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 : {1 2 3 4 5
No. of cans {280 298 305 360 400
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 weeks : 1 2 3 4 5
Number: 280 298 305 360 400
(2) Each trainee has to undergo a two week training.
(3) One employee is required to to train three trainee.
(4) Every trained worker producing one can/week but no production from trainers and trainees during training.
(5) Numbers of employee to be employed=150
(6) The production in any week not to exceed the cans required.
(7) Numbers of weeks for which newcomers would be employeed: 5, 4, 3, 2, 1
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
Grade 1 grade 2
Number of inspectors 9 11
Rate of checking 40 pieces/hour 30 pieces/hour
Inaccracy in checking 1-0.97=0.03 1-0.95=0.05
Cost of inacuracy in checking Rs. 3/piece Rs. 3/piece
Wage rate/hour Rs 5 Rs.5
Total pieces which must be inspected=2000
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) Total number of pieces that must be inspected in an 8 hour day constraints
40 X 8x1 + 30 X 8x2 >= 2000
(b) 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 12 0.8
B 20 1.7
C 45 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 / product type total
Constraints A B C
Production capacity (units) 50 25 30
Man hours per unit 0.8 1.7 2.5 100
Order commitment unit 20 15
Profit contribution (Rs./unit) 12 20 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 availibility (hours)
Preparation times (hrs) plant 1: 3 hrs/1000 gallons 1hrs/1 quintal 16
Plant 2: 2hrs/1000 gallons 1.5 hrs/quintal 16
Minimum daily production 10,000 gallons 8 quintal
Cost production plant 1: 15000 28000
Plant 2: 18000 26000
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) minimum daily production requirment constaints
x1+x3=>10
x2+x4=>8
and 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/ components
constraints c1 c2 total availibility
Budget (Rs) 10/units 40/units Rs 4000
Machine time 3 Hrs/unit 2 hrs/unit 2000 hrs
Assembly time 2 hrs/unit 3 hrs/unit 1400 hours
Selling price Rs 30 Rs 70
Cost price Rs 10 Rs 40
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 Part B Part C
Drilling 25 40 25
Shaping 25 20 20
Polishing 40 30 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 ( 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
and 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 / model Total availability (hrs)