lpp project

8/6/2019 LPP Project

1/23

Executive summary

Here we review different proposals of three articles based on inventory systems. All the three papers

discuss different techniques used in optimizing the inventory cost with incorporation of other costs like

handling cost, transportation cost, wasting cost etc. The Demand for each of these papers follows a

different pattern. The Paper Inventory model of deteriorated items with a constraint: A geometric

programming approach considers a deterministic and constant with time demand pattern. This is solved

by modified geometric programming (MGP) method and non-linear programming (NLP) method. This

paper compares the two methods i.e. MGP and NLP. Paper written by E.G Read and J.A.George is based

on deterministic and variable with time demand pattern which is solved by using dual dynamic

programming. This paper solves a LP for each period parametrically, and then uses a backwards

recursion, based on the marginal conditions of DP, to generate the optimal operating strategy for the

entire horizon in a convenient form. The third paper takes demand to be a random variable that obeys a

known, stationary probability distribution.


2/23

Table of Contents

1. Introduction

2. Problem Statement

3. Background of Methodology

4.1 Review of article 1

4.2 review of article 2

4.3 review of article 3

5. Conclusion


3/23

1. INTRODUCTION

Inventory too many small business owners is one of the more visible and tangible aspects of

doing business. Raw materials, goods in process and finished goods all represent various forms of

inventory. Each type represents money tied up until the inventory leaves the company as purchased

products. Likewise, merchandise stocks in a retail store contribute to profits only when their sale puts

money into the cash register.

In a literal sense, inventory refers to stocks of anything necessary to do business. These stocks

represent a large portion of the business investment and must be well managed in order to maximize

profits. In fact, many small businesses cannot absorb the types of losses arising from poor inventory

management. Unless inventories are controlled, they are unreliable, inefficient and costly. The inventory

control plays a significant role to take decisions, it is important to understand what should be stocked and

when should we re-ordered for stocks.

A good inventory system is a necessity for a fruitful production system in industries. But to

identify the right kind of production and inventory system for a manufacturing industry is very difficult.

In identifying the best model which makes the realization of the large investment is a difficult job, also

this cost remains fixed for a long period of time. Extensive research in optimization of the inventory cost

is taking place.

In another field where inventory management is important is in transportation they share a

common thread. Many companies have to answer the question inventory cost v/s actual transportation

cost is optimal. With many distribution centers and requirement of continuous replenishment in these

centers the optimized inventory levels and reducing cost functions like shortage cost and holding cost can

provide high profit margins to companies.

A good optimization of inventory for effective production and distribution increases the

expectations of customer and helps company exceed its reputation and maximize its profits.


4/23

2. PROBLEM STATEMENT

To create an optimized model for inventory control it is essential to understand the demand. The

complexity of a model depends on whether the demand is probabilistic or deterministic.

The demand pattern in general is categorized under the following four types:

1) Deterministic and constant (static) with time.

2) Deterministic and variable (dynamic) with time.

3) Probabilistic and stationary over time.

4) Probabilistic and non-stationary over time.

The three papers reviewed in order have incorporated demand pattern as deterministic and

constant, deterministic and variable and probabilistic and stationary respectively.

The three papers have a common goal of inventory control but with different areas of

concentrations. In one of the papers the inventory model is gauged in accordance with deteriorating items,

it first develops a EOQ model the problem is formulated with and without truncation on the deterioration

term and ultimately is converted to the minimization of a signomial expression with a posynomial

constraint. It is then solved by modified geometric programming (MGP) method and non-linear

programming (NLP) method. The problem is supported by numerical examples. The results from two

versions of the model (with and without truncation) and two methods (i.e. MGP and NLP) are compared.

We see that different models have developed over the years for Production/inventory systems.

Systems, in which items are produced, or collected, and stored for future use in a warehouse,

reservoir or stockpile, have proved to be one of the most fruitful areas for the application of

Bellman's method of dynamic programming (DP). The computational challenges arising from this

area have also served to stimulate many developments and refinments of the basic idea of DP. Though

there have been approaches involving NLP and MGP for a inventory model trying to provide the


5/23

production schedule. But, DP provides an optimal strategy for the production decision which should be

made, given any stock level in any period of the planning horizon.

Control of inventory is also of concern in the distribution center and the demand patterns are

similar in this case also. The problem is to determine the target inventory and the transportation

quantity in order to minimize the expectation of the sum of inventory related costs and transportation

costs. Simulation and linear programming are used to calculate the expected costs, and a random local

search method is developed in order to determine the optimum target inventory. A genetic algorithm is

also tested and compared with the proposed random local search method.

3. BACKGROUND OF METHODOLOGY

Successful inventory management involves balancing the costs of inventory with the benefits of

inventory. Many small business owners fail to appreciate fully the true costs of carrying inventory, which

include not only direct costs of storage, but also the cost of money tied up in inventory. This fine line

between keeping too much inventory and not enough draws attention for optimization. The reviewed

models of Inventory control in deteriorating items, production and distribution are carried out below

mentioned methodologies.

DETERIORATING ITEMS: NON LINEAR PROGRAMMING

The non-linear optimization problems have been solved by various non-linear optimization

techniques. Among those techniques, geometric programming (GP) is an efficient and effective method to

solve a particular type of non-linear problems.

Nonlinear programming is the process of solving a system of equalities and inequalities,

collectively termed constraints, over a set of unknown real variables, along with an objective function to


6/23

be maximized or minimized, where some of the constraints or the objective function are nonlinear.

Modified Geometric Programming (MGP) is a class of nonlinear optimization

PRODUCTION & INVENTORY SYSTEMS: DYNAMIC PROGRAMMING

Dynamic programming is a method of solving complex problems by breaking them down into

simpler steps. It is applicable to problems that exhibit the properties of overlapping sub

problems and optimal substructure. When applicable, the method takes much less time than naive

methods.

INVENTORY-DISTRIBUTION SYSTEM: LINEAR PROGRAMMING, GENETIC ALGORITHM

Linear programming (LP) is a technique for optimization of a linear objective function, subject

to linear equality and linear inequality constraints. Informally, linear programming determines the way to

achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model and

given some list of requirements represented as linear equations.

Genetic algorithm (GA) is a search technique used in computing to find the exact or approximate

solutions to optimization and search problems. Genetic algorithms are categorized as global search

heuristics.

4.1 REVIEW ON ARTICLE 1:Inventory model of deteriorated items with a constraint: A geometric

programming approach

Summary: The problem considered is an inventory model for deteriorating items with limited storage

space. This model follows the EOQ model and is formulated with and without truncation on the

deterioration term and ultimately is converter to a minimization of inventory cost model. The method of


7/23

solving this model is by Geometric programming method and non-linear programming. An example is

considered to support the problem and the results from the two versions of the model are compared.

Before generating this model, the author makes some assumptions:

First, items deteriorate at a constant rate;

Second, the inventory system has an infinite rate of replenishment, constant demand rate and limited

storage space.

The replenishment happens in time period Ti. Figure 1 shows the relationship between qi(t), the inventory

level at t and t:

Figure 1

Parameter definitions:

y i: ith item

y Qi : Order quantity

y C0i: Purchasing cost

y C1i: Holding cost per unit quantity per unit time

y C3i: Set-up cost

y i: Constant rate of deterioration (0


8/23

y TC(T): Total cost

Model Formulation

From the curve in Figure 1, in every time period [0,Ti], we have:

0ii i i i

dqq D for t T

dtU ! e e

This equation means that the reducing rate of inventory at t equals to the deteriorating rate at t plus

demand rate at t. So,0

1log( )

iQ i i ii

i i

Q dt

U

U

! !

. Transfer it in term of T i, we have the purchasing

items:

( 1)i i

ii

i

Q eUU!

. So the purchasing cost is 0i ic Q

Same, the holding cost is1 20

( ) ( 1)i

i iQ

ii i i i i

i

c H q t dt e U U

U! !

The total cost= purchasing cost + set-up cost + holding cost. The objective is:

1

min ( ) [ ]i n

ii i

i i i

a e C b c

U

!

!

Where

0 1 0 1 132 2

, ,i i i i i i i i i ii i i i

i i i i i

c

c

c

c

c

a c b cU U U U U

! ! !

The only constraint is the total area of all items needed must be less than or equals to the available storage

area W. So, the problem is subjected too 1( ( 1))

i ini

i

i i i

e W

U

U! e

,0

i "

Solution method:

This paper uses two methods to solve this problem: Lagrangian multiplier method and Geometric

programming.

1. Lagrangian multiplier method


9/23

This problem is a non-linear programming problem which can be solved by Lagrangian multiplier

method.

The conditions are:

2( 1) 0i i i

i i i i i i i

i

La b e

e

U UU Px ! !x

0SP !

0P u

To show the optimal solution* * *

( , , )T SP is the global minimum of this problem, the author proves that all

principle minors in a bordered Hessian matrix are not positive but one of them is zero.

So, we can get the global minimum by using Lagrangian multiplier method.

2. Geometric programming approach

Since

lim(1 )i iT i i

Te

U J

J

U

Jpg!

for sufficient large value ofJ

Let0

1

( , ) ( , )n

i i i i i

i

TC T TC T cF F!

! , 1

( )

i ii n

i ii

i i

w Dk

w DWU

U!

!

The problem changes to

0

1

min ( , ) ( )n

i ii i i

i i i

a bT T

T T

JF F!

!

Subject to 11

n

i i

i

kJF

!

eand

1 1 1i ii iTU

F FJ

e

, 0, 1, 2,...,i i

T i nF " !

However, the primal function is hard to solve for its DD=(3n-1). We can solve its dual problem. Using

MGP method given by the author, the dual problem is

2

1

( 1) 0iTn

ii

i i i

D

ew W S

T

U

P U!

x! !

x


10/23

1 2 3 01 02

1 1 2 3 01 02

[( ) ( ) ( ) ( ) ( )i i i i in

w w w w wi i i i i i

i i i i i i

a b kMax dw

w w w w w

K Q U Q

!

!

! 4

Where

"

#

n

i

i

wK!

!,

$

%

$ &

i i iw w Q !

,

'

(

) ' )

( (

0, ,i i i i i i

1 1 1 1 1 1 J J J! ! !

.

First, dw can be showed in term of 2 iw

,3

iw

and then solve log(dw) to get optimal solution

4

w . So

4

4 4

5

6

i

i

i

naT

w TC!

Validation of example considered:

We use the data from this article

Using NLP method by Lingo we obtain the optimal solution:

7

8

989869 76 587 8T T! ! !

Using Geometric programming method by MATLAB, we obtain the optimal solution:

@

A

B, ,T T TC ! !


11/23

09-11-24 8:49 C:\Users\Lizhe\Documents\MATLAB\MGP.m 1 of

x0=

[5;

5;5

;5]

;

r=1

000

00;

options=optimset('Display','iter');

[x,fval] = fsolve(@myMGP,x0,options);

k1=2*350/(0.08*(600+2*350/0.08+4*450/

0.07));

k2=4*450/(0.07*(600+2*350/0.08+4*450/

0.07));

a1=52450;b1=52500;a2=74237.8;b2=74387.8;rr=x(2)+x(4);w011=r-1+r*x(1)+r*x(2);w012=r-1+r*x(3)+r*x(4);

dw=(a1/x(1))^(-x(1))*(b1/(1+x(1)))^(1+x(1))*(k1*rr/x(2))^x(2)*((w011+1)/w011)^(w011)*(0.08*(w011+1)/r)* (a2/x(3))^(-

x(3))*(b2/(1+x(3)))^(1+x(3))*(k2*rr/x(4))^x(4)*((w012+1)/w012)^(w012)*(0.07*(w012+1)/r)

T1=a1/x(1)/sqrt(dw)

T2=a2/x(3)/sqrt(

dw)

TC=sqrt(dw)*2-

3500-3857.1

%save as

myMGP.m

%function F

= myfun(x)

%r=100000;

%k1=2*350/(0.08*(600+2*350/0.08+4*450/

0.07));

%k2=4*450/(0.07*(600+2*350/0.08+4*450/

0.07));

%F = [log(x(1))-log(52450)+log(52500)-log(1+x(1))-r*log(r-

1+r*x(1)+r*x(2))+r*log(r+r*x(1)+r*x(2));

% -r*log(r-1+r*x(1)+r*x(2))+r*log(r+r*x(1)+r*x(2))+log(k1)-log(x(2))+log(x(2)+x(4));

% log(x(3))-log(74327.8)+log(74387.8)-log(1+x(3))-r*log(r-1+r*x(3)+r*x(4))+r*log(r+r*x(3)+r*x(4));

% -r*log(r-1+r*x(3)+r*x(4))+r*log(r+r*x(3)+r*x(4))+log(k2)-log(x(4))+log(x(2)+x(4));];


12/23

Solution report:

Norm of Firstorder Trustregion

Iteration Funccount f(x) step optimality radius

0 5 0.605897 0.116 1

1 10 0.326867 1 0.0988 1

2 15 0.0417747 2.5 0.0513 2.5

3 20 0.0128444 6.25 0.0117 6.25

4 25 0.00409206 6.25 0.00352 6.25

5 30 0.00178838 6.25 0.000482 6.25

6 35 0.000852625 6.25 0.000483 6.25

7 40 0.000413007 6.25 6.25e005 6.25

8 45 0.000252881 6.25 0.000327 6.25

9 50 0.000130925 6.25 0.000123 6.25

10 55 6.5438e005 6.25 0.000127 6.25

11 60 4.37216e005 6.25 0.000323 6.25

12 65 2.18878e005 6.25 5.59e005 6.25

13 70 1.1915e005 6.25 5.28e005 6.25

14 71 1.1915e005 11.8636 5.28e005 15.6

15 76 8.82761e006 2.9659 0.000138 2.97

16 81 8.412e006 0.00287294 7.26e006 2.97

17 86 6.98601e006 2.9659 0.000126 2.97

18 91 6.73471e006 0.00229538 7.03e006 2.97

19 96 4.72377e006 2.9659 7.57e005 2.97

20 101 4.62953e006 0.00118923 4.11e006 2.97

21 106 4.36433e006 2.91123 0.000111 2.97

22 111 4.19011e006 0.00149164 4.81e006 2.97

23 112 4.19011e006 2.06945 4.81e006 2.97

24 113 4.19011e006 0.517362 4.81e006 0.517

25 114 4.19011e006 0.12934 4.81e006 0.129

26 115 4.19011e006 0.0323351 4.81e006 0.0323

27 116 4.19011e006 0.00808378 4.81e006 0.00808

28 117 4.19011e006 0.00202094 4.81e006 0.00202

29 118 4.19011e006 0.000505236 4.81e006 0.000505

30 119 4.19011e006 0.000126309 4.81e006 0.000126

31 120 4.19011e006 3.15773e005 4.81e006 3.16e005

32 121 4.19011e006 7.89432e006 4.81e006 7.89e006

33 122 4.19011e006 1.97358e006 4.81e006 1.97e006

34 127 4.19011e006 4.93395e007 4.09e007 4.93e007

Optimization terminated: firstorder optimality is less than options.TolFun.

dw =2.61+e007

T1 = 0.2323TC=2. C D 0e+003T2 = 0.2 E 0 T


13/23

4.2 REVIEW ON ARTICLE 2:Dual dynamic programming for linear production/Inventory systems

Summary: The problem considered is scheduling production of an inventoried item, using a production

technology described by a linear program. Traditional approaches involve using Linear programming to

find a schedule for a specific starting inventory, or using dynamic programming. This paper describes a

formation of a method that can be seen as a dual to a conventional Dynamic Program. In a conventional

dynamic program and essential grid of primal variables is chosen and for each one the optimal decision is

found, which gives the implied shadow prices, which in this case is a marginal value of stock. Here the

author chooses a grid of dual variables, the critical marginal stock values at which the production decision

changes, and for each one the corresponding stock level in the primal state space is found. The advantages

of the method, in terms of accuracy and efficiency, stem from the fact that it deals directly with the

critical values which determine the form of the optimal strategy, rather than trying to infer them by

interpolating on an arbitrary grid. This also eliminates any need to approximate solutions by successive

refinement of the grids, as is commonly done in Dynamic Programming. For the problem studied here the

critical values of the dual variable can be determined by parametric programming on the Linear Program

describing the production technology available in each period.

Assumption 1: The basic model which excludes wastage, discounting and holding costs is a linear

program which describes the production technology in each period.

Assumption 2: This paper concentrates on the problem of scheduling production of a single product,

which is produced in a factory whose technology is described by a Linear Program, and then stored for

future use if desired.


14/23

Assumption 3: The value of the end-of-horizon inventory (VT) ensures that stocks are maintained at a

reasonable level. Vt is required to be a concave, so that the marginal value curve (mT) is monotonically

decreasing.

Terminology:

y Xt : a vector of activity variables for inputs, intermediate commodities, processes etc. in period t,

y yt : output of the final product in period t,

y st : stock of the final product at the end of period t,

y dt : the demand for the final product in period,

y vt(st): the value of the end-of-horizon inventory,

y SMint, SMaxt : Minimum and Maximum stock levels at time t,

y Mt (st): Marginal value function,

y ht : holding cost per unit of inventory in period t,

y : discounting factor per period

y wt: wasting factor for period t.

Model formulation: A linear production model for a single product with multiple input resources and

multiple processes is considered. The objective is to minimize inventory cost which includes direct

holding costs for the inventory, wastage of goods in the inventory and discounting of revenue and costs.

In this method, first we consider the model excluding holding cost, wastage of inventory and discounting

of revenue to obtain a basic model.

Basic model:

The Basic model follows a linear program whose objective is to reduce the inventory cost. The constraints

being,


15/23

y Variables of input and output are constant.

y The summation of stock at end of a particular period and the output of the final product at that

particular period is equal to the demand.

y The stock at time t is within the range of stock.

Initially this LP is solved parametrically for demand, ranging within the predetermined lower and upper

limits of output for the product for each time period t = 1 . . . , T. This yields a series of optimal basic

solutions i = 1 . . . . . I (t), over the range of feasible output of the final product in t. These define a series

of distinct "production processes", with the marginal cost of production for process i being it, the dual

value of the constraint yt = dtat the ith basis.

Thus for the implied problem we have

Algorithm 1

y Solve the single-period sub-problem parametrically to produce a supply curve.

y Apply backward recursion to produce guidelines for each period in the planning horizon.

These guidelines, which completely characterize the optimal strategy, can be conveniently displayed on a

simple chart. It is observed that for this basic case the shape of the end-of-horizon marginal value function

between the guideline levels is largely irrelevant. Provided it is monotone so that we can identify

guidelines from the curve, the algorithm will produce optimal management guidelines for the entire

period. In fact, each guideline can be produced quite separately from the others, a potentially useful way

of producing approximate solutions for large problems.

From the marginal value curve as we continue with the process it yields the solution displayed as a chart

of production guidelines, for a particular time period. Thus, an optimal trajectory of production can be

traced from any initial stock level at a particular time period to the end of the planning horizon. Using this

data a steady state solution can be established provided the demand pattern is typical.


16/23

Holding cost, wastage and discounting:

The extend model is considered which includes the three types of inventory holding "cost"; direct holding

costs for the inventory, wastage of goods in the inventory and discounting of revenue and costs. The basic

structure of the problem is preserved and the same basic algorithm is applied, with some modifications to

the objective, constraints and the backward recursion formulae. Thus for this case we have:

Algorithm 2:

For each period t = T . . . 0:

y Produce a supply curve by solving its Linear Program model parametrically.

y Identify guidelines from the end-of-period value curve.

y Insert flats and Update the marginal value curve using backward recursion of the modified

equations.

This algorithm defines all of the marginal value curves and associated decisions exactly at each stage.

Thus, it is actually more accurate than the discrete approximation produced by conventional Dynamic

Program.

Validation of example chosen:

In order to explain the formulation of the inventory marginal supply curve for time period T in the basic

model an example is given in the paper. In order to validate this example we used solver to formulate the

LP of the basic model and from this we obtained the marginal values for each demand. From the solution

we see that the marginal value curve is obtained to be a monotone function and hence the assumption 3 is

a valid one. The LP of the given example is as shown below,

Objective: Minimize 4. x1+ . x2+ . x3+ . 2x4+0.2Sot+0.1 Tot+0.12Uot+0.1 Vot

Subject to

xl + x2 - SOT 4 0

x3 + x4 - TOT 4 0


17/23

x1 + 10x2 + 3x4 - UOT 4 0

3xl + x3 + x4 - VOT 4 0

x1 + x2 + x3 + x4 - y = O

y=d

SOT 0, TOT 0, UOT 0, VOT 0.

Objective F 0 0 30 0 0 0 10 0minz= G 20.H

x1 x2 x3 x4 Sot Tot Uot Vot

4.I H

G . P G . F G . P 2 0.2 0.1F G

0.12 0.1 G

1st cons P H -1 420


18/23

REVIEW ON ARTICLE 3: INTEGRATED OPTIMIZATION OF INVENTORY-DISTRIBUTION

SYSTEM BY RANDOM LOCAL SEARCH AND A GENETIC ALGORITHM

Summary: For a typical order up-to-R policy and transportation problem an integrated optimization model

is built. The problem is to find the target inventory and transportation quantity in order to reduce the cost

related with them. Linear programming is used to calculate the expected cost and random local search is

developed to get an optimum target inventory. The paper then compares a genetic algorithm to check the

effectiveness of this model, computational experiments are carried out. After taking appropriate

assumptions the model is developed. This model contains two decision variables the first is the target

inventory for replenishment and its value stays same once it is determined at the beginning of the

planning horizon. The second is the transportation quantity it must be determined by the given targetinventory. With suitable assumptions and well assigned notations the problem is solved in two phases,

first we determine the exact targeted inventoryRi which is calculated with the help of the initial inventory

given by Q* this gives the direction to vector R that minimizes F(R) because Q corresponds to R in the

original problem. A genetic algorithm is also created to find optimum R and this model is compared with

the random local search method. This Ri value is then utilized the main model to find out the optimum

values for the decision variables to minimize the overall function of transportation cost, holding cost &

shortage cost.

Assumptions: In order to get the integrated model we have to introduce assumptions and they are as

follows:

1) There exist multiple distribution centers and consumer points. Distribution centers transport

commodities ordered by consumers. Single item is treated in order to establish a basic model.

2) A distribution center can supply multiple consumers, and a consumer can be supplied by multiple

distribution centers.

3) The planning horizon consists of a series of short intervals each of which has the same length.

The length of an interval is small enough compared with the length of a review period.

4) Demand in each interval at each consumer point is assumed to be a mutually independent

random variable that obeys a known, stationary normal distribution.

) Order-up-to-R policy is adopted as the inventory policy at each distribution center. The length

of a review period is assumed to be a multiple of the length of an interval. Replenishment lead


19/23

time from plants or suppliers to each distribution center is deterministic and assumed to be a

multiple of the length of an interval.

) Commodities are transported to consumer points at the end of each interval. All shortages are

backlogged and the commodities are transported to the consumer points as soon as the inventory

is recovered.

Model Formulation:

The following notations are used for models to follow:

m: number of distribution centers

n: number of consumer points

i: distribution center index (i=1,2,,m)

j: consumer point index (j=1,2,,n)

U ij: unit transportation cost from distribution centeri to consumer pointj

Ri: target inventory for replenishment at distribution centeri

RiU: upper limit ofRi

R=(R1,R2,,Rm)T

U hi, U si: unit inventory holding cost and unit shortage cost at distribution centeri, respectively

K: number of intervals constructing planning horizon

: length of an interval

T: length of a review period (a multiple of)V

i: replenishment lead time from plants or suppliers to distribution center i (multiple of)

Djk: demand in interval kat consumer pointj (random variable)

j, j: mean and standard deviation ofDjk, respectively

djk: value ofDjkobserved in interval k

xik+: inventory level at the end of interval kin distribution centeri

xk+=(x1k

+,x2k+,,xmk

+)T

xik: backorder level (shortage level) at the end of interval kat distribution centeri

xk=(x1k

,x2k,,xmk

)T

yijk+: quantity transported from distribution centeri to consumer pointj in interval k

yijk

: quantity transported from distribution centeri to consumer pointj in interval k>kbecause of

shortage.

As mentioned earlier the model is formed in two phases the first phases concentrates in finding the

optimum target inventory for replenishment R*. The procedure to calculate F(R) is that, Fork=1, we start

withx0+=R,x0

=0 we generate random numbers djk after which we calculate the optimum value z* of

problem PK we need to make sure that K>K and then finally F(R) is the sum divide it by K.


20/23

Random Local Search Method: In this method we consider the following assumptions before proceeding:

1. j=0 for allj,W

i=0 for all i.

2. Each distribution center has initial inventory X i (i=1,2,,m).

3. Commodities are transported to meet demand k0j (j=1,2,,n) in time period T.

We introduce problem PLP to determine the transportation quantity and the inventory level that

minimize the sum of transportation costs and inventory holding costs only in time period T(length of a

review period) we use this PLP to calculate the optimum solution Q* . This value of Q helps us find the R

which is given by R=W(T

X

*) . Where W is the function of x which is equal to min of x andRi

U

Genetic Algorithm:

In another procedure we test genetic algorithm to find out the initial target inventory which is given by

representations for chromosomes, creations of initial parent set, crossover and mutation.

The following steps are used for the genetic algorithm

1. Representation. We consider a chromosome with length m whereRi is the ith gene. We use the same

notation as a vector R=(R1,R2,,Rm)T

to represent a chromosome.

2.Initial parent set. We create one parent P1=(R1,R2,,Rm)T by generating uniform random numbers Ri on

[0,RiU] (i=1,,m) and create initial parent set {P1,P2,,PN} by iterating thisNtimes, whereNis the

population size.

3.Y

rossover and mutation. We produce offspring with two-point crossover. Whenever an offspring is

produced, mutation is applied with probability q0. The operation of mutation is as follows.

Choose one gene of the chromosome at random.

If it is the hth element then replace it with a uniform random number on [0,RhU].

4. The parents of next generation.

The parents of next generation are selected as follows.

1) We first Calculate pl forl=1,,N.

2) Select one element from {P1,P2,,PN} with probabilitypl (l=1,,N). Select one more element

from the same parent set {P1,P2,,PN} with probabilitypl (l=1,,N). (Two selected elements are

allowed to be the same one.) Then create a pair of offspring by crossover.

3) Iterate the procedure, N/2 times.

4) Take the union of parent set and offspring set. Let the union be {Q1,Q2,,Q2N}. Arrange

2Nelements of the union in increasing order ofF(Ql) (l=1,2,,2N). Take the firstNelements

from the top and let them be the parents of the next generation. We iterate until Gth generation is

created.


21/23

After the calculation of the R value we sub it into the main model hence entering into the phase 2 of the

model given by

Subject to:

xik+0, i=1,2,,m,

xik0, i=1,2,,m,

yijk+0, i=1,2,,m;j=1,2,,n,

yijk0, i=1,2,,m;j=1,2,,n,

This gives us the final solutions of the optimal values for the decision variables i.exik+xik

yijk+yijk

This provides us with the most cost effective inventory-distribution system.

Computational Validation:

With a simple example

Let m=2, n=4, =1 day, T=10

days,K=2000,

h1=

h2=0.4 dollars/(unitday),

s1=

s2=4 dollars/(unitday),a

1=a

2=3 days,

1=2=3=4=10 units. 1=2=3=4= units, R1U=R2

U=. Let the matrix ofCij's be

We the optimum value of R to be (234, 234)T and the Zopt is approx 1 1 dollars after this we need to

calculate the transportation quantity from the problem Pk

we see that the the commodities are allowed to be transported from one distribution center to multiple

consumer points but it is prohibited to be transported from multiple distribution centers to one consumer

point, this is because there is an substantial amount of percentage cost that goes up if the latter is choosen.

Also with the help of a computational experiment it is found out that if the iteration number is very large

then the genetic algorithm provides a better performance, but the random local search method has betterperformance in smaller number of iterations.


22/23

CONCLUSION

Conclusion Article 1: Comparing this two method, NLP method has the better solution than MGP

method, which is different from the conclusion given by the author. In the article, MGP method has the

better solution than NLP. Considering that the author does not give any information of what solver his use

by NLP and MGP method. In this project, MATLAB is used for MGP method and Lingo is used for NLP

method. Although MGP method has a little worse solution due to this problem, MGP has its great

advantage: it changes this NLP problem into a much easier problem to solve. In this inventory model of

deteriorated items, when there are enormous types of inventory items, MGP will perform better.

Conclusion Article 2: The paper develops a hybrid technique, using LP and DP, to schedule production

of a single product over a planning horizon. This method generated optimal management strategy for all

periods in the form of a production chart by just solving a single period optimization model. The

production chart reveals a lot of information about the system and this approach requires little

computational time to perform post-optimal analysis. This approach has been applied in two real-life

problems, formulated as two-dimensional stochastic DPs. NLP can also be handled provided the functions

involved are differentiable.

Conclusion Article 3:

After reviewing the complete paper the conclusions I draw are:

1) An integrated model for inventory and distribution system is developed to reduce expected costs

of transporting the quantities holding the quantities and shortages.

2) The order up-to-R policy is taken into consideration with demand being probabilistic but

following a known variation.


23/23

3) Random search and Genetic algorithm are two methods used to find the optimal targeted

inventory.

4) Simple linear programming and simulation are used to calculate the overall cost.

) Through computational experiments we conclude about the feasibility of one distribution center

to many consumers is more feasible than vice versa

) The genetic algorithm proves to be better at larger iterations and random local search in smaller

iterations.