SCENARIO BASED STOCHASTIC LINEAR PROGRAMMING MODEL FOR MULTI PERIOD DISASSEMBLY LOT SIZING PROBLEMS UNDER RANDOM LEAD TIME

Scenario-Based Stochastic Linear Programming Model for Multi-Period Disassembly Lot-SizingProblems Under Random Lead Time || Group 03 Table of Contents I.. Scenario-Based Stochastic Line

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VIETNAM NATIONAL UNIVERSITY – HO CHI MINH CITY

INTERNATIONAL UNIVERSITY SCHOOL OF INDUSTRIAL ENGINEERING AND MANAGEMENT

SCENARIO-BASED STOCHASTIC LINEAR PROGRAMMING MODEL FOR MULTI-PERIOD DISASSEMBLY LOT-SIZING PROBLEMS UNDER

RANDOM LEAD TIME COURSE: INVENTORY MANAGEMENT

Lecturer: Dr Nguyen Van Hop

GROUP 03

Ho Chi Minh City, December 2020

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TIEU LUAN MOI download : skknchat@gmail.com

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Scenario-Based Stochastic Linear Programming Model for Multi-Period Disassembly Lot-Sizing

Problems Under Random Lead Time || Group 03

Table of Contents

I INTRODUCTION 3

1 System description 3

a) Introduction 3

b) Planned disassembly lead time (PLT) and Real disassembly lead time (DLT) 3

c) System study 3

2 Problem statement 4

3 Scope and limitations 5

II MATHEMATICAL MODEL 5

1 Indexes 5

2 Parameters 5

3 Uncertain parameter 5

4 Functions 6

5 Decision variables 6

6 Variables 6

7 Model 6

III DATA COLLECTION: 7

IV IMPLEMENTATION 9

1 Data input 9

a) Sets 9

b) Parameters 10

c) Decision Variables 10

2 Processing 10

a) The Objective function 10

b) The Constraints 11

V RESULT ANALYSIS 11

1 Binary indicator of disassembly in period t: δt 11

2. Parent items’ quantity received in period t: xt 12

3 Inventory level of leaf item I at the end of period t and scenario s: , , +

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4 Backordered quantity of leaf item I at the end of period t and scenario s: , , −

13 VI RECOMMENDATION ON IMPROVEMENT 14

1 Value of beginning backorder 14

2. Inconsistency in the definition of x t 14

VII REFERENCES 14

CONTRIBUTION 14

APPENDIX 15

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b) Planned disassembly lead time (PLT) and Real disassembly lead time (DLT)

To acknowledge how the disassembly system operates, the definition of PLT and DLT

must be clear at first

PLT is known in advance

In this study, we do not know when end-of-life products will be received after disassembling since the DLT is not known in advance For these reasons, we define DLT as below:

considered as an independent random discrete variable with known probability distributions.After each disassembly order release of the end-of-life products, a real disassembly lead time is noted It may be less, equal, or greater than the planned disassembly lead time

c) System study

This research paper, titled Scenario – based stochastic linear programming model for multi

–period disassembly lot – sizing problems under random lead time, presents a Reverse

– Material Requirements Planning ( R-MRP) model with the objective of minimizing theaverage total cost (ATC) dealing with a two – level product structure under uncertaintydisassembly lead time of the end – of – life product and known demand for eachpart/component over the planning horizon for the first time in disassembly lot –sizingproblem The end – of – life product forms the first level and the leaf items presenting theparts/components to be requested and not disassembled forms the second level Theirrelationship can be described by a tree diagram as shown in Figure 1 below The number inparentheses represents the disassembly yield or the number of components obtained by thedisassembly operation of a unit of the parent element

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Figure I.1: Two – level disassembly structure under uncertain environment

To meet the components requirements over the planning horizon, the amount of end – of –life products are launched for disassembling at the period j – PLT where PLT is theplanned disassembly lead time Once the end – of – life product is started in disassembly, arandom real disassembly lead time is found The disassembled end – of – life products arereceived in period t = j – PLT + with is a bounded discrete variable that varies between { −

… + } If a bad management policy of real disassembly lead time is applied, there are twocases that lead to either inventory cost or backorder cost:

• PLT >: The disassembled order started too early, the components arrive before

the expected delivery date to the customers, and therefore the stock level increases,which implies a cost of additional storage per component

generated

2 Problem statement

As mentioned above, although many research papers mentioned the R-MRP model on theproduct recovery process, none of them addressed the stochastic disassembly lead time.Therefore, the author aims to present a new scenario – based stochastic linear programmingmodel to integrate uncertainty in decision making for the first time in the disassembly lot –sizing problem

To specify, the overall compositions of the mathematical model includes:

horizon and the total disassembly quantities are unlimited in each period

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}

The problem is to determine the optimal order release quantity for root items for each period with the objective of minizine the average total cost (ATC) compounded by the sum of average backordering and inventory holding cots for each component Ii, and setup cost for the end – of– life product over the planning horizon and all scenarios

3 Scope and limitations

The project deals with a two – level product structure under uncertainty disassembly leadtime of end-of-life product and known demand for each part/component over the planninghorizon The most important feature in this project is that the lead time is uncertain, whichhas not been addressed yet

Optimization of the process would minimize the average total cost including inventory costand backordering cost for the supplier/manufacturer As a result, the project is addressed tointegrate uncertainty in decision making for the first time in disassembly lot – sizingproblem Otherwise, other stages such as packing, loading, or transportation would be out

of the scope Moreover, the project is limited to planning only, as its efficiency level hasnot been evaluated in real life

In this project, CPLEX is chosen as the programming solver tool, which would lead to alimitation on input size and running time For larger instances, the program may result inprocessing aborted Besides, several additional algorithms need to be applied to furtherextend this sizing availability

1 Indexes

S: Index for scenarios s = 1,2,3, …, S

External demand for leaf item in period t

Planed disassembly lead timeBeginning inventory of leaf itemPer period inventory holding cost of one unit of leaf itemPer period Backordering cost of one unit of leaf item

Setup cost for parent item in period t

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Binary indicator of disassembly in period tInventory level of leaf item i at the end of periodand scenario Backordered quantity of leaf item i at the end of period t and scenarios

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The scenario-based stochastic linear programming model:

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of keeping backlogs and inventories per part and the expense of setting up the final product, through the planninghorizon and all scenarios.

• Constraint (1) applies to the retention of inventory movement of component in each scenario s at the end of each period t The value

of , , + takes its value when it is larger than 0; otherwise, the value will be 0.

• The value of inventory level at period t is equal to the inventory level of the previous period (t – 1), plus the number of units of leaf item received in period t, minus external demands in period t, minus the value of backorder in the previous period (t – 1).

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the end of each period t The value of , , − takes its value when it is larger than 0; otherwise, thevalue will be 0

• The value of backorder at period t is equal to the backorder of the previous period (t –

1), plus external demands in period t, minus the number of units of leaf item

received in period t, minus the inventory level in the previous period (t – 1)

•

= {

1; ℎ

0; ℎ

In order to run the model, we need to prepare the relevant data and input them into theproj.dat The data input is generated randomly, which revolves around the number of units(yield) of leaf items and its Holding cost and Backordering cost In addition, Setup cost andExternal demand are also given in advance

n = 3; // number of leaf items

SheetConnection Data ( "dat.xlsx" );

In terms of Item, the number of units of each leaf item, holding cost, and backordering costare then respectively sourced from the excel file The setup cost of the parent item in eachperiod and the External demand of the leaf item in each period are also derived from thesame file through the SheetRead command

SheetRead ( Data , "Data! C6:E6" ); // Holding cost

cost Setup from SheetRead (Data, "Data! C13:I13" ); //

External demand

DLT from SheetRead ( Data , "Data! D28:H30" ); //DLT

The project’s major testing data set is mostly available in the paper example The

collection of data includes the following contents:

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the parent item

This uncertain parameter is, according to the paper, an independent discrete random variablewith a known and bounded probability distribution Given − = 1 and + = 3, the value of DLT

equals {1, 2, 3} When the two-level product structure is being considered, in each

period, existing only one value of DLT and that value can be either 1, 2, or 3

In this case study, the following set of the real lead time of 3 scenarios is taken intoconsideration:

The data collection is input into an Excel file and checked for validity Since the sample size of the case study is not so large, the data validity checking can be conducted manually.After checking for data validity, the testing process is implemented by the CPLEX Solver for reviewing and other purposes

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❖ In general, if the number of scenarios is 5 and the range of lead time is between 1 and 3, the

total number of possible non-identical scenarios obtained will be 243 (35) However, any scenarios

can be repeated creating a larger size of S (the number of scenarios) With the known distribution

probability of each DLT value and number of S, a random set of scenarios can be generated using

another programming language

The first features included in the model are n - the number of leaf items, p - the number

of periods, and c – the number of scenarios, all of which are open for data substitution As

each item is assigned one specific number ranging from the first number of leaf items to the

nth, the set Item is established Similarly, set Period and Scenario represent the period

ranging from 1 to p and scenario 1 to c respectively The set Periodz also represents the

period, but also adding period number 0 and p+1, hence ranging from 0 to p+1 instead

// INPUTS

int n = ; // Number of leaf items

int p = ; // Number of periods

int c = ; // Number of scenarios

range Item = 1 n // Set of leaf items

range Period = 1 p // Set of periods

range Scenario = 1 c // Set of scenarios

range Periodz = 0 p 1 // include period 0

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b) Parameters

Each item participating in the disassembling process can be used to obtain a certain unit R

D is the external demand for a certain item in a period In order to prepare better, the firmmust forecast disassembly lead time, in this case, is PLT On the other hand, the real leadtime in different Periods of each scenario is DLT And M is a very large integer

Moreover, during the process, there are also holing cost and backordering cost of each leafitem And last but not least, the setup cost for parent item is obtain in each Period

int R[Item] = ; // Units obtained from

demand

int PLT = ; // Planned disassembly lead

time float H [ Item ] = ; // Holding cost

float A [ Item ] = ; // Backordering cost

float Setup [ Periodz ] = ; // Setup cost

int M = ; // Big M

int DLT [ Period ][ Scenario ] = ; // Real disassembly lead time // in C++

c) Decision Variables

There are 4 decision variables in this model, relatively the order quantity release x,

and Backorder, respectively, and finally the binary variable u to identify whether or not

to assembly an item at a period These mentioned variables are presented below:

dvar int+ x [ Periodz ]; // Order quantity release

dvar int Inventory [ Item ][ Periodz ][ Scenario ]; // Inventory level of item i

dvar int Backorder [ Item ][ Periodz ][ Scenario ]; // Backordered quantity of

dvar boolean u [ Period ];

2. Processing

a) The Objective function

The goal of the function is to hold the ATC to a minimum, along with the average expense

of maintaining backlogs and supplies per part I and the cost of preparing the finishedproduct, and all scenarios

sum ( in Period , i in Item , s in Scenario )

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The first equation refers to each scenario at the end of each period to the retention of the

inventory movement of component i Specifically, if the Inventory is equal to a

positive integer, the result will show that integer On the other hand, if it is equal to a

negative one, the result will show 0 instead

The second equation refers the back bound norm for each of the i leaf items at the end of

each i loop Similar to the first equation illustrating the Inventory level, if Backorder is

equal to a positive integer, the result will show that integer On the other hand, if it is equal

to a negative one, the result will show 0 instead

The last equation is the maximal design for the disassembly predictor for each period i

This equation ensure that x will only have value that is larger than 0 when the binary

variable u is equal to 1 for the same period t

// Eq(3)

forall (t in Period, s in Scenario)

u t ] - ( x t ]/ M ) >= 0 ;

V RESULT ANALYSIS

The resulting optimal objective value obtained is 192.667, which means that the possible

minimum average total cost (ATC) would be 192.667 (unit currency)

1. Binary indicator of disassembly in period t: δt

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