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CHAPTER 5

SUPPLIER SELECTION BY LEXICOGRAPHIC

METHOD USING INTEGER LINEAR PROGRAMMING

5.1 INTRODUCTION

The SCMS model is solved using Lexicographic method by using

LINGO software. Here the objectives are ranked and they along with its trade

off values are brought as constraints for evaluation of the subsequent

objectives. Since the objectives are brought into the model as constraints, the

purchase manager will be able to control the defectives and late deliveries of

the shipment to be received from various vendors and put it under certain

limit depending on the company’s policy prior to the selection, which is the

greatest advantage of using this model. Weber et al (2000) generated weights

for quantity allocation that is done in constraint method. But these cannot

control the defectives and late deliveries prior to the selection and quantity

allocation, because the defectives and late deliveries may exceed the company

policy entailing the reallocation of order quantity. This chapter explores the

use of Lexicographic approach for the proposed research and its advantages.

5.2 WHAT IS LEXICOGRAPHIC METHOD?

This method involves a process of ranking objectives according to

the most important criterion. The top subset is identified according to the first

most important criterion, and the alternatives in this subset are ranked

according to the second most important criterion, and so on, until a single

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alternative left or all criteria have been examined. In the lexicographic method

it is required to rank order the importance of each behavior relative to the

other behaviors. Assume that the behaviors are ranked in decreasing order of

importance O1, O2,...,On. Then a sequential elimination process is started by

solving the following sequence of problems until either a unique solution is

found or all the problems are solved:

P1 : max O1(x), x X

P2 : max O2(x), x X 1

.....

Pi : max Oi(x), x X i-1

Xi-1 = { x|x solves P i-1 }, i = 2,..., n+1,.....

The basic idea is that the first objective is used to screen the

solutions of the second objective and so on. A famous application is in

university admittance where students with highest grades are allowed in any

college they choose. The second best groups are allowed only the remaining

places and so on.

5.2.1 Applicability of lexicographic method to supplier selection

The lexicographic method involves solving multiple optimization

problems in sequence, rather than minimizing a single function once. The first

step in this method is to categorize objective functions into different levels in

order of importance from one to k, where k is the index for the final level.

Then the following problems are solved one after another in order of

importance while keeping optimized results, Fj ( X j* ), constrained for the

successive problems :

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Find x R n such that

Min F i ( x )

subject to

Fj ( x ) Fj ( xj * )

where j = 1,2 ….., i-1; i1 and i=1,2,…,k

As the method proceeds, the number of constraints grows up to k-1

until an optimal solution x is found.

In this research, on the first level the most significant criterion is

selected and then the suppliers are compared according to this criterion.

Hence initially the supplier selection criteria are ranked in order of

importance. Suppliers are first evaluated on the most important criterion. If a

supplier satisfies this criterion much better than the other suppliers then we

choose it, if not we compare the suppliers with respect to a second criterion.

Hence here once the designer ranks the objectives in order of importance, the

optimum solution X* is then found by minimizing the objective functions

starting with the most important and proceeding according to the order of

importance of the objectives. Let the subscripts of the objectives indicate not

only the objective function number, but also the priorities of the objectives.

Thus f1(X) and fk(X) denote the most and least important objective functions,

respectively. The first problem is formulated as

Minimize f1(X)

subject to

gj(X) 0, j=1,2,…,m

and its solution

X1* and f1

* = f1(X1*) is obtained.

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Then the second problem is formulated as

Minimize f2(X)

subject to

gj(X) 0, j=1,2,…,m

f1(X) f1*

The solution of the problem is obtained as X2* and f2* = f2(X2

*).

This procedure is repeated until all the k objectives have been

considered. The ith problem is given by

Minimize fi(X)

subject to

gj(X) 0, j=1,2,…,m

fl (X) fl*, l=1,2,…,i-1

and its solution is found as

Xi* and fi

* = fi(Xi*).

Finally, the solution obtained at the end (i.e., Xk*) is taken as the

desired solution X* of the original multi-objective optimization problem.

5.3 EVALUATION OF THE MODEL USING LINGO

The model furnished in the previous chapter is solved considering a

case from a leading textile machinery-manufacturing firm. As indicated in the

solution methodology the criteria selected or ranked with the first objective as

quality followed by delivery rate and cost.

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5.3.1 An introduction to LINGO

This research uses Lindo Systems’ Lingo 7.0 for solving this

model. LINGO is a simple tool for utilizing the power of linear and nonlinear

optimization to formulate large problems concisely, solve them, and analyze

the solution. LINGO provides a completely integrated package that includes a

powerful language for expressing optimization models, a full featured

environment for building and editing problems, and a set of fast built-in

solvers.

5.3.1.1 Lingo built in solvers

Unlike many modeling packages, all of the LINGO solvers are

directly linked to the modeling environment. This seamless integration allows

LINGO to pass the problem to the appropriate solver directly in memory

rather than through more sluggish intermediate files. LINGO has four solvers

it uses to solve different types of models. These solvers are direct solver,

linear solver, nonlinear solver, branch and bound manager. LINGO is

designed, so the process of solving the model requires as little input from the

user as possible. When the solve command is initiated, LINGO analyzes the

problem and, when possible, reduces the problem and even substitutes out

variables. Based upon the models structure, LINGO automatically selects the

appropriate solver and intelligently adjusts internal parameters.

5.3.1.2 Linearization

LINGO's linearization capabilities can automatically convert many

non smooth functions and operators to a series of linear, mathematically

equivalent expressions. Similarly, the product of a continuous and binary

variable can also be linearized. Many non smooth models may be entirely

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linearized. This allows the linear solver to quickly find a global solution to

what would have otherwise been an intractable problem.

The LINGO solvers are all part of the same program. To solve a

model, the direct solver first computes the values for as many variables as

possible. If the direct solver finds an equality constraint with only one

unknown variable, it determines a value for the variable that satisfies the

constraint. The direct solver stops when it runs out of unknown variables or

there are no longer any equality constraints with a single remaining unknown

variable. Once the direct solver is finished, if all variables have been

computed, LINGO displays the solution report. If unknown variables remain,

LINGO determines which solvers to use on a model by examining its

structure and mathematical content. For a continuous linear model, LINGO

calls the linear solver. If the model contains one or more nonlinear

constraints, LINGO calls the nonlinear solver. When the model contains any

integer restrictions, the branch-and-bound manager is invoked to enforce

them. The branch-and-bound manager will in turn call either the linear or

nonlinear solver depending upon the nature of the model.

The linear solver in LINGO uses the revised simplex method with

product form inverse. A barrier solver may also be obtained, as an option, for

solving linear models. On linear integer models, LINGO does considerable

preprocessing (i.e., adding constraint “cuts” to restrict the non-integer feasible

region). These cuts will greatly improve solution times for most integer

programming models.

5.3.2 Developing LINGO model

In general, an optimization model will consists of the following

three items

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Objective Function

The objective function is a formula that expresses exactly what it is

you want to optimize. In business oriented models, this will usually be a profit

function you wish to maximize, or a cost function you want to minimize.

Variables

Variables are the quantities that we have under our control. We

must decide what the best values of the variables are. For this reason,

variables are sometimes also called decision variables. The goal of

optimization is to find the values of a model's variables that generate the best

value for the objective function, subject to any limiting conditions placed on

the variables.

Constraints

Almost without exception there will be some limit on the values the

variables in a model can assume-at least one resource will be limited (e.g.,

time, raw materials, department's budget, etc.). These limits are expressed in

terms of formulas that are a function of the model's variables. These formulas

are referred to as constraints because they constrain the values the variables

can take.

LINGO provides very easy commands and macros for specifying

the model with its variables and constraints. Once the program is developed,

and on clicking solve, LINGO will invoke the appropriate internal solver to

begin searching for the optimal solution to the model.

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5.3.2.1 Monitoring the progress of the solver

When the solver starts, it displays a solver status window on the

screen which is useful for monitoring the progress of the solver and the

dimensions of the model.

Variable box

The Variables box shows the total number of variables in the

model.

Constraint box

The Constraints box shows the total constraints and the number of

these constraints that are nonlinear. A constraint is considered nonlinear if

one or more variables appear nonlinearly in the constraint.

Computation time

The Elapsed Runtime box shows the total time used so far to

generate and solve the model.

Optimizer Status box

The Optimizer Status box shows the current status of the optimizer

and a description of the fields.

Once the solver can no longer find better solutions to the model, it

will terminate in either the "Global Optimum" or "Local Optimum" state. If

the model does not have any nonlinear constraints, then any locally optimal

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solution will also be a global optimum. Thus, all optimized linear models will

terminate in the global optimum state.

5.3.2.2 Solution report

When LINGO is done solving the model, it indicates the number of

iterations it took to solve the model, the maximum profit, reduced costs, slack

or surplus and Dual Price. A variable’s reduced cost can be the amount by

which the objective coefficient of the variable would have to improve before

it would become profitable to give the variable in question a positive value in

the optimal solution. Or it can be the amount of penalty paid to introduce one

unit of that variable into the solution.

The Slack or Surplus column in a LINGO solution report tells how

close is to satisfying a constraint as equality. This quantity, on less than or

equal constraints, is generally referred to as slack. On greater than or equal

constraints, this quantity is called a surplus. If a constraint is exactly satisfied

as equality, the slack or surplus value will be zero.

5.4 VENDOR SELECTION AND ORDER ALLOCATION

WITHOUT CONSIDERING QUANTITY DISCOUNTS

A description on the step by step approach in evaluating this model

using LINGO is described in this section.

5.4.1 Step 1: Defining the model

There are three basic steps to define the model – definition of the

objective function, variables and constraints. As indicated in the previous

chapter, the following variables are used for defining the model

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VENDOR (I) - The name of the vendor

CAPACITY - The maximum capacity of components that a

vendor can supply

PERCENT_DEF - Percentage of defectives that is expected from the

vendor. It gives the number of defectives in a lot of

100. Here the values are represented as fractions

i.e., if percent defective is 2.5% for a vendor then it

is given as 0.025.

DELIVERY_LATE - This gives the number of items delivered late in a

lot of 100. And this is also represented in fractions.

COST - The cost of the component supplied from a

particular vendor

MIN_BUS - The minimum business that is required to be done

with the vendor.

MIN_QTY - The minimum quantity that a vendor expects to be

supplied to a manufacturer.

SELECT - A variable which determines the selection of vendor

from the list of vendors.

SUP_DET - This option denotes the order quantities allocated to

the specific vendor.

5.4.1.1 Objective function

Consider the model indicated in Chapter 3: Here the objective is to

minimize the cost, number of late deliveries and number of defectives. The

Lingo model window with the objective definition is as shown in Figure 5.1.

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Figure 5.1 Lingo model window for the objective function

5.4.1.2 Variables

The variables define what should be controlled. In the given

problem, the variable SELECT (I) determine, whether the vendor needs to be

selected or not. SUP_DET (I, J) determine the order quantities to be allocated

by Manufacturer J to vendor I. They are defined in Lingo as

@FOR( VENDORS( I): @BIN( SELECT( I)));

@FOR( LINKS( I, J): @GIN( SUP_DET(I,J)));

where SELECT (I) can take only binary values (1/0), BIN denotes Binary

Integer Number. SUP_DET (I, J) can take integer variables and hence

denoted as GIN (General Integer Number).

104

5.4.1.3 Constraints

Next the constraints must be input. These constraints are written

using the defined variables as indicated in Figure 5.2 in Lingo.

Figure 5.2 Lingo window indicating the constraints

5.4.2 Step 2: Defining the tradeoff values

The objectives are considered one by one with the trade-off applied

between the objectives which then help in achieving the final objective cost.

Considering only the quality objective, the vendors V1, V2, V4, and V5 are

selected with their order allocation (SUP_DET) as indicated in Table 5.1.

Total Supply = 600+211+543+700 = 2054.

Total Defectives = vendor

) i F(PERCENT_DE *] i SUP_DET[

= 600*0.025 +211* 0.045+543*0.035 +700*0.015

= 54 (Global optimum)

Non-Defectives = 2054 -54=2000.

105

Global optimal solution found at step: 1019 Objective value: 54.00000 Branch count: 693 Export Summary Report: Transfer Method: OLE BASED (Object Linking and Embedding) Ranges Specified: 2

SUP_DET SELECT

Ranges Found: 2 Range Size Mismatches: 0 Values Transferred: 14

Model Title: Vendor Selection Quality Variable Value Reduced Cost

SELECT( V1) SELECT( V2) SELECT( V3) SELECT( V4) SELECT( V5) SELECT( V6) SELECT( V7) SUP_DET( V1, S1) SUP_DET( V2, S1) SUP_DET( V3, S1) SUP_DET( V4, S1) SUP_DET( V5, S1) SUP_DET( V6, S1 SUP_DET( V7, S1)

1.000000 1.000000 0.0000000 1.000000 1.000000 0.0000000 0.0000000 600.0000 211.0000 0.0000000 543.0000 700.0000 0.0000000 0.0000000

0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.2500000E-01 0.4500000E-01 0.5000000E-01 0.3500000E-01 0.1500000E-01 0.6000000E-01 0.5900000E-01

Demand is met exactly with least number of defectives satisfying all the

constraints. As per the lexicographic method a constraint for number of

defectives is fixed in the verdict of purchase manger and company policy, and

the model is solved for the minimum number of late deliveries.

Table 5.1 Lingo solution report for quality objective

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5.4.2.1 Sensitivity analysis for determining the tradeoff values

Sensitivity analysis by varying the different objectives and

determining its solution helps in fixing the upper limit and the trade off

values. From the global optimum value, the quality is relaxed in steps of 5 and

its corresponding delivery is determined as in Table 5.2. The graph is plotted

as in Figure 5.3. It is found that there is no appreciable decrease in late

deliveries when quality objective is relaxed beyond 75. Say if the company is

also not willing to take more than 75 defectives; the quality is fixed to be 75.

Now for this quality, the delivery objective is taken and relaxed in steps of 5

to finds its corresponding cost objective.

.

Table 5.2 Various quality and delivery values without quantity

discounts using ILP

Quality (Number of defectives)

Late deliveries (Global optimum)

54 113.43 60 94.53 65 51.82 70 44.30 75 38.60 80 37.55 85 36.45

107

30405060708090

100110120

50 55 60 65 70 75 80 85 90Defectives

Late

del

iver

ies

Figure 5.3 Sensitivity analysis for defectives and late deliveries without

discounts using ILP

Table 5.3 shows the relaxation for late deliveries and its corresponding cost.

From Figure 5.4, there is no appreciable difference in price objective when

delivery objective is relaxed beyond 55. Hence the trade off values for quality

and delivery objectives are fixed to be 75 and 55 respectively. Based on

which the model is updated with these trade-off values as in Figure 5.5.

Table 5.3 Various quality, delivery and cost values without quantity

discounts using ILP

Quality (Number of defectives)

Number of late deliveries Cost (Rs.)

75 40 22891.00

75 45 22814.00

75 50 22513.75

75 55 22392.80

75 60 22392.80

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2230022400225002260022700228002290023000

35 40 45 50 55 60 65Late deliveries

Cos

t in

Rs.

Figure 5.4 Sensitivity analysis for late deliveries and cost without

discounts using ILP

Figure 5.5 Lingo model for considering objectives with trade-off

5.4.3 Step 3: Defining the input to the model

The next step is to input the data for solving the model. The data for

7 vendors collected from the textile machinery-manufacturing firm were

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indicated in chapter 3. These details are obtained through the past history of

the vendors. In Figure 5.6, capacity indicates the maximum capacity a vendor

can deliver, Percent_Def is the percentage of defectives, Delivery_Late is the

percentage of late deliveries and Cost is the cost per component. The

purchasing department also has a minimum and maximum business which is

the minimum and maximum order quantities that can be placed for each

vendor. For this problem the minimum and maximum business are 100 and

1200 respectively. Hence if a vendor’s capacity is beyond these limits, this

model takes care of allocating the quantities to the vendors appropriately.

Figure 5.6 Input data for LINGO software without price breaks

5.4.4 Step 4: Solve the model and analyze the status

Once the problem is formulated and the inputs are given, LINGO

will invoke the appropriate internal solver to begin searching for the optimal

solution for the model. When the solver starts, it displays a solver status

window on the screen as indicated in Figure 5.7.

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Here the solver status window indicates there are 14 variables and

32 constraints. The solver takes 6010 iterations and 6993 branches to get the

global optimum solution. The current value of the objective function is

22392.8 and also Objective value of best integer solution found is also

22392.8.

Figure 5.7 Lingo solver status window without considering quantity

discounts

5.4.5 Step 5: Examine the solution report

The solution report gives complete details on how the solution has

arrived like the number of iterations it took to solve the problem, the exact

step where it found the solution and the like as described in the Table 5.4. The

vendors selected and orders allocated are as indicated in Table 5.4.

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Table 5.4 Lingo solution report without considering quantity discounts

Solution report from LINGO Global optimal solution found at step : 6010 Objective value : 22392.75 Branch count : 6993 Model Title: : VENDOR SELECTION

Variable Value Reduced Cost

DEMAND(S1) 2000.000 0.0000000 SELECT(V1) 1.000000 0.0000000 SELECT(V2) 1.000000 0.0000000 SELECT(V3) 0.000000 0.0000000 SELECT(V4) 0.000000 0.0000000 SELECT(V5) 1.000000 0.0000000 SELECT(V6) 1.000000 0.0000000 SELECT(V7) 0.000000 0.0000000 MIN_BUS(V1) 100.0000 0.0000000 MIN_BUS(V2) 100.0000 0.0000000 MIN_BUS(V3) 100.0000 0.0000000 MIN_BUS(V4) 100.0000 0.0000000 MIN_BUS(V5) 100.0000 0.0000000 MIN_BUS(V6) 100.0000 0.0000000 MIN_BUS(V7) 100.0000 0.0000000 MAX_BUS(V1) 1200.000 0.0000000 MAX_BUS(V2) 1200.000 0.0000000 MAX_BUS(V3) 1200.000 0.0000000 MAX_BUS(V4) 1200.000 0.0000000 MAX_BUS(V5) 1200.000 0.0000000 MAX_BUS(V6) 1200.000 0.0000000 MAX_BUS(V7) 1200.000 0.0000000 MIN_QTY(V1) 100.0000 0.0000000 MIN_QTY(V2) 200.0000 0.0000000 MIN_QTY(V3) 250.0000 0.0000000 MIN_QTY(V4) 350.0000 0.0000000 MIN_QTY(V5) 100.0000 0.0000000 MIN_QTY(V6) 300.0000 0.0000000 MIN_QTY(V7) 250.0000 0.0000000 CAP(V1) 600.0000 0.0000000 CAP(V2) 750.0000 0.0000000 CAP(V3) 800.0000 0.0000000 CAP(V4) 750.0000 0.0000000 CAP(V5) 700.0000 0.0000000 CAP(V6) 950.0000 0.0000000 CAP(V7) 1000.000 0.0000000 COST(V1,S1) 10.00000 0.0000000 COST(V2,S1) 11.50000 0.0000000 COST(V3,S1) 12.00000 0.0000000 COST(V4,S1) 9.500000 0.0000000 COST(V5,S1) 10.50000 0.0000000 COST(V6,S1) 12.25000 0.0000000 COST(V7,S1) 15.00000 0.0000000 SUP_DET( V1, S1) 599.0000 10.00000 SUP_DET( V2, S1) 483.0000 11.50000

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Table 5.4 (continued)

Variable Value Reduced Cost

SUP_DET( V3, S1) 0.0000000 12.00000 SUP_DET( V4, S1) 0.0000000 9.500000 SUP_DET( V5, S1) 682.0000 10.50000 SUP_DET( V6, S1) 301.0000 12.25000 SUP_DET( V7, S1) 0.0000000 15.00000 PERCENT_DEF(V1,S1) 0.2500000E-01 0.0000000 PERCENT_DEF(V2,S1) 0.4500000E-01 0.0000000 PERCENT_DEF(V3,S1) 0.5000000E-01 0.0000000 PERCENT_DEF(V4,S1) 0.3500000E-01 0.0000000 PERCENT_DEF(V5,S1) 0.1500000E-01 0.0000000 PERCENT_DEF(V6,S1) 0.6000000E-01 0.0000000 PERCENT_DEF(V7,S1) 0.5900000E-01 0.0000000 DELIVERY_LATE(V1,S1) 0.3250000E-01 0.0000000 DELIVERY_LATE(V2,S1) 0.5250000E-01 0.0000000 DELIVERY_LATE(V3,S1) 0.6250000E-01 0.0000000 DELIVERY_LATE(V4,S1) 0.1500000 0.0000000 DELIVERY_LATE(V5,S1) 0.2000000E-02 0.0000000 DELIVERY_LATE(V6,S1) 0.2500000E-01 0.0000000 DELIVERY_LATE(V7,S1) 0.2350000E-01 0.0000000 Row Slack or Surplus Dual Price

OBJECTIVE 22392.75 0.0000000 LATE_DELIVERY 1.286000 0.0000000 QUALITY 10.00000 0.0000000 DEMAND_ROW( S1) 0.0000000 0.0000000 CAPACITY_ROW( V1) 1.000000 0.0000000 CAPACITY_ROW( V2) 267.0000 0.0000000 CAPACITY_ROW( V3) 0.0000000 0.0000000 CAPACITY_ROW( V4) 0.0000000 0.0000000 CAPACITY_ROW( V5) 18.00000 0.0000000 CAPACITY_ROW( V6) 649.0000 0.0000000 CAPACITY_ROW( V7) 0.0000000 0.0000000 MAX_BUS_ROW( V1) 601.0000 0.0000000 MAX_BUS_ROW( V2) 717.0000 0.0000000 MAX_BUS_ROW( V3) 0.0000000 0.0000000 MAX_BUS_ROW( V4) 0.0000000 0.0000000 MAX_BUS_ROW( V5) 518.0000 0.0000000 MAX_BUS_ROW( V6) 899.0000 0.0000000 MAX_BUS_ROW( V7) 0.0000000 0.0000000 MIN_QTY_ROW( V1) 499.0000 0.0000000 MIN_QTY_ROW( V2) 283.0000 0.0000000 MIN_QTY_ROW( V3) 0.0000000 0.0000000 MIN_QTY_ROW( V4) 0.0000000 0.0000000 MIN_QTY_ROW( V5) 582.0000 0.0000000 MIN_QTY_ROW( V6) 1.000000 0.0000000 MIN_BUS_ROW( V2) 383.0000 0.0000000 MIN_BUS_ROW( V3) 0.0000000 0.0000000 MIN_QTY_ROW( V7) 0.0000000 0.0000000 MIN_BUS_ROW( V1) 499.0000 0.0000000 MIN_BUS_ROW( V4) 0.0000000 0.0000000 MIN_BUS_ROW( V5) 582.0000 0.0000000 MIN_BUS_ROW( V6) 201.0000 0.0000000 MIN_BUS_ROW( V7) 0.0000000 0.0000000

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In the above solution report, it indicates that it takes 6010 steps to

solve the problem and the minimum cost that can be achieved is 22392.75.

Next it indicates the variables in the problem like Demand, Minimum and

Maximum Business values from the manufacturer, minimum quantity and

maximum capacity of the vendor, cost, percentage defectives and late

deliveries. Here the variable SUP_DET indicates the order allocated for that

particular vendor. Similarly the variable SELECT indicates if the particular

vendor has been selected or not. The value 1 in SELECT indicates the vendor

has been selected.

A variable’s reduced cost is the amount by which the objective

coefficient of the variable would have to improve before it would become

profitable to give the variable in question a positive value in the optimal

solution. For example, if a variable had a reduced cost of 10, the objective

coefficient of that variable would have to decrease by 10 units in a

minimization problem in order for the variable to become an attractive

alternative to enter into the solution. In this particular problem, the reduced

cost is the cost of per unit of that component from a particular vendor.

The Slack or Surplus column in the solution report tells how close

it is to satisfy a constraint as equality. This quantity, on less than or equal ( )

constraints, is generally referred to as slack. On greater than or equal ( )

constraints, this quantity is called a surplus. If a constraint is exactly satisfied

as equality, the slack or surplus value will be zero. If a constraint is violated,

as in an infeasible solution, the slack or surplus value will be negative.

Knowing this can help to find the violated constraints in an infeasible model-a

model for which there doesn’t exist a set of variable values that

simultaneously satisfies all constraints. Non-binding constraints, constraints

with a slack or surplus value greater than zero, will have positive, nonzero

values in this column.

114

The dual price is the amount by which the objective would improve

as the right-hand side, or constant term, of the constraint is increased by one

unit. In a minimization problem, the objective value would decrease if it was

to increase the right-hand side of a constraint with a positive dual price. Dual

prices are sometimes called shadow prices, because they tell us how much we

should be willing to pay for additional units of a resource.

The Table 5.5 indicates the vendors selected, order allocated,

defectives, late deliveries and cost as inferred from the solution report.

Table 5.5 Vendor selection and order allocation without quantity

discounts using ILP

Vendor number

Order allocation

Number of defectives

Number of late

deliveries

Total cost in Rs.

V1 599 15 19 5990

V2 483 22 25 5554

V5 682 10 1 7161

V6 301 18 8 3687

Total 2065 65 53 22392

Order – Defectives = 2065 – 65 = 2000 = Demand

From the Table 5.5 it is found that the order allocation for the

vendors is done considering the number of defectives also. Hence the demand

is exactly met when the number of defectives is eliminated. This is very

advantageous and useful in a manufacturing industry where the demand will

always be met.

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5.5 VENDOR SELECTION AND ORDER ALLOCATION

CONSIDERING QUANTITY DISCOUNTS

Vendors offer quantity discounts in order to increase their order

allocations by the manufacturer. Manufacturer in turn considers quantity

discounts by the vendor to minimize their total cost. In the case of quantity

discounts, the vendor offer price breaks based on the number of components.

This section deals with the evaluation of SCMS model considering quantity

discounts.

5.5.1 Step 1: Defining the Model Taking the 7 vendor data used for evaluation of the model without quantity discounts, the non linear nature of the problem due to the inclusion of price breaks is converted to a linear problem as explained below. Consider each range of discount offered by a vendor as an additional vendor with same Quality and delivery. Here since there are 7 vendors with each offering 2 price breaks, the total number of vendors now would be 7*2= 14 vendors. The corresponding additional vendors are: V1-V8, V2-V9, V3-V10, V4 -V11, V5-V12, V6-V13, and V7-V14. For example additional vendor V1-V8 has the same quality and delivery as V1 but has a different price break. By doing this piecewise linear problem is converted to a linear problem. To make this feasible necessary surrogate constraints are added viz., minimum quantity for V1 is made 100, and for V8 as 300 and capacity for V1 is made 300 and for V8 as 600. Similar transformation is done for all vendors which eventually converts the piecewise linear problem into a linear problem. Figure 5.8 indicates this model, wherein the change is in the inclusion of the additional vendors.

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Figure 5.8 LINGO model window for vendor selection using quantity

discounts

5.5.2 Step 2: Defining the Tradeoff values

There would be no change in the upper limit value of the quality

when considering quantity discounts, since for determining the late deliveries

for various values of quality is the same as was without discount model. But

in the case of determining the upper limit for late delivery objective, there will

be change in the value of the cost objective when the quality objective is fixed

as 75 and the late delivery objective is varied due to the inclusion of price

breaks.

5.5.2.1 Sensitivity analysis for determining the tradeoff values

From Table 5.6 and Figure 5.9, the sensitivity analysis indicates

that when the late delivery is increased beyond 55 there is no appreciable

difference in the cost objective. Although the analysis indicates that the cost

objectives becomes constant after the late delivery is relaxed beyond 65, the

upper limit for late delivery is fixed as 55, because of two reasons

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Table 5.6 Various quality, delivery and cost values with quantity

discounts using ILP

Quality (Number of defectives)

Late deliveries Cost (Rs.)

75 40 21369.50 75 45 21284.00 75 50 21284.00 75 55 20756.00 75 60 20735.00 75 65 20512.00 75 70 20512.00 75 75 20512.00 75 80 20512.00

204002050020600207002080020900210002110021200213002140021500

35 40 45 50 55 60 65 70 75 80 85

Late deliveries

Cos

t in

Rs.

Figure 5.9 Sensitivity analysis for late deliveries and cost with quantity

discounts using ILP

1. A company can have norm of not accepting more number of

late deliveries even if the cost is less, since it might affect the

production of the product.

118

2. There is no marginal difference in cost when late delivery is

relaxed beyond 55.

Hence the trade off values of quality and late delivery is fixed as 75 and 55

respectively

5.5.3 Step 3: Defining the input to the model

5.5.3.1 Price breaks

An important aspect in the quantity discount model is defining the

inputs. The price breaks offered by the 7 vendors are as indicated in

Figure 5.10 which is same as the input given in Table 3.8.

Figure 5. 10 LINGO input data window for quantity discount

119

5.5.4 Step 4: Solve the model and analyze the status

As indicated in the vendor selection without quantity discount

approach, LINGO solver solves this model choosing the appropriate solver

and produces the result as indicated in Figure 5.11. Here the solver status

window indicates there are 28 variables and 67 constraints. The solver takes

7605 iterations and 7398 branches to get the global optimum solution. The

current value of the objective function is 20756 and also Objective value of

best integer solution found is also 20756. This value is lower than the value

that is obtained without quantity discount model and hence considering

quantity discounts produces better cost value.

Figure 5.11 Lingo solver status window considering quantity discounts

5.5.5 Step 5: Solution report

The vendors selected and orders allocated are as indicated in

Table 5.7. The solution report indicates the vendors selected are V6, V8, V9

and V12. Hence the actual vendors selected are as indicated in the

Table 5.8, V6, V1, V2 and V5 along with their order allocations

120

Table 5.7 Lingo solution report considering quantity discounts

Solution report from LINGO Global optimal solution found at step : 7605 Objective value : 20756.00 Branch count : 7398 Variable Value Reduced Cost

DEMAND( S1) 2000.000 0.0000000 SELECT( V1) 0.0000000 0.0000000 SELECT( V2) 0.0000000 0.0000000 SELECT( V3) 0.0000000 0.0000000 SELECT( V4) 0.0000000 0.0000000 SELECT( V5) 0.0000000 0.0000000 SELECT( V6) 1.000000 0.0000000 SELECT( V7) 0.0000000 0.0000000 SELECT( V8) 1.000000 0.0000000 SELECT( V9) 1.000000 0.0000000 SELECT( V10) 0.0000000 0.0000000 SELECT( V11) 0.0000000 0.0000000 SELECT( V12) 1.000000 0.0000000 SELECT( V13) 0.0000000 0.0000000 SELECT( V14) 0.0000000 0.0000000 MIN_BUS( V1) 100.0000 0.0000000 MIN_BUS( V2) 100.0000 0.0000000 MIN_BUS( V3) 100.0000 0.0000000 MIN_BUS( V4) 100.0000 0.0000000 MIN_BUS( V5) 100.0000 0.0000000 MIN_BUS( V6) 100.0000 0.0000000 MIN_BUS( V7) 100.0000 0.0000000 MIN_BUS( V8) 100.0000 0.0000000 MIN_BUS( V9) 100.0000 0.0000000 MIN_BUS( V10) 100.0000 0.0000000 MIN_BUS( V11) 100.0000 0.0000000 MIN_BUS( V12) 100.0000 0.0000000 MIN_BUS( V13) 100.0000 0.0000000 MIN_BUS( V14) 100.0000 0.0000000 MAX_BUS( V1) 1200.000 0.0000000 MAX_BUS( V2) 1200.000 0.0000000 MAX_BUS( V3) 1200.000 0.0000000 MAX_BUS( V4) 1200.000 0.0000000 MAX_BUS( V5) 1200.000 0.0000000 MAX_BUS( V6) 1200.000 0.0000000 MAX_BUS( V7) 1200.000 0.0000000 MAX_BUS( V8) 1200.000 0.0000000 MAX_BUS( V9) 1200.000 0.0000000 MAX_BUS( V10) 1200.000 0.0000000 MAX_BUS( V11) 1200.000 0.0000000 MAX_BUS( V12) 1200.000 0.0000000 MAX_BUS( V13) 1200.000 0.0000000 MAX_BUS( V14) 1200.000 0.0000000 MIN_QTY( V1) 100.0000 0.0000000 MIN_QTY( V2) 200.0000 0.0000000 MIN_QTY( V3) 250.0000 0.0000000 CAP( V5) 399.0000 0.0000000

121

Table 5.7 (continued) Variable Value Reduced Cost

CAP( V6) 599.0000 0.0000000 CAP( V7) 649.0000 0.0000000 CAP( V8) 600.0000 0.0000000 CAP( V9) 750.0000 0.0000000 CAP( V10) 800.0000 0.0000000 CAP( V11) 750.0000 0.0000000 CAP( V12) 700.0000 0.0000000 CAP( V13) 950.0000 0.0000000 CAP( V14) 1000.000 0.0000000 COST( V1, S1) 10.00000 0.0000000 COST( V2, S1) 11.50000 0.0000000 COST( V3, S1) 12.00000 0.0000000 COST( V4, S1) 9.500000 0.0000000 COST( V5, S1) 10.50000 0.0000000 COST( V6, S1) 12.25000 0.0000000 COST( V7, S1) 15.00000 0.0000000 COST( V8, S1) 9.000000 0.0000000 COST( V9, S1) 10.00000 0.0000000 COST( V10, S1) 11.00000 0.0000000 COST( V11, S1) 9.000000 0.0000000 COST( V12, S1) 10.00000 0.0000000 COST( V13, S1) 11.50000 0.0000000 COST( V14, S1) 14.00000 0.0000000 MIN_QTY( V4) 350.0000 0.0000000 MIN_QTY( V5) 100.0000 0.0000000 MIN_QTY( V6) 300.0000 0.0000000 MIN_QTY( V7) 250.0000 0.0000000 MIN_QTY( V8) 300.0000 0.0000000 MIN_QTY( V9) 500.0000 0.0000000 MIN_QTY( V10) 500.0000 0.0000000 MIN_QTY( V11) 550.0000 0.0000000 MIN_QTY( V12) 400.0000 0.0000000 MIN_QTY( V13) 600.0000 0.0000000 MIN_QTY( V14) 650.0000 0.0000000 CAP( V1) 299.0000 0.0000000 CAP( V2) 499.0000 0.0000000 CAP( V3) 499.0000 0.0000000 CAP( V4) 549.0000 0.0000000 SUP_DET( V1, S1) 0.0000000 10.00000 SUP_DET( V2, S1) 0.0000000 11.50000 SUP_DET( V3, S1) 0.0000000 12.00000 SUP_DET( V4, S1) 0.0000000 9.500000 SUP_DET( V5, S1) 0.0000000 10.50000 SUP_DET( V6, S1) 308.0000 12.25000 SUP_DET( V7, S1) 0.0000000 15.00000 SUP_DET( V8, S1) 597.0000 9.000000 SUP_DET( V9, S1) 506.0000 10.00000 SUP_DET( V10, S1) 0.0000000 11.00000 SUP_DET( V11, S1) 0.0000000 9.000000 SUP_DET( V12, S1) 655.0000 10.00000 SUP_DET( V13, S1) 0.0000000 11.50000 SUP_DET( V14, S1) 0.0000000 14.00000 PERCENT_DEF( V1, S1) 0.2500000E-01 0.0000000 PERCENT_DEF( V2, S1) 0.4500000E-01 0.0000000

122

Table 5.7 (continued) Variable Value Reduced Cost

PERCENT_DEF( V3, S1) 0.5000000E-01 0.0000000 PERCENT_DEF( V4, S1) 0.3500000E-01 0.0000000 PERCENT_DEF( V5, S1) 0.1500000E-01 0.0000000 PERCENT_DEF( V6, S1) 0.6000000E-01 0.0000000 PERCENT_DEF( V7, S1) 0.5900000E-01 0.0000000 PERCENT_DEF( V8, S1) 0.2500000E-01 0.0000000 PERCENT_DEF( V9, S1) 0.4500000E-01 0.0000000 PERCENT_DEF( V10, S1) 0.5000000E-01 0.0000000 PERCENT_DEF( V11, S1) 0.3500000E-01 0.0000000 PERCENT_DEF( V12, S1) 0.1500000E-01 0.0000000 PERCENT_DEF( V13, S1) 0.6000000E-01 0.0000000 PERCENT_DEF( V14, S1) 0.5900000E-01 0.0000000 DELIVERY_LATE( V1, S1) 0.3250000E-01 0.0000000 DELIVERY_LATE( V2, S1) 0.5250000E-01 0.0000000 DELIVERY_LATE( V3, S1) 0.6250000E-01 0.0000000 DELIVERY_LATE( V4, S1) 0.1500000 0.0000000 DELIVERY_LATE( V5, S1) 0.2000000E-02 0.0000000 DELIVERY_LATE( V6, S1) 0.2500000E-01 0.0000000 DELIVERY_LATE( V7, S1) 0.2350000E-01 0.0000000 DELIVERY_LATE( V8, S1) 0.3250000E-01 0.0000000 DELIVERY_LATE( V9, S1) 0.5250000E-01 0.0000000 DELIVERY_LATE( V10, S1) 0.6250000E-01 0.0000000 DELIVERY_LATE( V11, S1) 0.1500000 0.0000000 DELIVERY_LATE( V12, S1) 0.2000000E-02 0.0000000 DELIVERY_LATE( V13, S1) 0.2500000E-01 0.0000000 DELIVERY_LATE( V14, S1) 0.2350000E-01 0.0000000 Row Slack or Surplus Dual Price

OBJECTIVE 20756.00 0.0000000 LATE_DELIVERY 0.2250000E-01 0.0000000 QUALITY 9.000000 0.0000000 DEMAND_ROW( S1) 0.0000000 0.0000000 CAPACITY_ROW( V1) 0.0000000 0.0000000 CAPACITY_ROW( V2) 0.0000000 0.0000000 CAPACITY_ROW( V3) 0.0000000 0.0000000 CAPACITY_ROW( V4) 0.0000000 0.0000000 CAPACITY_ROW( V5) 0.0000000 0.0000000 CAPACITY_ROW( V6) 292.0000 0.0000000 CAPACITY_ROW( V7) 0.0000000 0.0000000 CAPACITY_ROW( V8) 3.000000 0.0000000 CAPACITY_ROW( V9) 244.0000 0.0000000 CAPACITY_ROW( V10) 0.0000000 0.0000000 CAPACITY_ROW( V11) 0.0000000 0.0000000 CAPACITY_ROW( V12) 45.00000 0.0000000 CAPACITY_ROW( V13) 0.0000000 0.0000000 CAPACITY_ROW( V14) 0.0000000 0.0000000 MAX_BUS_ROW( V1) 0.0000000 0.0000000 MAX_BUS_ROW( V2) 0.0000000 0.0000000 MAX_BUS_ROW( V3) 0.0000000 0.0000000 MAX_BUS_ROW( V4) 0.0000000 0.0000000 MAX_BUS_ROW( V5) 0.0000000 0.0000000 MAX_BUS_ROW( V6) 892.0000 0.0000000 MAX_BUS_ROW( V7) 0.0000000 0.0000000

123

Table 5.7 (continued)

Row Slack or Surplus Dual Price MAX_BUS_ROW( V8) 603.0000 0.0000000 MAX_BUS_ROW( V9) 694.0000 0.0000000 MAX_BUS_ROW( V10) 0.0000000 0.0000000 MAX_BUS_ROW( V11) 0.0000000 0.0000000 MAX_BUS_ROW( V12) 545.0000 0.0000000 MAX_BUS_ROW( V13) 0.0000000 0.0000000 MAX_BUS_ROW( V14) 0.0000000 0.0000000 MIN_QTY_ROW( V1) 0.0000000 0.0000000 MIN_QTY_ROW( V2) 0.0000000 0.0000000 MIN_QTY_ROW( V3) 0.0000000 0.0000000 MIN_QTY_ROW( V4) 0.0000000 0.0000000 MIN_QTY_ROW( V5) 0.0000000 0.0000000 MIN_QTY_ROW( V6) 8.000000 0.0000000 MIN_QTY_ROW( V7) 0.0000000 0.0000000 MIN_QTY_ROW( V8) 297.0000 0.0000000 MIN_QTY_ROW( V9) 6.000000 0.0000000 MIN_QTY_ROW( V10) 0.0000000 0.0000000 MIN_QTY_ROW( V11) 0.0000000 0.0000000 MIN_QTY_ROW( V12) 255.0000 0.0000000 MIN_QTY_ROW( V13) 0.0000000 0.0000000 MIN_QTY_ROW( V14) 0.0000000 0.0000000 MIN_BUS_ROW( V1) 0.0000000 0.0000000 MIN_BUS_ROW( V2) 0.0000000 0.0000000 MIN_BUS_ROW( V3) 0.0000000 0.0000000 MIN_BUS_ROW( V4) 0.0000000 0.0000000 MIN_BUS_ROW( V5) 0.0000000 0.0000000 MIN_BUS_ROW( V6) 208.0000 0.0000000 MIN_BUS_ROW( V7) 0.0000000 0.0000000 MIN_BUS_ROW( V8) 497.0000 0.0000000 MIN_BUS_ROW( V9) 406.0000 0.0000000 MIN_BUS_ROW( V10) 0.0000000 0.0000000 MIN_BUS_ROW( V11) 0.0000000 0.0000000 MIN_BUS_ROW( V12) 555.0000 0.0000000 MIN_BUS_ROW( V13) 0.0000000 0.0000000 MIN_BUS_ROW( V14) 0.0000000 0.0000000

Table 5.8 Selected vendors and actual vendors with their allocations

Vendors V6 V8 V9 V12

Order quantity 308 597 506 655

Actual vendors V6 V1 V2 V5

The Table 5.9 indicates the vendors selected, order allocated,

defectives, late deliveries and cost as inferred from the solution report

124

Table 5.9 Vendor selection and order allocation with quantity discounts

Vendor no (Actual vendor)

Order allocation

Number of defectives

Number of late

deliveries Total cost

in Rs. V8(V1) 597 15 19 5373 V9(V2) 506 23 27 5060

V12(V5) 655 10 1 6550 V6 308 18 8 3773

Total 2066 66 55 20756 Order – Defectives = 2066 – 66 = 2000 = Demand

From Table 5.9 the demand is exactly met when eliminating the

number of defectives. The number of defectives and number of late deliveries

are within the specified limits in the model.

5.6 SUMMARY

This chapter described the evaluation methodology for the single

component multiple vendor selection model using Lexicographic method.

LINGO has in built solvers and these solvers were used to generate the

solution. The model was evaluated using two scenarios, one with discounts

and one without discounts. It is evident from the solutions that the objective

value generated using discounts was better than the without discount model.

This evaluation method had used the all-unit quantity discount model, where

if the order exceeded a middle order quantity, they were considered as

different vendor with a price equal to the discounted price. The next chapter

discusses the evaluation of the multi-component multi-vendor model

considering incremental discounts.