmax = max

i=1,...,Nwi

18 / 24


Numerical ExperimentsGA parameters

Parameter ValuePopulation size 100Number of generation 25Crossover rate 0.8Chromosome mutation rate 0.4Switch-mutation probability 0.5Swap-mutation probability 0.5Operation DetailInitial population Uniform random integer ∈ [1, |N|]Fitness Evaluation Queuing approximation and local searchSelection Tournament selectionCrossover One-point uniform

Mutation(s)SwitchSwap

(a) switch-mutation: The procedure in which a random gene is chosenand switched to the value of another gene in the chromosome with equalprobability.(b) swap-mutation: The procedure in which two genes are selected atrandom, and then they are swapped by value.

19 / 24


Numerical ExperimentsCost reduction

∆FlexibleGA represents the relative percentage difference between the GA-based

heuristic and the fully flexible design. ∆DedicatedGA denotes the relative percentage

gap between the total cost of the pooled and the dedicated designs.

GA can yield ∼44% and ∼21% cost savings in comparison with dedicated andfully flexible designs, respectively.

In only 11% of the cases (15 out 128) in the testbed, the fully flexible repairshop designs outperform the pooled designs in terms of the total system cost.

10 20

−20 %

0 %

20 %

40 %

60 %

80 %

100 %

∆Flexible

GA

5 10

−20 %

0 %

20 %

40 %

60 %

80 %

100 %

0.65 0.80

−20 %

0 %

20 %

40 %

60 %

80 %

100 %

1 100

−20 %

0 %

20 %

40 %

60 %

80 %

100 %

IND HPB

−20 %

0 %

20 %

40 %

60 %

80 %

100 %

10hmax 100hmax

−20 %

0 %

20 %

40 %

60 %

80 %

100 %

0.01f 0.1f

−20 %

0 %

20 %

40 %

60 %

80 %

100 %

10 20

−20 %

0 %

20 %

40 %

60 %

80 %

100 %

Number ofSKUs

∆Dedicated

GA

5 10

−20 %

0 %

20 %

40 %

60 %

80 %

100 %

Number ofInitial Servers

0.65 0.80

−20 %

0 %

20 %

40 %

60 %

80 %

100 %

Utilization Rate

1 100

−20 %

0 %

20 %

40 %

60 %

80 %

100 %

MinimumHolding Cost

IND HPB

−20 %

0 %

20 %

40 %

60 %

80 %

100 %

HoldingCost Variant

10hmax 100hmax

−20 %

0 %

20 %

40 %

60 %

80 %

100 %

Server Cost

0.01f 0.1f

−20 %

0 %

20 %

40 %

60 %

80 %

100 %

Cross-TrainingCost

the pooled designs found by proposed heuristic can yield ∼44% and ∼21% costsavings in comparison with dedicated and fully flexible designs, respectively. In only11% of the cases (15 out 128) in the testbed, the fully flexible repair shop designsoutperform the pooled designs in terms of the total system cost.

20 / 24


Numerical ExperimentsRuntime analysis

run all the experiments on a computer with 16 GB RAM and 2.8 GHz i7 CPU.

converges the best solution in 1456 CPU seconds. The worst case performance isaround 4000 CPU seconds.

the increasing problem size (number of SKUs and the number of initial servers)has a negative impact on the run time of the algorithm due to the increasingeffort to solve the multi-class multi-server queueing system approximation.

the cross-training cost factor indirectly affects the size of multi-class multi-serverqueueing problem that has to be solved several times during execution of thealgorithm, which results in longer run times.

10 200

1,000

2,000

3,000

4,000

Number ofSKUs

RunTim

e

(CPU

Secon

ds)

5 100

1,000

2,000

3,000

4,000

Number ofInitial Servers

0.65 0.800

1,000

2,000

3,000

4,000

Utilization Rate

1 1000

1,000

2,000

3,000

4,000

MinimumHolding Cost

IND HPB0

1,000

2,000

3,000

4,000

HoldingCost Variant

10hmax 100hmax

0

1,000

2,000

3,000

4,000

Server Cost

0.01f 0.1f0

1,000

2,000

3,000

4,000

Cross-TrainingCost

21 / 24


Numerical ExperimentsCross-training and capacity usage

“How much flexibility/cross-training is usually enough?”

When the pooling heuristic is used, the skill distribution per serverdepends on the number of SKUs in the system. It obeys alog-normal distribution.We observe that 90% of the servers are less than 50% cross-trained.

100% cross-training (fully flexible server) is observed when the pooling heuristicis used. These extreme cases usually occur: (i) when the cost of having an extraskill is relatively small compared to the cost of having an additional server (i.e.,the case of cross-training cost being equal to 0.01f ), and (ii) when the problemsize is small (i.e., the instances where N and M are equal to 10 and 5,respectively).

0 2 4 6 8 10 12 140

10

20

30

Average # of skills per server

4 6 8 10 120

20

40

60

# of server used

20 40 60 80 1000

10

20

30

40

Average % cross-training per server 22 / 24


Conclusions and Future Research

(i) which spare part types to pool together; (ii) how to assign servers(capacity) to each cluster; and (iii) how much spare inventory tokeep for each repairable type.

The pooled designs can result in cost savings of ∼44% and ∼21%compared to the conventional dedicated and fully flexibleparts-server assignments, respectively.

The advantages of pooled repair shop designs diminish whencross-training costs (skill cost) increase.

Partial flexibility achieved by up to 50% cross-trained servers isusually sufficient for the optimal repair shop performance.As future research:

Improving the performance of the GA by developing new mutationand crossover operationsIntegrating pooling decision with static and dynamic routing andprioritization rules in the part repair processesUsing simulation instead of queuing based approximations

23 / 24


Thank you for your attention ...

Questions, Comments, Suggestion?

24 / 24

Documents

A Genetic Algorithm for Integrated Decisions on Spare Part ... · OutlineIntroductionThe ProblemMathematical ModelSolution AlgorithmComputational StudySummary A Genetic Algorithm