optimization of food processing operations in s.r.’s canneries, allahabad by linear programming
TRANSCRIPT
PROJECT REPORT
ON
“OPTIMIZATION OF FOOD PROCESSING OPERATIONS IN S.R.’s CANNERIES, ALLAHABAD BY LINEAR PROGRAMMING”
Submitted to the Allahabad Agricultural Institute-Deemed University, in partial fulfillment of the requirements for the award of the degree of
BACHELOR OF TECHNOLOGY
IN
FOOD TECHNOLOGY (PROCES AND FOOD ENGINEERING)
BY
MANISHAKAR
SHALINI BAJPAI
COLLEGE OF AGRICULTURAL ENGINEERING & TECHNOLOGY
ALLAHABAD AGRICULTURAL INSTITUTE- DEEMED UNIVERSITY
ALLAHABAD-211007 (U.P.) INDIA
2007
CERTIFICATE
Certify that the project titled “OPTIMIZATION OF FOOD PROCESSING
OPERATIONS BY LINEAR PROGRAMMING” submitted to the College of
Agricultural Engineering of Technology, Allahabad Agricultural Institute-Deemed
University, Allahabad in partial fulfillment of requirement for the award of degree
Bachelor of Technology in Food Technology (Process and Food Engineering) is bonafide
record of research carried out by Mr. Manishakar & Ms. Shalini Bajpai under my
supervision and guidance.
Recommended for Acceptance
…………………………………
Prof. (Dr.) Tufail Ahmad
Evaluated by Examination Committee
Prof. (Dr.) M. Imtiaz Prof.
Chairman External Examiner
Prof. (Dr.) Tufail Ahmad Prof.
Member Member
Approved
Prof. (Dr.) M. Imtiaz
Dean
College of Agricultural Engineering of Technology
SELF ATTESTATION
This is to certify that we have personally worked on the project titled
“OPTIMIZATION OF FOOD PROCESSING OPERATIONS BY
LINEAR PROGRAMMING”. The data mentioned in the project have
been generated during the work and are genuine. Data / information
obtained from other agencies have been duly acknowledged. None of the
findings or information pertaining to the work has been concealed. The
results embodied in this project report have not been submitted to any
other university or institution for the award of any degree.
Date: Manishakar
Place: Shalini Bajpai
CONTENTS
CHAPTER PARTICULARS PAGE No.
ACKNOWLEDGEMENT i
ABSTRACT ii
LIST OF TABLES iii
LIST OF ABBREVIATIONS iv
I INTRODUCTION 1
1.1 Scope 3
1.2 Importance 6
1.3 Justification 7
1.4 Objectives 8
II REVIEW OF LITERATURE 9
III THEORETICAL CONSIDERATIONS 11
3.1 Definition 11
3.2 Basic Terminology 11
3.3 Specifications in Linear Programming 13
3.4 General Mathematical Model 15
IV METHODOLOGY 16
4.1 Data Collection 16
4.2 Data Analysis 16
4.3 Formulation of Objective Function and 20
Linear Constraints
4.4 Conversion of Function and Constraints 20
into Mathematical Equations
V RESULTS AND DISCUSSION 22
REFERENCES 23
APPENDIX 26
ACKNOWLEDGEMENT
We would like to express heartfelt gratitude to Prof. Tufail
Ahmad, Allahabad Agricultural Institute-Deemed University, Allahabad
for his valuable guidance, constructive criticism and encouragement
towards the successful completion of the project work. We are grateful to
Er. T.R.Genitha (Assistant Professor), Department of Food Technology
(Process & Food Engg.), who extended every possible help. We are
grateful to Prof. M. Imtiyaz, Dean, College of Agricultural Engineering &
Technology, Allahabad Agricultural Institute-Deemed University,
Allahabad for his cooperation.
We would also like to thank Mr. Vinay Tandon, owner of
S.R.’s Canneries for his permission and cooperation for collection of data
from S.R.’s Canneries.
We are also thankful to our friends Vikash Rajiv and Ashish
for their kind help and advice. Finally we are thankful to our parents &
family who were constant source of encouragement during the course of
this work.
Manishakar
Shalini Bajpai
ABSTRACT
Studies were done on the data collected from S.R.’s Canneries for evaluating the
optimal solution using simplex method. The studies included formulating the objective
function and constraints from collected data and achieving optimal solution by the
application of simplex method. The studies revealed that more profitable products should
be manufactured in large quantities to get maximum profit and simplex method helps in
finding the optimum no. of units to be manufactured for getting maximum profit.
LIST OF TABLES
TABLE No. TITLE PAGE No.
4.1 Data for different products 19
5.1 Optimal value of decision variable and 22
objective function
A1. Optimal Solution using Simplex method 23
LIST OF ABBREVIATIONS
MRP : Maximum Retail Price
LPP : Linear Programming Problem
Max. : Maximum
Min. : Minimum
et. al. : etalibi and others
etc. : et cetra
Opt : optimum
LP : Linear Programming
min. : Minutes
hr. : Hour
g. : Gram
kg. : Kilogram
Rs. : Rupees
CHAPTER I
INTRODUCTION
Optimization is the art & science of allocating scare resources
to the best possible effect. Optimization is called into play every day in
questions of industrial planning, resource allocation, scheduling, decision
making etc. An edge in maximizing profits or minimizing costs can often
mean the difference between success & failure in business.
Optimization Modeling requires appropriate time. The general
procedure that can be used in the process cycle of modeling is to
Describe the problem
Prescribe a solution
Control the problem by assessing / updating the optimal solution
continuously, while changing the parameters and structure of the
problem.
Optimization problems are made up of four basic ingredients:
1. An objective function that is to be minimized or maximized. That is
the quantity to be maximized or minimized expressed in
mathematical form is called objective function.
2. Controllable inputs are the set of decision variables which affect the
value of objective function. In the manufacturing problem, the
variables might include the allocation of different available
resources or the labor spent on each activity. Decision variables are
essential. If there are no variables, the objective function and the
problem constraints cannot be defined.
3. The uncontrollable inputs are called parameters. The input values
may be fixed number associated with the particular problem. These
values are called as parameters of the model.
4. Constraints are relations between decision variables and the
parameters. A set of constraints allows some of the decision
variables to take on certain values and exclude others.
Linear Programming deals with a class of optimization
problems, where both the objective function to be optimized and all the
constraints, are linear in terms of the decision variables. Linear
Programming is the most commonly applied form of constrained
optimization.
The main elements of linear programming (constrained optimization)
problems are:
Variables (decision variables): The values of the variables are
unknown before solving the problem. The variables usually
represent the decisions that can be controlled. The goal is to find
values of the variables that provide the best value of the objective
function.
Objective function: This is a mathematical expression that
combines the variables to express the goals. For example, it may
represent profit. Usually, it is required to either maximize or
minimize the objective function.
Constraints: These are mathematical expressions that combine
the variables to express limits on the possible solutions. For
example, they may express the idea that the number of workers
available to operate a particular machine is limited, or that only a
certain amount of steel is available per day.
1.1 SCOPE
1.1.1 Production Management
1. Product Mix: A company can produce several different
products, each of which requires the use of limited production
resources. In such cases, it is essential to determine the
quantity of each product to be produced knowing its marginal
contribution and amount of available resources used by it . The
objective is to maximize the total contribution, subject to all
constraints.
2. Production Planning: This deals with the determination of
minimum cost production plan over planning period of an item
with a fluctuating demand, considering the initial number of
units in inventory, production capacity, constraints on
production, manpower and all relevant cost factors. The
objective is to minimize total operation costs.
3. Assembly-line Balancing: This problem is likely to arise when
an item can be made by assembly different components. The
process of assembling requires some specified sequences. The
objective is to minimize the total elapse time.
4. Blending Problems: These problems arise when a product can
be made from a variety of available raw materials, each of
which has a particular composition and price. The objective
here is to determine the minimum cost blend, subject to
availability of the raw material, and minimum and maximum
constraints on certain product constituents.
5. Trim Loss: When an item is made to a standard size, the
problem that arises is to determine which combination of
requirement should be produced from standard materials in
order to minimize the trim loss.
1.1.2 Financial Management
1. Portfolio Selection: This deals with the selection of
investment activity among several other activities. The
objective is to find the allocation which maximizes the total
expected return or minimizes risk under certain limitations.
2. Profit Planning: This deals with the maximization of the
profit margin from investment in plant facilities and
equipment, cash in hand and inventory.
1.1.3 Marketing Management
1. Media Selection: Linear programming technique helps in
determining the advertising media mix so as to maximize the
effective exposure, subject to limitation of budget, specified
exposure rates to different market segment, specified
minimum and maximum number of advertisement in various
media.
2. Travelling Salesman Problem: The problem of a salesman is to
find the shortest route from a given city, visiting each of the
specified cities and then returning to the original point of
departure, provided no city shall be visited twice during the
tour. Such type of problem can be solved with the help of the
modified assignment technique.
3. Physical Distribution: Linear Programming determine the
most economic and efficient manner of locating
manufacturing plants and distribution for physical
distribution.
1.1.4 Personnel Management
1. Staffing Problem: Linear Programming is used to allocate
optimum manpower to a particular job so as to minimize the
total overtime cost or total manpower.
2. Determination of Equitable Salaries: Linear Programming
technique has been used in determining equitable salaries and
sales incentives.
3. Job Evaluation and Selection: Selection of suitable person for
a specified job and evaluation of job in organizations has been
done with the help of linear programming technique.
1.2 IMPORTANCE
Linear Programming helps in attaining the optimum use of
productive resources. It also indicates how a decision –maker
can employ his productive factors effectively by selecting and
distributing (allocating) these resources.
Linear Programming technique improves the quality of
decisions. The decision-making approach of the user of this
technique becomes more objective and less subjective.
Linear Programming Techniques provide possible and
practical solutions since there might be other constraints
operating outside the problem which must be taken into
account. Just because we can produce so many units does not
mean that they can be sold. Thus, necessary modification of
its mathematical solutions is required for the sake of
convenience to the decision-maker.
Highlighting of bottlenecks in the production processes is the
most significant advantage of this technique. For example,
when a bottleneck occurs, some machines cannot meet demand
while other remains idle for some of the time.
Linear Programming also helps in re-evaluation of a basic
plan for changing conditions. If conditions change when the
plan is party carried out, they can be determined so as to
adjust the remainder of the plan for best results.
1.3 JUSTIFICATION
Linear Programming is a mathematical modeling technique
very useful for economic allocation of scarce or limited resource such as
labor, material, machine, time, warehouse space, capital, energy etc. to
several completing activities such as product, services jobs, new
equipment project etc. on the basis of a given criterion of optimality.
1.4 OBJECTIVES
1. To study the various parameters required for optimization.
2. To analyze the problem in order to maximize profit.
3. To obtain optimization by using Linear Programming.
CHAPTER II
REVIEW OF LITERATURE
Lagrange (1762) solved tractable optimization problem with simple
equality constraints.
Gauss (1820) solved Linear System of equation by what is now called
caussian elimination.
Dantzig (1947) invented Simplex Method.
Gilbert (1950) studied the Optimization of machining parameters in
turning with respect to maximum production rate and minimum production
cost as criteria.
Brewer (1966) suggested the use of Lagrangian Multipliers for
Optimization of the constrained problem of unit cost, with cutting power
as the main constraint.
Fiacco and Mc. Cormic (1968) introduced the Interior Point Method.
Bhattacharya et al (1970) optimized the unit cost for turning, subject to
the constraints of surface roughness and cutting power by the use of
Lagrange’s Method.
Walvekar and Lambert (1970) discussed the use of geometric
programming to selection of machine variables. They optimized cutting
speed and feed rate to yield minimum production cost.
Karmarkar (1984) applied the Interior Method to solve Linear Programs
adding his innovative analysis.
Adema et al (1991) Achievement test construction using 0-1 Linear
Programming.
Agapiou (1992) formulated single-pass and multi-pass machining
operations. Production cost and total time were taken as objective and
weighing factor was assigned to prioritize the two objectives in the
objective function. Several physical constraints were considered applied in
his model.
Prasad et al (1997) reported the development of an Optimization module
for determining process parameters for turning operations as part of a PC-
based generative CAPP system. The formulated models are solved by the
combination of Geometric and Linear Programming techniques.
CHAPTER III
THEORETICAL CONSIDERATION
3.1. DEFINITION
The word Optimization means the art and science of
allocating scarce resources to the best possible effect. Optimization
Technique are called into play every day in question of industrial
planning, resource allocation, scheduling, decision-making etc. For
example, how does a food processing company decide where to buy raw
material, where to ship it for processing, what products to convert it to,
where to sell final product and at what prices. Many of the large scale
optimization techniques in general use today can trace their origin to
methods developed during world war-II to deal with the massive logistical
issue raised by huge armies having millions of men and machines.
Linear programming deals with a class of optimization
problems, where both the objective function to be optimized and all the
constraints, are linear in terms of the decision variables. Linear
programming is the most commonly applied form of constrained
optimization.
3.2. BASIC TERMINOLOGY
3.2.1 Analysis
Moving from the real world problem to the algorithm model
or solution technique is known as analysis.
3.2.2 Numerical methods
Moving from the algorithm, model or solution technique to
the computer implementation is generally the province of numerical
methods.
3.2.3 Verification
Moving from computer implementation back to the algorithm,
model, or solution technique is called verification.
3.2.4 Validation
It is the process of making sure that the model or solution
technique is appropriate for the real situation.
3.2.5 Sensitivity analysis
It looks at the effect of the specific data on the results.
3.2.6 Variable
The values of the variables are not known when one starts the
problem. The variables usually represent things that can adjust or control,
for example the rate at which to manufacture items. The goal is to find
values of the variables that provide the best value of the objective
function.
3.2.7 Objective function
This is a mathematical expression that combines the variables
to express the goal. It may represent profit, for example. It is required to
either maximize or minimize the objective function.
3.2.8 Constraint
These are mathematical expressions that combine the variables
to express limits on the possible solutions. For example they may express
the idea that the number of workers available to operate a particular
machine is limited, or that only a certain amount of steel is available per
day.
3.3 SPECIFICATIONS IN LINEAR PROGRAMMING
3.3.1 Decision Variable
Various alternatives for arriving at the optimal value of the
objective function are evaluated. Obviously, if there are no alternatives to
select, LPP is not required. The evaluation of various alternatives is
guided by nature of objective function and availability of resources. For
this, certain variables are produced termed as Decision variables usually
denoted as x 1 , x2 ,x3 ,…..x n etc.
The value of these variables represents the extent to which
each of these is to be performed.
3.3.2 Objective function
The objective or goal function of each linear programming
problem is expressed in terms of decision variables to optimize the
criterion of optimality (also called as measure of performance) such as
Profit, Cost, Revenue, Distance etc.
Opt (Max. or Min.) Z = C 1x1+C2x2+………. +C nxn
Where Z is measure of performance variable which is a
function of x 1 , x2…... , x n .
Quantities C 1 , C2 ,….., Cn are parameters which represent the
contribution of units of respective variables x 1 ,x2…..,x n to
measure the performance of Z.
The optimal value of the given objective function is obtained by
‘SIMPLEX METHOD’.
3.3.3 Constraints: The importance of setting accurate and reasonable
practical constraints cannot be overstated, as these are essential
components in LP test assembly because they are the means by which test
specifications are fully met. Constraints should be closely checked before
any attempt is made to solve the model.
3.4 GENERAL MATHEMATICAL MODEL
Finding values of decision variables x1, x2,………., xn so as to
Optimize (Maximize or Minimize)
Z=c1x1 + c2x2 +………+ cnxn
Subject to linear constraints,
a11x1 + a12x2 +…………+ a1nxn(< , = , >)b1
a21x1 + a22x2 +…………+ a2nxn(< , = , >)b2
……………………………………………………………..
……………………………………………………………..
am1x1 + am2x2 +…………+ amnxn(< , = , >)bm
and,
x1,x2,……….,xn>0
where x1, x2,………., xn ≥ 0
CHAPTER IV
METHODOLOGY
S.R.’s Canneries is a Food Processing Industry involved in
canning and production of tomato ketchup, mixed pickle, mixed jam and
orange marmalade etc. With a strong rural and co-operative base, the
company is engaged in strengthening the co-operative food distribution
system. Being a leading food processing company in Uttar Pradesh its
products are sold to Taj group of hotels, railways, airlines etc.
4.1 DATA COLLECTION
In the first phase of present study, the review of company’s
current system and survey was conducted for existing optimization
problem in S.R.’s Canneries. The raw materials, work in process, labor
distribution and finished goods data were collected by regular visits to the
company.
4.2 DATA ANALYSIS
4.2.1 Tomato Ketchup
Capacity of Packaging Machine = 4000/hr
Labor requirement for the whole process = 8
Number of packaging machine = 2
Cost of 1 kg. foil = Rs. 168
100g. foil prepares = 140 packs
Failure rate = 1.5hr / 10hr
MRP of 15g. packs = Rs. 2
Profit/ 15g. pack = 10%
4.2.2 Mixed Pickle
Capacity of packaging machine = 100cups/min
Labour requirement for whole process = 8 labors
Number of packaging machines = 2
Machine Capacity = 40500 cups/day
Production requirement = 30000 pieces/ day
Failure rate = 2 hr/ 2 months
Cost/ 15 g. pack = 70 paise
Profit/15 g. pack = 7%
4.2.3 Mixed Jam
Labor requirement = 6 labor
Failure rate = 2hr/ 6 months
Production requirement per day = 20000 cups per day
Cost/ 15g. pack = 65 paise
Profit/ 15 g. packs = 9%
4.2.4 Orange Marmalade
Labor requirement = 6 labor
Failure rate = 2 hr/ 6 months
Production requirement Per day = 5000 cups/day
Cost per 15g. pack = 55 paise
Profit/ 15 g. = 11%
Table 4.1 Data for Different Products
Resources/
Constraints
Tomato
Ketch-
up
Mixed
Pickle
Orange
Marmalade
Mixed
Jam
Total/Availability
Production
Capacity(Units*) 380 405 270 -
Order
Commitment(Units*) 300 300 50 200 -
Labour hour/Unit .2 .1876 .2111 199.5
Packaging or Filling
Time/Unit*
(hours)
.2125
.0233
.0351 .0351
8.075
9.4666
9.4888
Cost/Unit*
(Rs.)
65 70 55 65 -
Profit/Unit*
(Rs.)
6.5 4.9 6.05 5.85 -
*1unit ≡ 100 pieces
4.3 FORMULATION OF OBJECTIVE FUNCTION AND LINEAR
CONSTRAINTS
Maximize,
Z = 6.5x1 + 4.9x2 + 6.05x3 + 5.85x4
Subject to Linear Constraints,
.2x1 + .1876x2 + .2111x3 + .2111x4 ≤ 209
.0212x1 ≤ 8.075
.0233x2 ≤ 9.4666
.0351x3 + .0351x4 ≤ 9.4888
1x1 ≥ 300
1x2 ≥ 300
1x3 ≥ 50
1x4 ≥ 200
4.4 CONVERSION OF FUNCTION AND CONSTRAINTS INTO
MATHEMATICAL EQUATIONS
Maximize, Z = 6.5x1 + 4.9x2 + 6.05x3 + 5.85x4 – Ma1 – Ma2 – Ma3 – Ma4
Subject to Linear Constraints,
.2x1 + .1876x2 + .2111x3 + .2111x4 + s1 = 209
.0212x1 + s2 = 8.075
.0233x2 + s3 = 9.4666
.0351x3 + .0351x4 + s4 = 9.4888
1x1 – s5 + a1 = 300
1x2 – s6 + a2 = 300
1x3 – s7 + a3 = 50
1x4 – s8 + a4 = 200
Where,
s1, s2, s3, s4 are Slack Variables.
s5, s6, s7, s8 are Surplus Variables.
a1,a2,a3,a4 are Artificial Variables.
M is a positive co-efficient of high numerical value.
CHAPTER V
RESULTS AND DISCUSSION
After analysis of data and consideration of conditions,
Simplex Method was used to solve the Linear Programming Problem.
Table 5.1 Optimal Value of Decision Variables and Objective Function
Sl. No. Decision Variable /Objective Function Calculated
Optimal Value
(units*)
1. x1 380.8962
2. x2 353.1593
3. x3 70.3361
4. x4 200
5. Z Rs.5801.8392
* 1 unit ≡100 pieces
From the calculation of optimal solution using Simplex
method it was found that by producing 380896 pieces of tomato ketchup
(15 g. sachet), 353159 pieces of mixed pickle (15 g. sachet), 70336 pieces
of orange marmalade and 20000 pieces of mixed jam (15 g. sachet), a
maximum profit, Rs. 5801.8392, can be obtained.
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APPENDIX
A1. OPTIMAL SOLUTION USING SIMPLEX METHOD
Cj 6.5 4.9 6.05 5.85 0 0 0 0 0 0 0 0 0 -M -M -M -M
S.V. X1 X2 X3 X4 S1 S2 S3 S4 S5 S6 S7 S8 a1 a2 a3 a4 S.Q.
0 S1 .200 .1876 .2111 .2111 1 0 0 0 0 0 0 0 0 0 0 0 199.5 997.5
0 S2 .0212 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 8.075 380.8962
0 S3 0 .0233 0 0 0 0 1 0 0 0 0 0 0 0 0 0 9.4666 -
0 S4 0 0 .0351 .0351 0 0 0 1 0 0 0 0 0 0 0 0 9.4888 -
-M a1 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 0 300 300
-M a2 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 300 -
-M a3 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 50 -
-M a4 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 200 -
Zj -M -M -M -M 0 0 0 0 M M M M -M -M -M -M -850M
Cj - Zj 6.5+M 4.9+M 6.05
+M
5.85
+M
0 0 0 0 -M -M -M -M 0 0 0 0
0 S1 0 .1876 .2111 .2111 1 0 0 0 .200 0 0 0 -.200 0 0 0 139.5 660.82
0 S2 0 0 0 0 0 1 0 0 .0212 0 0 0 -.0212 0 0 0 1.715 -
0 S3 0 .0233 0 0 0 0 1 0 0 0 0 0 0 0 0 0 9.4666 -
0 S4 0 0 .0351 .0351 0 0 0 1 0 0 0 0 0 0 0 0 9.4888 270.3361
6.5 x1 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 0 300 -
-M a2 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 300 -
-M a3 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 50 50
-M a4 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 200 -
Zj 6.5 -M -M -M 0 0 0 0 -6.5 M M M 6.5 -M -M -M 1950-
550M
Cj - Zj 0 4.9 6.05 5.85 0 0 0 0 6.5 -M -M -M -M 0 0 0
26
+ M +M +M -6.5
27
0 S1 0 .1876 0 .2111 1 0 0 0 .200 0 .2111 0 -.200 0 -.2111 0 128.945 610.82
0 S2 0 0 0 0 0 1 0 0 .0212 0 0 0 -.0212 0 0 0 1.715 -
0 S3 0 .0233 0 0 0 0 1 0 0 0 0 0 0 0 0 0 9.4666 -
0 S4 0 0 0 .0351 0 0 0 1 0 0 .0351 0 0 0 -.0351 0 7.7338 220.336
6.5 x1 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 0 300 -
-M a2 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 300 -
6.05 x3 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 50 -
-M a4 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 200 200
Zj 6.5 -M 6.05 -M 0 0 0 0 -6.5 M -6.05 M 6.5 -M 6.05 -M 22252.5
0-.500
M
Cj - Zj 0 4.9 +
M
0 5.85 +
M
0 0 0 0 6.5 -M 6.05 -M -M -
6.05
0 -M - 6.5 0
0 S1 0 .1876 0 0 1 0 0 0 .200 0 .211 .2111 -.200 0 -.2111 -.211
1
86.725 462.286
0 S2 0 0 0 0 0 1 0 0 .0212 0 0 0 -.0212 0 0 0 1.715 -
0 S3 0 .0233 0 0 0 0 1 0 0 0 0 0 0 0 0 0 9.4666 406.2918
0 S4 0 0 0 0 0 0 0 1 0 0 .0351 .0351 0 0 -.0351 -.035
7
0.7138 -
6.05 x1 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 0 300 -
-M a2 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 300 300
6.05 x3 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 50 -
5.85 x4 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 200 -
Zj 6.5 -M 6.05 5.85 0 0 0 0 -6.5 M -6.05 -5.85 6.5 -M 6.05 5.85 3422.5
Cj - Zj 0 4.9
+M
0 0 0 0 0 0 6.5 -M 6.05 5.85 -M -
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0 S1 0 0 0 0 1 0 0 0 .200 .1876 .2111 .2111 -.200 -.187
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0 S2 0 0 0 0 0 1 0 0 .0212 0 0 0 -.0212 0 0 0 1.715 80.896
0 S3 0 0 0 0 0 0 1 0 0 .0233 0 0 0 -.023
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0 S4 0 0 0 0 0 0 0 1 0 0 .0351 .0351 0 0 -.0311 -.0357 0.7138 -
6.5 x1 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 0 300 -
4.9 x2 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 300 -
6.05 x 3 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 50 -
5.85 x 4 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 200 -
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0 S5 0 0 0 0 0 47.16
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0 S3 0 0 0 0 0 0 1 0 0 .0233 0 0 0 -.023
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0 S4 0 0 0 0 0 0 0 1 0 0 .0351 .0351 0 0 -.0351 -.0351 0.7138 20.3369
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