Q1: a company engaged in producing tinned food has 300 trained employee on the rolls each of whon
can produce one can of food in a week. Due to the developing taste of public fo0r this kind of food, the
company plans to add the existing labor force by employing 150 people in phased manner, over the
next five weeks. The newcomers have to undergo as two week training programme before being put to
work. The training is to be given by employee from the existing ones and it is known that one
employee can train three trainees. Assume that there would be no production from the trainers and the
trainees during training period as the training is off the job. However the trainee would be remunerated
at the rate of 300 per week, the same rate as for the trainers.
The company has booked the following numbers of cans to supply during the next five weeks:
Weeks :
No. of cans
{1
{280
2
298
3
305
4
360
5
400
Assume that the production in any week would not be more than the numbers of cans ordered for so
that every delivery of food would be ‘fresh’.
Formulate a LP model to develop a training schedule that minimise the labor cost over the five period.
LP model formulation: The data of the problem is summerised as given below:
(1)
Cans supplied
(2)
(3)
(4)
Each trainee has to undergo a two week training.
One employee is required to to train three trainee.
Every trained worker producing one can/week but no production from trainers and trainees
during training.
Numbers of employee to be employed=150
The production in any week not to exceed the cans required.
Numbers of weeks for which newcomers would be employeed: 5, 4, 3, 2, 1
(5)
(6)
(7)
weeks :
Number:
1
280
2
298
3
305
4
360
5
400
From the given information you may be observe the following facts:
(One)
workers employed in the beginning of the first week would get salary for all the five weeks
those employed at the second week would get salary for 4 week and so on.
(Two) The value of the objective the function would be obtained by multiplying it by 300 because
each person would get a salary of 300 per week.
(Three) Inequalities have been used in constraints because some workers might remain idle in some
week(s).
Decision variables: let
x1, x2, x3, x4, x5= number of trainees in the beginning of week 1,2,3,4, and 5 respectively.
The LP model
Minimize total labor force z= 5x1+4x2+3x3+2x4+x5
Subject to the constraints
(a)
capacity constraints
300-x1/3 >= 280
300-x1/3-x2/3>=298
300+x1-x2/3-x3/3>=305
300+x1+x2+-x3/3-x4/3>=360
300+x1+x2+x3+-x4/3-x5/3>=400
(b)
new recruitment constraints
x1+ x2+ x3+ x4+ x5=150
and
x1, x2, x3, x4, x5>=0
Q.2: A company has two grades of inspectors 1 and 2 who are to be assigned for a quality
control inspection. It is required at least 2000 pieces to be inspected per 8-hour day. A grade
one inspector can check pieces at the rate of 40 per hour, with an accuracy of 97%. A grade 2
inspector checks at the rate of 30 pieces per hour with an acuracy of 95%.
The wage rate of grade 1 inspector is 5 per hour while that of grade 2 inspector is Rs 4 per
hour. An error made by an inspector cost Rs 3 to the company. There are only 9 grade 1
inspector and 11 grade 2 inspector available in the company. The company wishes to assign
work to the available inspectors so as to minimized the total cost of the inspection. Formulate
this problem as a linear programming module.
LP model formulation:
the data of the problem is summerised as follows:
Inspectors
Number of inspectors
Rate of checking
Inaccracy in checking
Cost of inacuracy in checking
Wage rate/hour
Total pieces which must be inspected=2000
Grade 1
grade 2
9
40 pieces/hour
1-0.97=0.03
Rs. 3/piece
Rs 5
11
30 pieces/hour
1-0.95=0.05
Rs. 3/piece
Rs.5
Decision variables:
let x1, x2= number of grade 1 and grade 2 inspectors to be assigned
for inspection, respectively.
The LP model
Hourly cost of each of grade 1 and 2 inspectors can be computed as follows:
For inspector grade 1:Rs (5+3X40X0.03)=Rs. 8.60
For inspector grade 2:Rs.(4+3X30X0.05)=Rs. 8.50
Based on the given data the linear programming problem can be formulated as follows:
Minimize ( daily inspection cost) z= 8(8.60x1 + 8.50x2) = 68.80x1+68.00x2
Subject to the constraints
(a)
(b)
Total number of pieces that must be inspected in an 8 hour day constraints
40 X 8x1 + 30 X 8x2 >= 2000
Numbers if inspector of grade 1 and grade 2 avaivlable constraints
x1=< 9; x2=< 11
x1, x2 >=0
Q.3: A manufacturing company engaged in producing three types of products: A, B and C. the
production department daily produces component sufficient to make 50 units of A, 25 units of B and 30
units of C. The management is confronted with with problem of optimizing the daily production of
products in assembly department where only 100 man-hours are available daily to assemble the
products. The following additional information is available.
Type of product profit contribution per unit of product (Rs.) Assembly time per product (hrs)
A
B
C
12
20
45
0.8
1.7
2.5
THE COMPANY HAS a daily order commitment for 20 units of product A and total of 15 units of B
and C products. Formulates this problem as an LP model so as to maximize the total profit.
LP model formulation: the data of the problem is summarised as follows:
Resources /
Constraints
A
Production capacity (units)
Man hours per unit
Order commitment unit
Profit contribution (Rs./unit)
50
0.8
20
12
product type
B
C
25
1.7
15
20
total
30
2.5
100
45
Decision variables: let x1, x2, x3=numbers of units of products A, B and C to be produced respectively
The LP model
Maximize (total profit) Z = 12x1 + 20x2 + 45x3
Subject to the constraints
(a)
labor and material constraints
0.8x1+1.7x2+2.5x3 =<100
x1
=<50
x2
=<25
x3=<30
(b)
order commitment constraints
x1
>=20
x2
+x3
>=15
x1,x2,x3>=0
Q.4: A company has two plants each of which produces and supply and supplies two products: A and
B. The plant can each work upto 16 hours a day. In plant 1, it takes three hours to produce and pack
1000 gallon of A and and 1 hour tp prepare and pack I quintal of B. in plant 2, it takes 2 hour to
produce and pacj 1000 gallons of A and 1.5 hours to prepare and pack a quintal of B. in plant 1 it cost
Rs. 15000 to prepare and PACK 1000 gallons of A and Rs. 28000 to prepare abd pack a quintal of B.
Whereas these cost are Rs 18000 and Rs 26000 respectively in plant 2. The company is obliged to
produce daily at least 10000 gallons of milk and 8 quintal of B.
Formulate this problem as an LP model to find out as to how the company should organize its
production so that required amounts of the two products be obtained at a minimum cost.
LP model formulation :
the data of the problem as summarized as follows:
Resources/
product
total
Constarints
A
B
Preparation times (hrs)
Plant 2: 2hrs/1000 gallons
Minimum daily production
plant 1: 3 hrs/1000 gallons
1.5 hrs/quintal
16
10,000 gallons
8 quintal
Cost production
Plant 2: 18000
plant 1: 15000
26000
availibility (hours)
1hrs/1 quintal
16
28000
Decision variables:
let
X1, x3= quantity of product A (in 000 gallons) to be produced in plant 1 and 2 respectively
X2,x4 = quantity of product B (in quintal) to be produced in plant 1 and 2 respectively
LP model
Minimize (total cost) Z= 15000x1+28000x2+18000x3+26000x4
Subjecte to the constraints
(i)
Preparation time constarints
3x1+x2=<16
2x3+1.5x4=<16
(ii)
and
minimum daily production requirment constaints
x1+x3=>10
x2+x4=>8
x1,x2,x3,x4>= 0
Q.5:
An electronic company is engaged in the production of two component c1 and c2 used in
radio sets. Each units of C1 cost the company Rs 5 in material, while each of C2 cost the company Rs
25 in wages and Rs 15 in material. The company sells both product on period credit terms but the
company’s labor and material expenses must be paid in cash. The selling price of C1 is Rs 30 per units
and of C2 is Rs 70 per units. Because of the string monopoly of the company for these components it
is assumed that the company can sell at the prevaling prices as many units as it produces. The
company’s production capacity is however is limited by two considerations. First at the beginning of
period 1 the company has initial balance of Rs. 4000 (cash plus bank credit plus collection from past
credit sales). Second the company has available in each period 2000 hours of machine time and 1400
hours of assembly time. The production of c1 requires 3 hours of machine time and 2 hoyrs of
assembly time whereas the production of c2 each c2 requires 2 hours of machine time and 3 hours of
assembly time. Formulate this problem as a Lp model so as to maximize the total profit of the
company.
LP model formulation
The data of the problem is summarized as follows:
Resources/
constraints
c1
components
c2
total availibility
Budget (Rs)
Machine time
Assembly time
Selling price
Cost price
10/units
3 Hrs/unit
2 hrs/unit
Rs 30
Rs 10
40/units
2 hrs/unit
3 hrs/unit
Rs 70
Rs 40
Rs 4000
2000 hrs
1400 hours
Decision variable:
let
X1, x2= number of units component C1 and C2 to be produced respectively
The LP model
Maximize ( total profit) Z= Selling price- cost price
=(30-10)x1 + (70-40)x2 =20x1+30x2
subject to constraints
(1)
the total budget available constraints
10x1+ 40x2=<4000
(2)
Production time constarints
3x1+2x2=< 2000
2x1+3x2=<1400
x1, x2>= 0
Q.6:
A company produces 3 types of parts for automatic washing machine. It purchases casting of
the parts from a local foundary and then finishes the part of drilling and shaping and polishing
machine.
The selling price of part A, B and C respectively are Rs 8, Rs 10 And Rs a4. All parts made
can be sold. Casting for the parts A, B and C cost Rs. 5, 6 and 8 respectively.
The shop posses only one of each type machine. Cost per hoyr to run each of the machine are
Rs 20 for drilling, Rs 30 for shaping and Rs 30 for polishing. The capacity ( part per hour ) for each
part on each machine are shown in the following table:
Machine
capacity per hour
Part A
Drilling
Shaping
Polishing
25
25
40
Part B
40
20
30
Part C
25
20
40
The management of the shop wants to know how many parts of each type it should produce per hour in
order to maximize profit for an hour’s run. Formulate this problem as an LP model.
LP model formulation : Decision variables Let
x1,x2,x3 number of type A, B and C parts to be produced per hour respectively
profit must allow not only for the cost of the casting but for the cast od drilling , shaping and polishing.
Since 25 types A parts per hour can be run on drilling machine at a cost of Rs 20, then Rs 20/25= 0.80
is the drilling cost per type A part. Similar reasoning for the shaping and polishing gives
profit per type A part= ( 8-5 )-( 20/25+30/25+30/40)=0.25
profit for type B part ( 10-6 )- ( 30/40+30/20+30/30)=1
Profit of type C part ( 14-10 ) - ( 20/25+30/20+30/40)=0.95
On the drilling machine one type A part consume 1/25 th part of available hour , a type B part
consume 1/40th and a type C part consume a/25 of an hour. Thus the drilling machine constraints is
X1/25 +x2/40 +x3/25=<1
Similarly other constraints can bwe established
The LP model
Maximize ( Total Profit ) Z= 0.25x1+1.00x2+0.95x3+
Subject to constraints
(i)
Drilling machine constraints
and
( x1/25+x2/40/x3/25)=,1
(ii)
Shaping machine constraints
( x1/25+x2/20+x3/20=<1
(iii)
Polishing machine constraints
( x1/40+x2/20+x3/40)=<0
x1, x2, x3 >=0
Q.7:
A tape recorder campany manufacturer models A , B and C which have profit contribution per
unit of Rs. 15, Rs 40 and Rs 60 respectively. The weekly minimum production requirement are 25 units
for model A, 130 units of model for B and and 55 units of model for unit C. each type of recorder
requires a certain amountr of time for the manufacturing of component parts, for assembling and for
packing.. specifically, a dozen units of model A requires 4 hours for manufacturing, 3 hours for
assembling and 1 hours for packing. The corresponding figure for a dozen units of model B are 2.5, 4
and 2 and for a dozen units for model C are 6, 9 and 4 . during the forthcoming week the company has
available 130 hours of manufacturing and 170 hours of assembling and 52 hours of packaging time
formulate this problem of production scheduling as an LP model so as to maximize profit.
LP model formulationL the data of the problem is summarised as follows:
Resource /
Constraints
model
A
B
C
Total availability (hrs)
production requirement (units)
manufacturing time per dozen
Assembling time ( per dozen)
Packaging time (per dozen)
Contribution per unit (Rs)
25
4
3
1
15
130
2.5
4
2
40
55
6
9
4
60
130
170
52
Decision variables let x1,x2,x3= units of model A, B and C produced per week respectively.
The LP model
Maximize (total profit ) Z= 15x1+40x2+60x3
Subject to constraints
(i)
(ii)
And
Minimum production requirment constraints
X1
>=25
X2 >=130
X3 >=55
Manufacturing time constraints
(4x1/12 +2.5x2/12+6x3/12=<130
(iii)
Assembling time constraints
3x1/12+4x2/12+9x3/12 =<170
(iv)
Packaging time constraints
X1/12+2x2/12+4x3/12=<52
x1, x2,x3 >= 0
Q.8: In a chemical industry two product A and B are made involving two operation. The production of
B result is also in a by-product C at no extra cost. The product A can be sold at the profit of Rs. 3 per
unit and B at a profit of Rs. 8 per unit. Some of this by product can be sold at a unit price Rs. 2, the
remainder has to be destroyed and the destruction cost is Rs 1 per unit. Forecast show that upto 5 unit
of C can be sold. The company get 3 units ofd C for each units of B produced. The manufacturing time
are 3 hours per unit for A on operation one and two respectively, and 4 hours and 5 hours per unit for
B on operation one and two respectively. Because the product C results from producing B,no time is
used in producing C. the available times are 18 hours and 21 hours of operation one and two
respectively. Formulate an LP model in order to determine the quality of A and B which should be
produced keeping C in mind to take the highest profit
LP model formulation : the data of the problem is summerized as follows:
Time ( hrs required by
Constraints
availibility
A
Operation one
3
Operation two
3
By product of B
Profits per unit (Rs) 3
B
4
5
1
8
C
3
2
18hrs
21hrs
5units
Decision variables: let
X1,x2,x3= unit of product A, B, C to be produced respectively
X4= units of product C destroyed
The Lp model
Maximize (total profit ) Z= 3x1+8x2+2x3-x4
Subject to the constraint
(i)
for two products A and B
3x1+4x2=<18
3x1+5x2=<21
(ii)
for by-product C
-3x2+x3+x4
x1,x2,x3,x4
x3=<5
=0
>=0
Q.9:
Consider the following problem faced by a production planner in a soft drink plant. He has
two bottling machine A and B. A is designed for 8 ounce bottles and B for 16 ounce bottles. However
each can be used on both types with some loss of efficiency. The manufacturing dat ais as follows:
machine
A
B
8 ounce bottles
100/minutes
60/minutes
16 ounce bottles
40/minutes
75/minutes
The machine can be run 8 hour per day, 5 days per week. Profit on an 8ounce bottle is 15 paise and on
an 16 ounce bottle is 25 paise. Weekly production of the drink can not be exceed 3,00,000 ounce and
the market can absorb 25,000 8 ounce bottles and 7,000 16 ounce bottles per week. The planner wishes
to maximize his profit subject, of course to all the production and marketing restriction. Formulate this
problem as an LP model in order to maximize total profit.
LP model formulation:
Constraints
Machine A time
Machine B time
Production
Marketing
profit/units (Rs.)
the data of the problem is summarised in the following table:
Production
8 ounce bottles 16 ounce bottles
availability
100/minute
60/minutes
1
1
0.15
8X5X60=2400 minutes
8X5X60=2400 minutes
300,000 units/week
25000 units/week
7000 units/week
7000 units/week
40/minute
75 minutes
1
1
0.25
decision variables: Let
x1= units of 8 ounce bottles to be produced weekly
x2= units of 16 ounce bottles to be produced weekly
the LP model
Maximize ( total profit ) Z= 0.15x1+0.25x2
Subject to constraints
(i)
(ii)
(iii)
machine time constraints
x1/100+x2=</2400 ; x1/60+x2/75=<2400
production constarints
x1+x2 =< 3,00,000
marketing constarints
x1=< 25,000
x2=<7,000
x1,x2>=0
and
Q.10: A complete unit od certain product consist of four units of component A and three units of
component B. the two component ( A and B ) are manufactured from two different raw materials of
which 100 units and 200 units respectively available. Three department are engaged in the production
process with each department using a different method of manufacturing the component per production
run and the recoulting units are given below:
Department
input per run (units)
Raw material
raw material
I
II
1
7
5
2
4
8
3
2
7
output per run (units)
component
component
A
B
6
4
5
8
7
3
formulate this problem as linear programming model so as to determine the number of productions for
each department which will maximize the total number of complete units of the final product.
Lp model formulation:
Let
X1,x2,x3= number of production run for department 1, 2, 3 respectively
Since each unit of the final product requires 4 units of component A and 3 units of component B,
therefore maximum number of units of the final product cannot exceed the smaller value of
{total number of units of A products/4 ; total number of value B/3}
or
{6x1+5x2+7x3/4 and 4x1+8x2+3x3/3}
also if y is the number component units of final product, then obviously, we have
6x1+5x2+7x3/4>=y
and
4x1+6x2+3x3/4>=y
the LP model
Maximize Z= Min [ 6x1+5x2+7x3/4; 4x1+6x2+3x3/4]
Subject to the constraint
and
(i)
raw material constraints
7x1+4x2+2x3 =<100 (raw material I)
5x1+8x2+7x3 =<200 ( raw material II)
(ii)
Number of component units of final product constraints
6x1+5x2+7x3-4y>=0
4x1+8x2+3x3-4y>=0
x1,x2,x3 >=0
Q.11: A manufacturer of biscuit is considering four type of gifts pack containing three types of
biscuits. Orrange cream (OC), chocolate cream (CC) and wafers (W). a market research study
conducted recently to asses the preference of the consumer shows the following types of asortments to
be in good demand:
Assortment
contents
selling price
Per kg (Rs)
A
not less than 40% of OC
Not more than 20% of CC
Any quantity of W
20
B
not less than 20% of OC
Not more than 40% of CC
Any quantity of W
Not less than 50% of OC
Not more than 10% of CC
Any quantity of W
25
No restriction
12
C
D
22
For the biscuit the manufacturing cost and capacity are given below:
Type of biscuit
plant capacity
OC
CC
W
8
9
7
200
200
150
manufacturing cost (Rs/Kg.)
Formulate an LP model to find the production scvhedule which maximize the profit assuming that there
are no market restrictions.
LP model formulation:
Decision variable Let
For gift pack A
xaoc= number of kg of OC in A
xa,cc= number of kg of cc in A
xa,w= number of kg of W in A
For gift pack B
xboc= number of kg of OC in b
xb,cc= number of kg of cc in b
xb,w= number of kg of W in b
For gift pack C
xcoc= number of kg of OC in C
xc,cc= number of kg of cc in C
xc,w= number of kg of W in C
For gift pack d
xdoc= number of kg of OC in D
xd,cc= number of kg of cc in D
xd,w= number of kg of W in D
The LP model
Maximize (total profit) Z=20(xa,oc+xa,cc+xa,w)+25(xb,oc+xb,cc+xb,w) +
22(xc,oc+xc,cc+xc,w) + 12(xd,oc+xd,cc+xd,w) –8(xa,oc+xb,oc+xc,oc+xd,oc)9(xa,cc+xb,cc+xc,cc+xd,cc)-7(xa,w+xb,w+xc,w+xd,w)
Subject to the constrains
(a)
plant capacity constraints constraints
xa,oc+xb,oc+xc,oc+xd,oc=<200
xa.cc+xb,cc+xc,cc+xd,cc=<200
xa,w+xb,w+xc,w+xd,w=<150
specification constraints
For the gift pack A
Xa,oc>=0.40(xa,oc+xa,cc+xa,w)
Xa,cc=<0.20(xa,oc+xa,cc+xa,w)
For gift pack B
Xb,oc>=0.40(xb,oc+xb,cc+xb,w)
Xb,cc=<0.20(xb,oc+xb,cc+xb,w
For gifts pack C
Xcoc>=0.40(xc,oc+xc,cc+xc,w)
Xc,cc=<0.20(xc,oc+xc,cc+xc,w)
Q.12: Abc company manufacturer three grade of paints : Venus, Dianna and Aurora. The plant
operates on a three shift basis and the following data are available from the production recordes:
Requirment
Of resources
Venus
Special additive (kg./litre)
Milling kilolitres/ machine shift
Packing kilolitres/shift
0.30
2.00
12.00
Grade
Dianna
0.15
3.00
12.00
Availibility ( capacity/month)
Aurora
0.75
5.00
12.00
600 tonnes
100 machine shifts
80shifts
There are no limitations on the other resources. The particulars of sales forecasts and estimated
contribution to overheads and profit are given below
Maximum possible sales
Per month (kilometeres)
Contribution (Rs./ kilometres )
Venus
Dianna Aurora
100
4000
400
3500
600
2000
Due to commitment already made a minimum of 200 kilolitres per month of Aurora has to be
necessarily supplied next year.
Just as the company was able to to finalise the monthly production programme for the next 12
month, an offer was received from the nearby competitor for hiring 40 machines shift per month of
milling capacity for grinding diana paint, that could be spared for at least a year. However due to the
additional handling the profit margin of the competitor involved by using this facility the contribution
from the dianna will get reduced by Rs. 1 per litre.
Formuluate this problem as an LP model for determining the monthly production programme
to maximize contribution.
LP model formulation: Decision variable
let
X1=quantity of venus (kilolitres) produced in the company
X2=quantity of Dianna (kilolitres) produced in the company
X3=quantity of Dianna (kilolitres) produced in the company
X4= quantity of Aurora (kilolitres) produced in the company
The Lp model
Maximize (total profit) Z= 4000x1+3500x2 +(3500-1000)x3+2000x4
Subject to constrains
(a)
special additive constraint
0.30x1+ 0.15x2+0.15x3+0.75x4=<600
(b)
Own milling facility constraints
X1/2+x2/3+x4/5=<100
(c)
Hired milling facility constraints
X3/3 =<40
(iv)
(d)
and
packing constraints
x1/12+(x2+x3)/12+x4/12=<80
Marketing constraints
X1=<1000 (venus)
X2+x3=<(dianna)
200=<x4=<600 (Aurora)
x1,x2,x3,x4>=0
Q.13: Four products has to be processed through the plants, the quantities required for the next
production period being:
Product1
2000 units;
product 3
3000 units
Product 2
3000 units;
product 4
6000 UNITS
THERE ARE THREE PRODUCTION LINES ON WHICH THE PRODUCT COULD BE Processed.
The rates of the production in units per day and the total available capacity in days are given below.
The cost of the using lines is Rs 600, Rs 500 and Rs 400 per day respectively.
Production
Lines (days)
1
150
2
200
3
160
total
2000
product
1
2
100
500
100
760
80
890
3000
3000
maximum
3
4
400
20
400
20
600
18
6000
line
formulate this problem as an LP model to minimize the cost of operation.
Lp model formulation:
Describe variable Let
Xij=number of units of product I (I=1,2,3,4 ) produced on production line j (j=1,2,3)
The LP model
Minimize (total cost ) Z
4
4
4
i 1
i 1
i 1
Z  600 xi1  500 xi2  400 xi3
Subject to the following constrain
(i) Production constrain
(b)
Line capacity constrain
3
3
j 31
j 31
j 1
j 1
 x1 j  2000;  x2 j  3000
 x3 j  2000;  x4 j  3000
Line capacity constrain s
x11 x 21 x31 x 41



 20
150 200 500 400
x12 x 22 x32 x 42



 20
200 100 760 400
x13 x 23 x33 x 43



 18
160 80 890 600
And
Xij >= 0 for all I and j
Q.14: XYZ company produces an automobile spare part. The contract that it has signed with a large
truck manufacturer calls for the following 4 month shipping schedule
Month
January
February
March
April
No. of parts to be shipped
3000
4000
5000
5000
The company can manufacture 3000 parts of engimes per month on a regular time basis and 2000 parts
per month on overtime basis. Its production cost is 15,000 for a part produced in regular time and
25,000 for a part produced in overtime . its monthly inventory holding cost is 500.
Formulate this problen as an LP model to minimize the overall cost.
LP model formulation: Decision variable
Lwt
Xijk= No. of units of automobiles spare part manufactured in month I (I= 1, 2, 3, 4 ) using
shift j ( j= 1,2) and shipped in month k (k=1,2,3,4)
The LP model
Minimized total cost Z= regular time production basis + overtime production cost + one month
inventory cost + two month inventory cost + three month inventory cost
{ 15000( x111+ x112+ x113+ x114+ x212+ x213+ x214+ x313+ x314+ x414)} + { 25000 ( x121+
x122+x123+x124+x222+x223+x224+x323+x324+x424)} + {
500(x112+x122+x213+x223+x314+x324)} + { 1000(x113+x123+x214+x224)} + 1500(x114+x124)}
Subject to the constrain ts
(a)
monthly regular timwe production constraints
x111+x112+x113+x114 =<3000
x212+x213+x214
=<3000
x313+x314
=<3000
x414
=<3000
(b)
monthly overtime production constrains
x121+x122+x123+x124 =<2000
x222+x223+x224
=<2000
x323+x324
=<2000
x424
=<2000
(c)
And
Monthly demand constraints
X111+x121
X112+x122+x212+x222
X113+x123+x213+x223+x313+x323
X114+x124+x214+x224+x313+x324+x414+x424
Xijk>=0 for all I, j, k
=3000
=4000
=5000
=5000
marketing
Q.15: An advertising company wishes to plan an advertising campaign in three different mediatelevision, radio and magazine. The purpose of the advertising is to reach as many potential customer
as possible. Result of market study are given below:
Television
Cost of advertising
Prime day
prime time
radio
magazine
40,000
75,000,
30,000
15,000
9,00,000
5,00,000
2,00,000
Number of potential customer
Reached per unit
4,00,000
Number of women customers
Reached per unit
3,00,000
4,00,000
2,00,000
1,00,000
The company does not want to spend more than 8,00,000 on advertising. It is further required that
(a)
at least 2 million expouser take place among women
(b)
advertising on TV be limited to Rs. 5,00,000.
(c)
At least 3 advertising units to be bought on prime day and two units during prime
time; and
(d)
The number of advertisement units on radio and magazine should each be between 5
and 10.
LP model formulation : Decision variables Let
X1,x2,x3 and x4= numbers of advertising units bought in prime time in TV, radio, and magazine
respectively
He LP model
Mximize (total potential customer reached) Z= 400000x1+900000x2+500000x3+200000x4
Subject to the constraints
(a)
advertising budget constrain
40000x1 + 75000x2+30000x3+15000x4 =<800000
(b)
number of women customer reached by the advertising compaign constraint
300000x1+400000x2+200000x3+100000x4>=2000000
(c)
TV advertising constraint
40000x1 + 75000x2 =<500000
x1>=3
x2>=2
(d)
radio and magazine advertising constraint
5=<x3=<10
5=<x4=<10
x1,x2,x3,x4>=0
and
Q.16: An advertising agency is preparing an advertising campaign for a group of agencies. These travel
agencies have decided that their target customer should have the following characteristics with
importance (weightage) as given.
Age
Annual income
Female
Characteristics
25-40 years
Above Rs. 60,000
Married
weightage (%)
20
30
50
The agency has made a careful analysis of three media and has complied the following data
Data item
women magazine Radio TV
Reader characteristics
(a)
Age: 25-40
80
70
60
(b)
Annual Income above 2000 60
50
45
(c)
Female/married 40
35
25
Cost per advertisement
9500
25000 100000
Minimum number of advertisement allowed 10
5
5
Maximum number of advertisement allowed 20
10
10
Audience size (1000’s)
750
1000
1500
The budget for launching the advertisement campaign is of 5,00,000. Based on the available data
formulate an LP model for the agency to maximizethe expected effective resources exposures.
LP model formulation: Decision variable Let
X1,x2,x3= number of advertisement media :women magazine, radio and TV respectively.
The effectiveness coeffecient corresponding to each of the advertising media is calculated as follows:
Media
Effectiveness Coefficient
Women’s magazine
0.80(0.20) + 0.60(0.30)+0.40(0.50)=0.54
Radio
0.70(0.20)+0.50(0.30)+0.35(0.50)=0.46
TV
0.60(0.20)+0.35(0.30)+0.25(0.50)=0.38
The coeffecient of the objective function I,e, effective expousre for all the three media employed can
be computeed as follows
Effective expousre = effectiveness X audience size
Where effectiveness coefficiency is weightage average of audience characteristics
Thus the effective exposure of each media is as follows
Women magazine =0.54 X 7,50,000 = 4,05,000
Radio
=0.46X 10, 00, 000 = 4,60,000
TV
=0.38 X 1,50,000 = 5,70,000
The LP model
Maximize (effective exposure ) Z= 4,05,000x1 + 4,60,000x2 + 5,70, 000x3
Subject to the constraints
(a)
budget constraint
9500x1 + 25,000x2 + 1,00,000x3 =< 5,00,000
(b)
Minimum nO. of advertisement allowed constraints
X1>=10; x2>=5; x3>=5
(c)
And
Maximum number of advertisement constraint
X1=<20; x2=<10; x3=<10
x1,x2,x3>=0
Q.17: A businessman is opening a new restaurant and has budget Rs. 5,00,,000 for advertisemnet in the
coming month. He is considered four types of advertisement
(a)
30 second TV commercial
(b)
30 second radio commercial
(c)
half page advertisement in new paper
(d)
full page advertisement in a weekly magazine which will appear four times during
the coming month
the owner wishes is reach families with income of both over and under 50,000.
The number of exposure to families of each type and the cost of each of the media is shown below:
Media
cost of advertisemet
exposure to family with exposure to families with
( Rs)
annual income over 50,000
annual income under 50,000
TV
40,000 2,00,000
3,00,000
Radio 20,000 5,00,000
7,00,000
Newspaper
15,000 3,00,000 1,50,000
Magazine
5,000 1,00,000 1,00,000
To have a balanced campaign the owner has determined the following restriction:
(a)
no more than 4 Tv advertisement
(b)
no more than 60% of all the advertisement in news paper and magazine
(c)
there must be at least 30,00,000 exposure to the families with income over 50,000
(d)
there must be at least 45,00,000 exposure to the families with income under Rs. 50,00
formulate this problem an LP model to determine the number of each type of advertisement to purchase
respectively
LP model formulation:
Decision variable Let
X1,,x2,x3 and x4 = no. of TV ,radio, newpaper and magazine afdvertisement to purchase respectively
The LP model
Maximize (total nomber of exposure of both groups) Z
= (2,00,000+3,00,000)x1 + (5,00,000+7,00,000)x2
+(3,00,00+1,50,000)x3+(1,00,000+1,00,000)x4
=5,00,000x1+12,00,000x2+4,50,000x3+2,00,000x4
Subject to the constraints
(a)
(ii)
(b)
or
(c)
And
Available budget constraint
40,000x1 + 20,000x2 + 15,000x3 + 5,000x4 =<5,00,000
Maximum Tv advertisement constraints
x1=<4
maximum newspaper and magazien constarints
{(x3+x4)/(x1,x2,x3,x4)}=<0.60
-0.6x1-0.6x2+0.4x3+0.4x4=<0
Maximum magazine advertisement constarints
X4=<4 (because magazine wil appear only four times in th next month)
x1,x2,x3 and x4 >= 0
Finance
Q.18: an enginnering comapany is planning to diversify its operation during the year 1996-97. The
company has allocated capital expenditure budget equal to Rs 5.15 crore in th year 1996 and Rs. 6.50
crore in the year 1997. Th ecompany has five investment projects under consideration. The estimated
net returns at present value and expected cash expenditure of each project in the two year are as
follows:
Project estimated net returns
cash expenditure
( in ‘000 Rs.)
In ‘000 Rs.
Year 1998
year 1999
A
240
120
320
B
390
550
594
C
80
118
202
D
150
250
340
E
182
324
474
Assume that the returns from a particular project would be in direct proportion to the investment in it,
so that for example, if in a project say A, 20% (of 120 in 1996 and 320 in 1997) is invested then th
resulting net returns in it would be 20% ( of 240). This assumtion also implies that individuality of the
project should be ignored. Formulate the capital budgeting as an LP model to maximize the net returns.
LP model formulation: decision variable lLet
X1, x2, x3, x4, and x5 = proportion of investment in the project A, B, C, D and
respectively
The LP model
Maximize (net returns ) Z= 240x1+ 390x2 + 80x3+ 150x4 + 182x5
Subject to the constraints
(a)
capital expenditure budget constraints
120x1+550x2+118x3+250x4+324x5=<515 [for year 1996]
320x1+594x2+203x3+340x4+474x5=<650 [for year 1997]
(b)
And
0-1 integar requirment constarints.
X1=< 1; x4=<1
X2=<1; x5=<1
X3=<1
x1,x2,x3,x4,x5>=0
Q.19: XYZ is an invetment company . to add in its investment decision, the company has developed
the investment alternative for 10 year period as a given in the following table. The return on investment
is expressed as an annual rate of return on the invested capital. The risk coefficient and growth
potential are subjective estimates made by the portofoilio manager of the company. The terms aof
investment is the average length of time period required to realized the return on th investyment as
indicated:
Investment
alternative
length
of investment
annual rate)
Risk coeeficient grwoth potential
of return (year )
returns (%)
A
3
0
4
1
B
C
D
E
F
CASH
7
8
6
10
3
0
12
9
20
15
6
0
5
4
8
6
3
0
18
10
32
20
7
0
The objective of the company is to maximize the return on its investment. Thuis guidelines for
selecting the portofolio are:
(i)
the averager length of the investment for the portofolio should not exceed 7 years.
(ii)
the average risk for the portofolio should not exceed 5
(iii)
the average growth potential for the portofolio should be at least 10%
(iv)
at least 10% of all available funds must be retained in the form of cash at all times.
Formulate this problem as an Lp model so as to maximize the return
LP model formulation: decision variable Let
Xj= proportion of funds to be invested in the jth investment alternative (j=1,2……7)
The LP model
Maximized (total return) Z=0.03x1+ 0.12x2+0.09x3+0.20x4+0.15x5+0.06x6+0.00x7
Subject to the constraints
(i)
length of investment constraints
4x1+7x2+8x3+6x4+10x5+5x6+0x7=<7
(ii)
Risk level constarints
X1+5x2+4x3+8x4+6x5+3x6+0x7=<5
(iii)
growth potential constraints
0z1+0.18x2+0.10x3+0.32x4+0.20x5+0.07x6+0x7=<0.10
(iv)
cash requirment constarints
x7>=0.10
(v)
proportion of funds constraints
(vi)
x1+x2+x3+x4+x5+x6+x7=1
and
x1,x2,x3,x4,x5,x6,x7 >=0
Q.20: An invetor has invetment opportunities available at the bergining of each of the next five year,
and also has a total of Rs 5,00,000 vaialable for the invetment vat the beginning of the first year. A
summary of the finincial characteristics of the three invetment alternative is presented in th efollowing
table:
investmen
alternative
Allowable size oft return %
initial investment (Rs)
timing of return
possible
immediate reinvetment
1
2
3
1,00,000
unlimited
50,000
1 years later
2 years later
3 years later
yes
yes
yes
19
16
20
this investor wishes to determine the investment plan that will maximo=ized th amount of money
which can be accumulated by the beginning of 6th year in future. Formulate this problem as an LP
model so as to maximize total return
LP modeule formulation: Decision variables Let
Xij=amount to be invested in investment alternative (I=1,2,3) at the beginning of the year
_j=1,2,…5)
Y1=amount not inveted in any of the investment alternatives in period j
LLP model
Minimize total return Z = 1.19x15+1.16x24+1.20x33+y5
Subject to the constraint
(i)
yearly cash flow constraints
x11+x21+x31+y1=5,00,000 (year I)
-y1-1.19x11+x12+x22+x32+y3=0 (year 2)
-y2-1.16x21-1.19x12+x13+x23+x33+y3=0 (year 3)
-y3-1.20x31-1.16x22-1.19x13+x14+x24+x34+y4=0 (year 4)
-y4-1.20x32-1.16x23-1.19x14 +x15+x25++x35+y5=0 (year 5)
(ii)
size of invetment constraints
x11=< 1,00,000; x31=<50,000
x12=<1,00,000; x32=<50,000
x13=< 1,00,000; x33=<50,000
x14=<1,00,000; x34=<50,000
x15=< 1,00,000; x35=<50,000
and Xij; Yi>= 0 for all I-1,2,3,; j=1,2,…5
remark: to formulate the first set of constraints of yearly cash flow the following situation is adopted:
(invetment alternatives/x12+x22+x32+y2) = (invetment alternatives/ y1+1.19x11
or –y1-1.19x11_x12+x22+x32+y2=0
agriculture
Q.21: A cooperative form owns 100 acres of land and haas 25,000 in funds for invetment the form
member can produce a total of 3500 man hours worth of labour during the months of sep-may and
4000 man hours during the month of june-august. If any of these man hours are not needed some
members of the firm will use them to work on a neighbouring farm for Rs 2 per hour during
september-may and Rs 3 per during june-august. Cash income can be obtained from three main crop
and two type of livestock: dairy cows and laying hens. No invetment funds are needed for this crops.
However each cow will require an invetment outlay of Rs 3200 and each hen Rs 15
Moreever each cow will require 15 acres of land, 100 man hours during the summer. Each
cow will product net annual cash income of 3500 for the farm. The correspondingfigure for each hen
are: no acrage, 0.6 man hours during sep-may, 0.4 man hours during june-augsut and an annual net
cash income of Rs 200. The chicken house canm accomadate a maximum of 4000 hens and the size of
the cattle-shed limits the members to a maximum of 32 cows
Estimatwes man -hours and income per acre planted in each of the three crops are :
Paddy bajra
jowar
Man hours
September – May
40
20
25
June _ August
50
35
40
Net annual cash income 1200
800
850
The cooperative farm wishes to determine how much acrage should be planted in each of the
crops and how many cows and hens should be kept to maximize its net cash income.
LP model formulation: the data of the problem is summarized as follows:
constraints
Crop
extra gpurs
total availability
cows
hens
paddy bajra
jawar sep-may june-aug
Man-hours
Sep_may
100
0.6
40
20
25
1
3500
June –Aug
50
0.4
50
35
40
1
4000
Land
1.5
1
1
1
100
Cows 1
32
Hens
1
4000
Net annual cash
Income ( Rs.)
3500
200
1200
800
850
2
3
Decision variable: let
X1 and x2 = number of dairy cows and laying hens respectively
X3,x4,x5 = average of paddy, bajra and jawar crop
X6= extra man hours utilized in sep-may
X7=extra man hours utilized in june aug.
The LP model
Maximized net cash income Z= 3500x1+200x2+1200x3+800x4+850x5+2x6+3x7
Subject to the constarints
(i)
(ii)
(iii)
and
man jours constaints
100x1+0.6x2+40x3+20x4+25x5+x6=3500 (Sep-may duration)
50x1+0.4x2+50x3+35xy+40x5+x7= 4000 (June-Aug during)
land availability constarints
1.5x1+x3+x4+x5 =<100
livestock constarints
x=<32 (dairy cows)
x2=<4000
(laying hens)
x1,x2,x3,x4,x5,x6,x7>=0
Q.22: A certain farming organisation operates three farms of comparable productivity. The output
of each farm is limited to both bythe usable acrage and by the maount of water available for irrigation.
The data for the upcoming season is shown below:
farm
usable acrage
1
2
3
400
600
300
water available
in acre feet
1500
2000
900
the organisation is considering the crops for planting which differs primarily in their expected profit per
acre and in their consumption of water. Furthermore the total acrage that can be devoted to each of the
crop is limited by the amount of appropriate harvesting equipment available.
Crop
maximize average
A
B
C
700
800
300
5
4
3
water consumption
In acre feet
expected prifit
per acre (Rs)
4000
3000
1000
In order to maintain uniform workload among the farms it is the policy of the organization that the
percentage of the usable acerage planted by the same at each farms. However any combination of the
crops may be growth at any farms . the organisation wishes to how much of each crop should be
planted at the respective farms in order to maximize the expeted profit .
Formulate this problem as an LP model in order to maximize expected total profit.
LP model formulation:
the data of the problem is summarized below:
Crop
3
crop
requirment (In acres)
x12
x22
x32
400
1500
x13
x23
33
600
2000
farms
12
A
x11
B
x21
C
x31
Usable acrahge
Water available
Decision variables:
700
800
300
300
900
expected
proffit?acre
4000
3000
1000
let
Xij= nun=mber of acrage to be allocated to crop I(I=,2,33) to farm j(j=1,2)
The LP model
Maximize (net profit) Z= 4000(x11+x21+x13)+3000(x21+x22+x23)+1000(x31+x32+x33)
Subject to the constarint
(i)
crop requirment nconstarint
x11+x12+x13=<700
x21+x22+x23=<800
x31+x32+x33=<300
(ii)
available acrage constraints
x11+x21+x31=<400
x12+x22+x32=<600
x13+x23+x23=<300
(iii)
(iv)
and
water available constraints
5x11+4x21++3x31
5x12+4x22++3x32
5x13+4x23++3x33
=<1500
=<2000
=<900
social equality constraints
(a)
(x11+x21+x31)/400
=(x12+x22+x32)/600
(b)
(x12+x22+x32)/600
=(x13+x23+x33)/300
(c)
(x13+x23+x33)/300
=(x11+x21+x31)/400
Xij>= 0 for all the I= 1,2,3: and j=1,2,3,
transportation
Q.23: ABC manudfacturing company wishes to develop a monthly production schedule for the
next thnree months. Depending upon the salews commitment the company can either keep the
production constant allowing fluctuation in inventory or inventories can be maintained at a constant
level with fluctuating production. Fluctuating production made overtime work necessary the cost of
which is estimated to be the doubble the normal production csot of Rs 12 per unit. Fluctuating
inventories result in an inventory carrying cost of Rs 2 per unit/month if the company fails to fulfil its
sales cimmitent it incurs a sales shortage of Rs 4 per unit/month. The production capacity for the next
three month are shown below:
Production capacity
sales
Month regular overtime sales
1
50
30
60
2
50
0
120
3
60
50
140
formulate this problem as an LP model so as to minimized the total cost.
LP model formulation: the data of the problem is summarized as follows.
Production capacity
sales
Month regular overtime
1
50
30
60
2
50
0
120
3
60
50
40
normal prod. Cost
Re 12 per unit
overtime cost
Re 24 per unit
inventory carrying cost
Re 2 per unit per month
shortage cost
Re 4 per unit per month
assume five sources of supply: three regular and two overtime (because second month overtime
production is zero) production capacity. The demand for thje three month will be the sales during these
months.
All supplies against order have to be made and can be made in subsequent month if not possible,
during the month of order with additional cost equivalent to the shortage cost I.e. in month 2. The
culumative production of month 1 and 2 in regullar and overtime is 130 units while the order are for
180 units. This balance can be supplied during month 3 at an additional production cost of Rs. 4.
The information given can now be presented in matrix form as follows:
M1
M2
M3
M1(OT)
M2(OT)
M1
12
16
20
24
32
m2
14
12
16
26
28
m3
16
14
12
28
24
production (supply)
50
50
60
30
Sales demand
60
120
140
Decision variable : Let
Xij = amount of commodity sent from source of supply I(1,2,…,5) to destination j(=1,2,3)
The LP model
Minimize (total cost) Z= 12x11 + 14x12 + 16x13 + 16x21 + 12x22 + 12x22 + 14x23 + 20x31 + 16x32
+ 12xx33 + 24x41+ 26x42 + 28x43 + 32x51 + 28x52 + 24x53
Subject to the constarints
(i)
production supply constarints
x11+x12+x13= 50
x21+x22+x23= 50
x31+x32+x33= 60
x41+x42+x43= 30
x51+x52+x53= 30
(ii)
sales demabd constraints
x11+x21+x31+x41+x51= 60
x12+x22+x32+x42+x52= 120
x13+x23+x33+x43+x53= 40
xij>= 0 for all I, j
and
personal
Q.24: evening shift resident doctor in a government hospital work five consequive day and have two
consecutive days off. Their five days of work can start on any day of the week and schedule rotates
indefinetely. The hospital require the following minimum number of doctors working:
sun
mon
tues
wed
thurs
fri
sat
35
55
60
50
60
50
45
nomore than 40 doctors can start their five working days on the same day. Formulate a general linear
programming model to minimize the number of doctors employed by the hospital.
LP model formulation : decision variable let
Xj=number of doctor who start their duty on day j of the week.
The LP model
Minimize total number of doctor Z=x1 + x2 + x3 + x4 + x5 + x6 + x7
Subject to the constarints
AND
x1 + x4 + x5 + x6 +x7>=35
x2 + x5 + x6 + x7 +x1>=55
x3 + x6 + x7 + x1 +x2>=60
x4 + x7 + x1 + x2 +x3>=50
x5 + x1 + x2 + x3 +x4>=60
x6 + x2 + x3 + x4 +x5>=50
x7 + x3 + x4 + x5 +x6>=45
xj>= 0 for all j
Q.25: A machine tools company conduct a job training programme for machinist. Trained machinist
are used as a teacher in the programme at the ratio of one for every ten trainees. The training
programme lasts for one month. From past experience it has been forund that out of ten trainee hired,
only seven complete the programe sucessfully and t he rst are released.
Trained machinist are also needed for machinig and company’s requirement for the next three
months are as follows: January 100, February 150, March 200. In addition the company requires 250
machinist by april. There are 130 trained machinist available at the beginning of the year. Payro;; per
month are:
Each tarinee
: Rs 1400
Each trained mechinist
(machining and teaching)
: Rs. 1900
each trained mechinist
( idle)
1700
formulate this problem as an LP model so as to optimize the cost of hiring and training schedule and
the compan’r requirments.
LP model formulation:
decision variable let
X1 ,x2 = trained machinist teaching and idle in January respectively
X3, x4= trained machinist teaching idle in February respectively
X5, x6= trained machinist teaching idle in march respectively
The LP model
Minimize (total cost) Z=cost of training program (teacher and trainees) + cost of idle
mechanist + cost of mechinist doing machine work (constant)
= 1400 ( 10x1+10x3+ 10x5) + 1900 ( x1+x3+x5) +1700(x2+x4+x6)
subject to constraints
(i)
total number of trained machinist available at the beginning of january
=number of machinist doing machining + teaching + Idle
130= 100 + x1 +x2
x1 + x2=30
(ii)
total trained machinist available at the beginning of february
=number of mechinist in january + joining after training program
130+7x1= 150 +x3+x4
7x1-x3-x4=20
In january there are 10x1 trainees in the program and out of those only 7 x1 will become trained
machinist
(iii)
total trained machinist available at the beginning of march
=number of machinist in Jan. + Joining after training programme in Jan And Feb.
130 + 7x1+ 7x3 = 200 + x5 + x6
7x1+ 7x3-x5-x6=70
(iv)
company requires 250 trained mechanist by april
130 + 7x1 + 7x3 + 7x5=250
7x1 + 7x3 + 7x5=120
x1, x2, x3 x4, x5, x6 >=0
and
Q.26: The super bazar in a city daily needs between 22 and 30 worker depending on the time of the
day. The rusg hours are between noo and 2 PM. The table below indicates the number workers needed
at various hours that the bazar is open.
Time period
no. of worker NEEDED
9AM-11AM
11AM-1PM
1PM-3PM
3PM-5PM
22
30
25
23
the super bazar now employes 24 full time worker but need a few part time worker also. A part time
worket must put in exactly 4 hours per day, but can start any time between 9 AM and 1PM full time
worker work from 9 AM to 5 PM but are allowed a hour for lunch ( half of the full timers eat at 12 AM
the other half at 1 PM) full timers thus provided 35 hours per week of productive labor time.
The management of the superbazar limits part time hours to a maximum of 50% of the day’s
total requirement .
Part timers earn 28 per day on the average, while full timers earn 90 per day in salary and
benefits on the average . the management want to set a schedule that would minimize total manpower
costs.
Formulate this problem as an LP model so as to minimize total manpower daily cost.
LP model formualtion: decision variables Let
Y= full time worker
Xj=part timer worker starting at 9AM, 11AM, and 1PM respectively (j=1,2,3)
The LP model
Minimize (total daily manpower cost) Z= 90(y+28(x1+x2+X3)
SUBJECT TO CONSTRAINTS
Y+x1>=22
[ 9AM-11AM need]
(1/2)y+x1+x2>=30
[11AM-1PM need]
(1/2)y+x2+x3>=25
[1PM-3PM need]
y+x3>=22
[ 3PM-5PM need]
y =<24 [full time available]
4(x1+x2+x3)=<0.50(22+30+25+23)
[ part timers time cannot exceed 505 of total hours required each day]
and
y, Xj>=0 for all j
Problem on production
Q.27 A company sells two differenet product A and B, making a profit of Rs 40 and Rs 30
respectively per unit on them . they ar eproduced in a common production process and are sold in two
different markets. The production process has a total capacity of 30,000 man hours. It takes three hours
to produce a unit if A and one hour to produce a unit of B. the market has been surveyed and
company’s official feels that the maximum number of units of A that can be sold is 8,000 units and
that of B isd 12,00 units. Subject to these limitations, products can be sold in any combitations.
Formulate this problem as a Linear programming model.
Q.28 The manger of an oil refinery must decided on the optimal mix of two possible blending
processes of which the input and output per production run are given as follows:
Process
Inputs (unit)
Output
(units) Crude A
Crude B Gasoline X
Gasoline Y
1
5
3
5
8
2
4
5
4
4
the maximum amount available of crude A and B are 200 units and 150 units respectively. Market
requirement shown that at least 100 units of gasoline X and 80 units of gasoline Y must be produced.
The profit per production run from process 1 and process 2 are Rs 300 and Rs 500 respectively.
Formulate this problem as a linear programming model
Q.29 A firm places an order for a particular product at the beginning of each month and the product is
received at the end of the month. The firm sells during the month from the stock and it c an sell any
quantity.
The prices at which the firm buys and sell vary every month. The following table
shows the projected buying and selling prices for the next four months:
Month
Apri
May
June
July
selling price
(during the months)
90
60
75
purchasing price
(beginning of the month)
75
75
60
-
The firm has no stock on hand as on April 1, and does not wish to have any stock at the end of
july. The firm has a warehouse of limited size, which can hold a maximum of 150 units of the
product.
The problem is to determine the number opf units to buy and sell each month to maximize the
profit from its operation. Formulate yhis problem as an linear programming problem
Q.30 A manufacturer produces three model (I, II, III) of a certain product product. He uses two type
of raw material (A and B) of which 4000 and 600 units respectively are available. The raw material
requirement per unit of the three model are as follows:
Raw material
A
B
requirement per unit of given model
I
II
III
2
3
5
4
2
7
The labor time of each unit of model I is twice that of model II and three times that of model
III. The entire labor force of the factory can produce the equivalent of 2500 units of model I. A market
surveys indicate that the minimum demand of the three model are : 500, 500 and 375 units respectively.
However the ratio of the numbers of units produced must be equal to 3: 2: 5. Assume that the profit
per units of models I, II,III is Rs 60, 40 and 1000 respectively.
Formulate this problem as an Lp model in order to determine the number of units of each
product which will maximize profit.
Q.31 consider a small plant which makes two type of automobiles part say A and B. It buys casting
that are mechined, bored and polished. The capacity of machine is 25 hour for A and 24 per hour for
B. capacity of boring is 28 per hours for A and 35 per hour for B and the capacity of polishing is 35
per hour for A and 25 per hour for B. casting for part A cost Rs 2 and sells for Rs. 5 each and those of
for part B cost is Rs. 3 and sell for Rs 6 each. The three machiness have running cost of Rs. 20, Rs 10
and Rs. 17.50 per hour. Assume that any combination of part A and B can be sold, formulate this
problemas an LP model to determine the product mix which maximize profit.
Q.32 On october 1, a company received a contract to supply 6,000 units of specialized product. The
terms of contract require that 1,000 units be shipped in october, 3,000 units in november and 2,000
units in december. The company can produce 15,000 units per month on regular time and 750 units per
month on overtime. The manufactureing cost per item produced during regular time is Rs. 3 and the
cost per item during overtime is Rs 45. The monthly storage cost is Rs. 1. Formulate this problem as
an LP model so as to get optimum schedule.
Q.33 A wine maker has stock of three different wineswith the following characteristics:
Wine
A
B
C
Proofs
27
33
32
Acids %
0.32
0.20
0.30
specific gravity
1.70
1.08
1.04
stock(gallons)
20
34
22
A good dry table wine should be between 30 and 31 degree proof, it should contain at least
25% acid and should have specific gravity of at least 1.06. the wine maker wishes to blend
the three types of wine to produce a as large a quantity as possible of a satisfactory dry table
wine. However his stock od wine A must be completely used in the blend because further
storage would cost it to deteriorate. What quantity of wine B and C should be used. Formulate
this problem as an LP model.
Q.34 the XYZ company assembles and marketing two types of transistor radio, A and B. presently
200 radios of each type are manufactured per week. You are advised to formulate the production
schedule which will maximize the profits in the kight of the following information::
Type
Total componet
cost per radio (Rs)
Man hours of
assembly time
per radio
A
B
200
160
12
6
Average man-minute
of inspection and
correction time per
radio
10
35
Selling price per
radio (Rs)
400
280
The company employs 100 assembler who are paid Rs. 10 per hour actually worked and who will
work upto a maximum of 48 hours per week. The inspectors who are presently four,have agreed to
aplan whereby they average 40 hours per week each. However, the four inspector have certain other
administrative duties which have been found to take up an average of 8 hours per week.
Each radio of either type requires one speaker, the type being the same for each radio. However,the
company can obtain a maximum supply of 600 in any one week.Their cost has been included in the
components cost given for each radio in the table above.Only speaker actually used need paid for. The
only other cost incurred by the company are fixed overheads of Rs.20,000 per week.
Q.35 ABC foods company is developing a low calorie hihg protein diet supplement called Hi=Pro.
The specification for Hi-pro have been established by a panel of medical expert. These specification
along with the calorie , protein and vitamin content of three basic food are given in the following table:
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shamshad