Thesis on Eit

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ECONOMICS OF INTERNATIONALTRANSPORTATION

TITLE OF THE PROJECT

OPTIMIZATION OF TRANSPORTATION COST OFLCD TV

SUBMMITED BY: SUBMITTED TO:

Dr. NAVAL KARRIR

RAJEV KUMAR PAL

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INDEX

S.No.

Topic PageNo.

1 Executive Summary 32 Introduction 4

3 Background 5

4 Transportation Problem 6

5 Approach and Methodology 8

Phase I Obtaining an initial feasible solution

6 Northwest-Corner Method 8

7 The Minimum Cell Cost method 9

8 Vogels Approximation Method (VAM) 10

Phase II Moving towards optimality

9 MODI Method 11

10 Stepping-Stone Method 1511 Analysis 17

12 Sensitivity Analysis 18

13 Conclusion 21

14 References 22

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EXECUTIVE SUMMARY

Today electronic products have become the main power of national economygrowth. All the manufacturers have set their primary markets to meet theconsumers demand. Taking the current LCD TV manufacturers as an example,they have to emphasize on the product design, performance and the reduction of cost. Therefore, nowadays the most important task for LCD TV industry is thatthe international manufacturer shall fix a suitable winwin price and productivecapacity for itself as well as for the Original Equipment Manufacturing (OEM)when the OEM has received the order, so that both sides could construct a long-established relation and they could reach the object of maximized profit.

This paper, reports the results from a series of various methods regarding how

the transportation cost of SONY LCD TV (50) could be optimized.Manufacturers have managed to break through the constraints of cost andcomplex cooperation relationship between partners, through strategic co-ordination and integration.

Effective distribution of SONY LCD TV improves all of the five performance

dimensions (cost, flexibility, responsiveness, delivery, and financial return),although the degrees of improvement on different performance dimensions aredifferent to some extent.

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INTRODUCTION

In the fiscal year 2008, ending in March 2009, Sony incurred its largestoperational loss in its history of $2.4 billion. Sharp incurred a net loss of $1.3billion in fiscal 2008, ending March 2008, citing stagnant consumption, fiercecompetition. (Source has given in references pg.)

The struggle for growth during the economic downturn forced, many

companies has stetted the target to reduce their prices in order to increase salesand to maintain their market share on the basis of contract manufacturers for cost-reduction opportunities.

To succeed in this shifting environment, Sony needs to build strong links withsuppliers. It is important that Sony and its suppliers share policies, strategiesand technology. Collaboration between Sony and its suppliers should ultimatelybe aimed at earning customer approval. This goal must form the common baseof Sony's procurement activities and its suppliers' sales activities. Sony callssuppliers capable of maintaining this kind of collaborative relationship"partners."

The overall structure of the SONY LCD TV supply chain is shown the

integration of the SONY LCD TV supply chain influences manufacturerscapability to satisfy market requirements. The integration relationship involvessuppliers, manufacturer, and firms downstream. In the co-ordination betweenmanufacturer and suppliers, the manufacturer sets up production planning bydemand forecasting or firm order from certain customers, and then places ordersto all Consumer demand and customer requirements are pushing the industryvery hard in the areas of product quality, price and delivery lead times, with

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customers expecting manufacturers to provide the production flexibility torespond to fluctuations in demand.

BACKGROUND

SONY LCD TV (50) which has total cost at factories Rs. 40,000

Transportation Mode Truck (By Road) which includes the Cost Factors (Transportation CostsRs.220/km., Admin. Cost, [email protected]%, Loading Charges, Octrai, Taxes (toll taxes+ excise duty + etc), and Warehouses charges, Unloading Charges etc.)

A liquid crystal display (LCD ) is a thin, flat panel used for electronically displayinginformation such as text, images, and moving pictures. It hasno problemto transportif it's stillin the factory packaging. Mostmanufacturersgo through a decent amount of work to makesure their packaging can stand up to fairly stressful conditions. For large LCD's it is harder todo so; however, it is usually at least packaged so that it can stand on any side. If there is no'this side up' on the box, then it is safe to transport on its side. (They are careful to note if damage will occur when transported on one side, because they would be liable for damageotherwise.) Without factory packaging-- The screens are fairly delicate to puncture andshould not be transported face-down or at a weird orientation.Factory packaging equalizespressure on all sturdy surfaces to minimise the risk of damage, and without packagingthere is none of this protection . Some of the packers movers Kanpur, Noida, Ghaziabadand Faridabad also offer warehousing and storage facilities to the customers. They have wellbuilt, spacious and moisture free warehouse well guarded by the companies employed guardsand workers. Warehouse are friendly for the storing all kinds like household items, industrialgoods, commercial goods, office goods and many other lovable commodities of thecustomers.

Table (A) Distance and Warehouses between Destination and Sources

S.N. Distance (Km) W1 W2 W31 GZB-Dehradun (216km.) Saharanpur

2 GZB-Mumbai (1418 km.) Bhopal Pune

3 GZB-Chennai (2114 km.) Bhopal Nagpur Vijayawada

4 Delhi-Dehradun (235km.) Saharanpur

5 Delhi-Mumbai (1407 km.) Bhopal Pune

6 Delhi-Chennai (2095km.) Bhopal Nagpur Vijayawada

7 Kanpur-Delhi (575 km.) Delhi Saharanpur

8 Kanpur- Mumbai (1288 km.) Nagpur Pune

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mailto:[email protected]%25mailto:[email protected]%25


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9 Kanpur- Chennai (1885 km.) Nagpur VijayawadaTRANSPORTATION PROBLEM

A destination may receive its demand from many sources.So, The objective is to determine the transporting plan, to meet all demands but not exceedany supply, to minimize the total transportation cost.The standard transportation model seeks to find a transportation plan for SONY LCD TV(50) from Ghaziabad, Delhi and Kanpur warehouses toDehradun, Mumbai and Chennaistores.The data in the model includes:

(1) The amount of supply at each source and the demand at each destination;(2) The unit transportation cost of theSONY LCD TV (50)from each source to each

destination.Table (1)

Warehouse/Stores

Dehradun Mumbai Chennai Supply

Ghaziabad Rs.2667 Rs.3781 Rs.4525 150 UnitsDelhi Rs.2768 Rs.3760 Rs.4500 200 UnitsKanpur Rs.3473 Rs.3635 Rs.4267 150 UnitsDemand 100 Units 200 Units 200 Units 500 Units

Transportation Problem is a special case of Linear Programming:

The LP formulation of the TP problem is:

Let,

Xij = quantity of SONY LCD TV (50) transport from source i to destination j

Cij = per unit transporting cost from sources i to destination j (fromGhaziabad, Delhi andKanpur warehouses toDehradun, Mumbai and Chennaistores)

Si be the row i total supply (where i= 1, 2, 3,)

Dj be the column j total demand (where j= 1, 2, 3)

For this type of problem all units are available

3 Warehouse and 3 Stores

No. of Variables is 9

No. of Constraints is 6 (Constraints are for warehouses capacity and stores demand)

To solve the transportation problem by its special purpose algorithm, it is required that thesum of the supplies at the warehouses equal the sum of the demands at the stores.

Si(i=1,2,3)= Dj(j=1,2,3)= 500 units6


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LP Formulation

The linear programming formulation in terms of the amounts shipped from the origins to thedestinations, Xij, can be written as:

Objective function:

Minimize Z = 2667X11 +2768X21 +3437X31 + 3781X21 + 3760X22 + 3635X23 + 4525X31+4500X32 +4267X33

Subject to the constraints:

X11+X21+X31> 100

X12+X22+X32> 200

X13+X23+X33> 200

X11+X12+X13< 150

X21+X22+X23< 200

X31+X32+X33< 150With non negativity condition: X11,X12,X13,X21,X22,X23,X31,X32,X33> 0

Network Representation:

Figure (1)

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Dehradun

(S1) 150 units

Mumbai

(S2) 200 units

Chennai

(S3) 150 units

Ghaziabad

(D1) 100 units

Delhi

(D2) 200 units

Kanpur

(D3) 200 units


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(Warehouses) (Stores)

APPROCH AND METHODOLOGY

The transportation problem is solved in two phases:

Phase I obtaining an initial feasible solution

Phase II moving toward optimality

In Phase I , the Minimum-Cost Procedure can be used to establish an initial basic feasiblesolution without doing numerous iterations of the Simplex Method.

There are three different ways:

Northwest corner method

The Minimum cell cost method

Vogels approximation method (VAM)

Northwest corner method :

The North West corner method is easy to use and requires only simple calculation. As themethods name implies, we start work in the northwest corner or the upper left cell. Make anallocation to this cell that will use either all the demand for that row or all the supply for thatcolumn, whichever are smaller as cell (1, 1) 100 units (Rs. 2667). We see that, site1s supplyis smaller than Dehraduns demand. This eliminates column site 1 from further considerationbecause we used all its demand and now repeat the above steps, we have the followingtableau and Stop since all allocated have been assigned.

Initial tableau of NW corner method

Table (2)

Warehouse(i)/Stores(j)


Ghaziabad Rs.2667 Rs.3781 Rs.4525 150 Units

Delhi Rs.2768 Rs.3760 Rs.4500 200 Units

Kanpur Rs.3473 Rs.3635 Rs.4267 150 Units

Demand 100 Units 200 Units 100 Units 500 Units

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100 50

150 50

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The initial basic feasible solution is:

2667X11 +3781X21 + 3760X22 + 4500X32 +4267X33 =

2667(100) +3781(50) + 3760(150) + 4500(50) +4267(150) = Rs.18,84,800

Minimum cell cost method :

The initial basic feasible solution obtained by this method usually gives a lower beginningcost, so start with lowest cost(Rs.2667)entry 100 units in the cell (1, 1) and allocated asmuch possible, i.e., X11= 100 units. The next lowest cost (Rs.2768) lies in the cell (2, 1), somake no allocation, because the demand from Dehradun Store was already used in the cell (1,1). The next lowest cost (Rs.3635) lies in the cell (3, 2), so allocated X32 =150 units.Similarly for cell (2, 2) allocation is X22 = 50 units, X31 =50 units, X32= 150 units.

Initial tableau of the Minimum cell cost method

Table (3)








2667X11 +3760X22 + 3635X23 + 4525X31 +4500X32 =2667(100) +3760(50) + 3635(150) + 4525(50) +4500(150) = Rs.19,01,200

The cost is more by Rs.1, 16,400, as compared to the cost obtained by Northwest corner method.

Vogels approximation method (VAM) : 9

100

50

50

15

150


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In this method there are some steps to obtaining initial basic feasible solution.

Step 1: for each column and row, determine its penalty cost by subtracting their two of their least cost

Step 2: select row/column that has the highest penalty cost in step 1Step 3: assign as much as allocation to the selected row/column that has the least cost

Step 4: Block those cells that cannot be further allocated

Step 5: Repeat above steps until all allocations have been assigned

Initial tableau of the Vogels approximation method (VAM)

Table (4)

Penalty (2768-2667)=101

(3635-3760)=125

(4500-4267)=233


Dehradun Mumbai Chennai Supply Penalty Penalty

Ghaziabad Rs.2667 Rs.3781 Rs.4525 150 Units (3781-2667)=1114

(4542-3781)=744

Delhi Rs.2768 Rs.3760 Rs.4500 200 Units (3760-2768)=792

(4500-3760)=740

Kanpur Rs.3473 Rs.3635 Rs.4267 150 Units (3635-3473)=162 (4267-3635)=632Demand 100 Units 200 Units 200 Units 500 Units


2667X11 +3781X21 + 3760X22 + 4500X32 +4267X33=

2667(100) +3781(50) + 3760(150) + 4500(50) +4267(150) = Rs.18, 84,800

Initial solution from:

Northeast cost, total cost = Rs.1884800

The min cost, total cost = Rs.1901200

VAM, total cost = Rs.1884800

It shows that the 1st and 3rd method has the min cost. So, this will be used to obtaining optimalsolution.

Phase II

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100 50

150 50

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In Phase II , the Stepping Stone Method, using the MODI method for evaluating the reducedcosts may be used to move from the initial feasible solution to the optimal one.

Modified distributed method (MODI) :

It is a modified version of stepping stone methodMODI has three important elements:

(1) It determines if a tableau is the optimal one

(2) It tells you which non-basic variable should be firstly considered as an entry variable

(3) It makes use of stepping-stone to get its answer of next iteration

Modified distributed method (MODI) for Initial tableau of the Vogels approximation

method (VAM): Initial tableau of the Vogels approximation method (VAM)

Table 4(a)

Vi V1 V2 V3

Ui Warehouse(i)/Stores(j)


U1 Ghaziabad Rs.2667 Rs.3781 Rs.4525 150 Units

U2 Delhi Rs.2768 Rs.3760 Rs.4500 200 Units

U3 Kanpur Rs.3473 Rs.3635 Rs.4267 150 Units


Lets test to see whether the current tableau (4a) represent the optimal solution. We can dothis because of duality and sensitivity analysis to interpret. We can introduce twoquantities Ui,

and Vj, where Ui, is the dual variable associated with row i and Vj, is the dual variable

associated with column.From duality theory:

Ui + Vj, = Cij ..eq (1) Represent all Basic Variables

We can compute all Ui, and Vj, values from the initial tableau (4a) using eq (1):

U1 + V1 = Rs.2667 (C11)................................................. (1)11

100 50

150 50

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U1 + V2 = Rs.3781(C12). (2)

U2 + V2= Rs.3760(C22).. (3)

U2 + V3 = Rs.4500(C23). (4)

U3 + V3= Rs.4267(C33).. (5)

Since, there are five equations and six unknowns (because we added a non basic variable). Tosolve these equations, it is necessary to assign only one of the unknown a value of zero. Wecan arbitrarily assign a value to one of the unknown. A common method is to choose the rowwith the largest number of allocations i.e. (U1= 0).

Using substitute calculates the Basic Variables:

Basic Variables: ( U1= 0, U2= -21, U3= -254 and V1= 2667, V2=3781, V3=4521)

To recognize whether this tableau represents the Optimal Solution;

For every Non basic Variable (those cells without any allocations)

Cij Ui Vj = K ij > 0..eq (2) (represent Non-Basic Variable)

For cell (1, 3) C13 U1 V3 = K 13= 4.eq (a)

For cell (2, 1) C21 U2 V1= K 21= 122..........................................eq (b)

For cell (3, 1) C31 U3 V1= K 31 = 1060.......................................eq (c)For cell (3, 2) C32 U3 V2 = K 32 = 108.........................................eq (d)

Non Basic Variables: ( K 13= 4, K 21= 122, K 31 = 1060, K 32 = 108 > 0)

Eq (2) is true in every case of eq (a), eq (b), eq (c), eq (d), because these variables aresatisfied with non negativity condition (K 13, K 21, K 31, K 32> 0). So the current tableaus(4a) represent the optimal solution and Rs.18, 84,800 is the lowest cost of

transportation.

Modified distributed method (MODI) for Initial tableau of the Minimum cell cost method:

Table 3(a) Initial tableau of the Minimum cell cost method

Vi V1 V2 V3

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Ui Warehouse(i)/Stores(j)


U1 Ghaziabad Rs.2667 Rs.3781 Rs.4525 150 Units

U2 Delhi Rs.2768 Rs.3760 Rs.4500 200 Units

U3 Kanpur Rs.3473 Rs.3635 Rs.4267 150 Units


From duality theory: (source given in references)

Ui + Vj, = Cij ..eq (1) Represent all Basic Variables

We can compute all Ui, and Vj, values from the initial tableau (3a) using eq (1):

(Where Ui, is the dual variable associated with row i and Vj, is the dual variable associatedwith column and Cij represent the cost of the allocation cell)

Ui + Vj = Cij (Represent the all basic variables)

U1 + V1 = Rs.2667 (C11)................................................. (6)

U1+ V3 = Rs.4525(C13).. (7)

U2 + V2= Rs.3760(C22).. (8)

U2 + V3 = Rs.4500(C23). (9)

U3 + V2= Rs.3635(C32).... (10)

Such that, there are five equations and six unknowns (because we added a non basicvariable). To solve these equations, it is necessary to assign only one of the unknown a value

of zero. (Let U1=0) because U1 has largest number of allocations.Basic Variables

U1= 0, U2= -25, U3= -50 and V1= 2667, V2=3785, V3=4525

To recognize whether this tableau represents the Optimal Solution;

For every Non basic Variable (those cells without any allocations)

Cij Ui Vj = K ij > 0.....eq (2) (represent Non-Basic Variable)

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For cell (1, 3) C13 U1 V3 = K 12= -4.... (e)

For cell (2, 1) C21 U2 V1= K 21= 126.................................................................................. (f)

For cell (3, 1) C31 U3 V1= K 31= 885.................................................................................. (g)

For cell (3, 2) C32 U3 V2 = K 33 = -208............................................................................... (h)

Non Basic Variables

K 12= -4,K 21= 126, K 31 = 856, K 32 = -208

Eq (2) is true in the case of eq (f) and eq (g) i.e. (K 21= 126 >0, K 31= 885>0)

Eq is not true in the case of eq (e) and eq (h) i.e. (K 12= -4 < 0, K 33 = -208 < 0) thiscondition does not satisfies with the non negativity conditions, so the current tableau (..)does not represent the optimal solution.

Stepping Stone Method

Table 3(b)







For simplex (Minimization Problems), Chose the most negative reduced costK 33= -208determined by cell (3, 3) as the new entering variable to reduce the cost by allocating this cell(3, 3) and to increase the value as much as possible so place the (+) in this cell (3, 3). (Thestepping stone path for this cell (3, 3) is (2, 2), (2, 3), (3, 2), (3, 3) Table ()). For equilibriumin everything place (-) in the cell (2, 3) and place (+) in cell (2, 2), this indicate the equality inthis row and then Place (-) in cell (3, 2). The allocations in cell (2, 3) and cell (3, 2)(subtraction cells) are 150 and 150 respectively. Thus the new solution is obtained byreallocating 150 on the stepping stone path. Thus for the next tableau:

X 33 = 0 + 150 = 150 (0 is its current allocation)14

100 50

50 15

150

-4

126856

-208


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X 23 = 150 - 150 = 0

X 22 = 50 + 150 = 200

X 32 = 150 - 150 = 0 (blank for the next tableau)

Table 3(c)




Delhi Rs.2768 Rs.3760(+)

Rs.4500(-)

200 Units

Kanpur Rs.3473 Rs.3635(-) Rs.4267(+) 150 Units


Table 3(d)







Modified distributed method (MODI)

Table 3(e)

15

-4100 50

50 15126

150856

-208

100

20

150

50


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Cij Ui Vj = K ij ..eq (2) empty cell (represent non-basic variable)

C12 U1 V2 = K 12= 0.. (i)

C21 U2 V1= K 21= 0........................................................................................................... (j)

C23 U2 V3= K 23= 0........................................................................................................... (k)

C31 U3 V1= K 31 = 0........................................................................................................... (l)

C32 U3 V2 = K 32 = 0.......................................................................................................... (m)

Analysis:

Basic Variables

U1= 0, U2= -851, U3= -146 and V1= 3619, V2=3781, V3=5351Non Basic Variables

K 12= K 21= K 23= K 31= K 32= 0

K ij = 0 the solution is optimal and unique and this satisfied the non negativity conditionfor the transportation problem.

Total transportation cost:

2667X 11 + 3760X 22 + 4525X 32 +4267X 33

2667(100) + 3760(150) + 4525(50) +4267(150) = Rs.1086160

SENSITIVITY ANALYSIS

Sensitivity Analysis investigates the change in the optimum solution resulting from making

changes in parameters of the linear programming of transportation problem, So the changes

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in coefficients of (Cij) Cost Factors (Transportation Cost @ Rs. 180.00 per km, Insurance @2.7%, Tax- Toll Tax, Excise Duty @ 3%Warehouse Charges, And Unloading Charges) of the

Objective Function:

Minimize C = 2970.80X 11 +3005.00X 21 +3665.00X 31 + 5349.40X 12 + 5315.60X 22 +5105.40X 32 + 6628.80X 13 +6597.00X 23 +6387.20X 33

Table (1a)



Ghaziabad Rs.2970.80 Rs.5349.40 Rs.6628.80 150 Units

Delhi Rs.3005.00 Rs.5315.60 Rs.6597.00 200 Units

Kanpur Rs.3665.00 Rs.5105.40 Rs.6387.20 150 Units


NW Corner Method:








2970.80X11 +5349.40X12 + 5315.60X22 + 6597.00X23 +6387.20X33

2970.80(100) +5349.40(50) + 5315.60(150) + 6597.00(50) +6387.20(150) = Rs.26181440

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Minimum Cell Cost Method:








2970.80X11 +5315.60X22 + 5105.40X32 + 6628.80X13 +6597.00X23

2970.80(100) +6628.80(50) + 5315.60(50) + 6597.00(150) +5105.40(150) = Rs.2649680

VAM:

Penalty 34.20 210.20 421.00


Dehradun Mumbai Chennai Supply Penalty Penalty

Ghaziabad Rs.2970.80 Rs.5349.40 Rs.6628.80 150 Units 2378.80 1279.80

Delhi Rs.3005.00 Rs.5315.60 Rs.6597.00 200 Units 2310.60 1281.40

Kanpur Rs.3665.00 Rs.5105.40 Rs.6387.20 150 Units 1440.40 1070.60



2970.80X11 + 5315.60X22 + 6628.80X13 +6387.20X33

2970.80(100) +6628.80(50) + 5315.60(200) +6387.20(150) = Rs.2681060

Northeast cost, total cost = Rs.2618140 19

10 50

50 15

15

100 50

200

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The min cost, total cost = Rs.2649680

VAM, total cost = Rs.2681060

And now, to obtaining optimal solution select the Minimum Cell Cost Method tableau

Modified distributed method (MODI) for Initial tableau of the Minimum cell cost method:







Basic Variables

U1= 0, U2= -31.80, U3= -242 and V1= 2970.80, V2=5347.40, V3=6629.20

Non Basic Variables

K 12= 2,K 21= 66, K 31 = 836.20, K 33 = .40

K ij > 0 the solution is optimal and unique and this satisfied the non negativity conditionfor the transportation problem, therefore an optimal basic feasible solution willmaintain its optimality if the change in C ij.

CONCLUSION

.The LCD-TV market will continue to grow robustly during the downturn in2009; the operational and financial challenges caused by the recession are

forcing many OEMs to reconsider their internal expansion plans and20

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outsourcing strategies, as well as to initiate changes that are having animmediate impact on the supply chain

Sony believes that cooperation as true partners is critical to supplying productsthat achieve high levels of customer satisfaction. Sony and its suppliers work together in a wide range of areas - by combining technological skills thatcomplement each other, building powerful supply chains, preserving andenhancing the quality of parts, strictly complying with relevant laws andregulations and contributing to society as a whole.

In this paper, a particular method for evaluating the sensitivity analysis of supply and demand values was presented.Supposing the balance in the algorithms of transportation problems, thesensitivity analysis of transportation problems is, in one hand, a simultaneousanalysis of right-hand-side parameters and on the other hand, theimplementation of the balanced equation. Like a constraint, this balanced

equation directly affects the parameters whose changes are important. Thus, inthe sensitivity analysis of supply and demand in transportation problem,because of the functional relation between supply and demand parameters, wealways face the simultaneous changes of parameters.

REFERENCES

S.D.Sharma, Operation Research,(Theory, Methods and Applications),

2009, pg.TP.1-TP8121


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Ford, L.R., and D.R. Fulkerson,Flows in networks , Princeton Univ.Press, 1962.

H. Arsham, A.B. Kahn,Simplex-type algorithm for generaltransportation problems

An alternative to stepping-stone , J. Opl Res Soc, 40 (1989), 581-590. NERALI, L. (1979):Modified Lagrangians and optimization of

costs , Ph. D. Thesis, Faculty of Economics, University of Zagreb, Zagreb

http://www.rpi.edu/dept/math/math-

programming/cplex90/amplcplex90userguide.pdf Hamdy A Taha, 1997,Duality and Sensitivity Analysis , pg. no. 115-

162

http://www.faqs.org/abstracts/Electronics-and-electrical-industries/Sony-

arms-security-sets-with-network-functions-digital-

recorders.html#ixzz0ZVBqpMTW

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