Assignment II

Linear Programming Problems

Simplex Method

Q1.    A company manufactures 3 type of parts which use precious platinum and gold. Due to shortage of these precious metals, the govt. regulates the amount that may be used per day. The relevant data with respect to supply requirement and profit are summarized in the table shown below:

          Product      Platinum              Gold required      Profit

          A                 2                           3                           500

          B                 4                           2                           600                                         

          C                 6                           4                           1200

          Daily allotments of platinum & gold are 160 gm and 120 gm. How should the company divide the supply of scare precious metals? What is the optimum profit?

Ans:  Maximize (Z) = 500x₁+600x₂+1200x₃         

Subject to constraints  2x₁+4x₂+6x₃≤160, 3x₁+2x₂+4x₃≤120,

Non Negative Restrictions x₁, x_2, x₃≥0

Z=32800 X₁=8, X₂=0, X₃=24

Q2     A manufacture has two products A & B, both of which are made in steps by machine 1 & machine 2. The process time per hundred for two products on two machines are (set up times are negligible)

          Product                Machine I                      Machine II

          A                           4 HOURS                       5 HOURS

          B                          5 HOURS                       2 HOURS

          For the coming period, machine 1 has 100 hours and machine 2 has available 80 hours. The contributors for product A is Rs. 10 per 100 units and for Product B is Rs. 5 per 100 units. The manufacture in market, which can absorb both products as much he can produce for immediate period ahead. Determine how much of product A and B he should produce to maximize his contribution.

Ans:  Maximize (Z) = 10x₁+5x₂            

Subject to constraints  4x₁+5x₂≤100, 5x₁+2x₂≤80,

Non Negative Restrictions x₁, x_2, ≥0

Z=2900/17, X₁=200/17, X₂=180/17,

Q3     A firm produces three products A B & C, each of which passes through three department: Fabrication, finishing & Packaging. Each unit of product A requires 3,4 & 2 hours; a unit of b requires 5,4 & 4 hours while each unit of product C requires 2,4 and 5 hours respectively in the three department. Every day 60 hours are available in the fabrication department, 72 hours in the finishing department and 100 hours in the packaging department. If the unit contribution of product A is Rs 5, of Product b is Rs 10, and of product C is Rs. 8, determine the number of units of each of the products, that should be made each day to maximize the total contribution. Also determine if any capacity would remain un utilities

Ans:  Maximize (Z) = 5x₁+10x₂+8x₃               

Subject to constraints  3x₁+5x₂+2x₃≤60, 4x₁+4x₂+4x₃≤72, 2x₁+4x₂+5x₃≤100

Non Negative Restrictions x₁, x_2, x₃≥0 Z=160 X₁=0, X₂=8, X₃=10

Q4.    A farmer has 1000 acres of land on which he can grow corn, wheat or soyabean. Each acre of corn costs Rs 100 for preparing requires 7 man-days of work an yield a profit of Rs. 40. An acre of soabean costs Rs. 70 to prepare, require 8 man-days of work and yields a profit of Rs. 20. An acre of wheat cost Rs 120 to prepare require 10 man days of work and yield a profit of Rs. 40. If the farmer has Rs. 1,00,000 for preparation and can count on 8000 man days of work, how many acres should allocated to each crop to maximize profits.

Ans:  Maximize (Z) = 30x₁+40x₂+20x₃        

Subject to constraints  10x₁+12x₂+7x₃≤1,00,000 7x₁+10x₂+8x₃≤8000,

x₁+x₂+x₃≤1000

Non Negative Restrictions x₁, x_2, x₃≥0

Z=32500 X₁=250, X₂=625, X₃=0

Q5.    A manufacture of leather belts makes three types of belts A B C which are processed on three machines m1, m2 and m3. Belt a requires 2 hours on machine m1 and3 hours on machine m3. belt b requires 3 hours on machine m1, 2 hours on machine m2, and 2 hours on machine m3. There are 8 hour of time per day available on machine m1, 10 hours of time per day available on machine m2 and 15 hours of time per day available on machine m3. The profit gained from belt A is Rs. 3.00 per unit, from belt b is Rs. 5.00 per unit, from belt C is Rs 4.00 per unit, what should be the daily production of each type of belts so that the profit is maximum.

Ans:  Maximize (Z) = 3x₁+5x₂+4x₃   

Subject to constraints  2x₁+3x₂≤8, 2x₂+5x₃≤10, 3x₁+2x₂+4x₃≤15

Non Negative Restrictions x₁, x_2, x₃≥0

Z=765 X₁=89/41, X₂=50/41, X₃=62/41

Q6.    XYZ firm manufactures and sells two products Alpha & Beta. Each unit of Alpha requires 1 hours of machining and 2 hours of skilled labour, whereas each unit of Beta uses 2 hours of machining and 1 hours of labour. For the coming month the machine capacity is limited to 720 machine hours and skilled labour is limited to 780 hours. Not more than 320 unit of Alpha can be sole in the market during a month. Develop a suitable model that will enable determination of the optimal product mix and determine the optimal product mix and the maximum contribution. Unit contribution from Alpha is Rs. 6 and from Beta is Rs. 4.

Ans:  Maximize (Z) = 6x₁+4x₂

Subject to constraints  x₁+2₂≤720, 2x₁+x₂≤780, x₁≤320

Non Negative Restrictions x₁, x_2, ≥0

Z=2560 X₁=280, X₂=220

Q7.    Following data are available for a firm which manufactures three items A B & C

          Product                                    Time required                          Profit

                                                Assembly              Finishing                                                    a                             10                         02                         800

                   B                          04                         05                         600                                c                             05                         04                         300                      

          Capacity                         2000                     1009 

Express the above data in the form of LPP to maximize the profit from its production and solve it by simplex method.

Ans:  Maximize (Z) = 800x₁+600x₂+300x₃             

Subject to constraints  10x₁+4x₂+5x₃≤2000, 2x₁+5x₂+4x₃≤1009,

Non Negative Restrictions x₁, x_2, x₃≥0

Z=200600 X₁=142, X₂=145, X₃=0

Q8.    A company makes two kinds of leather belts. Belt a is a high quality belt and belt b is of lower quality. The respective profits are Rs 4 & 3 per belt. The production of each of type A requires twice as much time as a belt of type B, and if all belts were to type B, the company could make 1000 per day. The supply of leather is sufficient for only 800 belts per day(both A and B Combined). Belt A requires a fancy buckle and only 400 per day are available. There are only 700 buckles a day available for belt B. Formulate this problem as an LP Model and solve it by simplex Method.

Ans:  Maximize (Z) = 4x₁+3x₂               

Subject to constraints 2x₁+x₂≤1000, x₁+x₂≤800, x₁≤400, x₁≤700

Non Negative Restrictions x₁, x_2, ≥0

Z=2600 X₁=200, X₂=600

Q9.    A Furniture manufacturing company plans to make two products chairs and tables from its available resources which consist of 400 board feet of mahogany timber and 450 man hours of labour. It knows that to make chair requires 5 board feet and 10 man hours and yields a profit of Rs. 45 while each table uses 20 board feet and 15 man-hours and has a profit of Rs. 80. How many of each of the product, the manufacturer should make in order to maximize his profit.

Ans:  Maximize (Z) = 45x₁+80x₂        

Subject to constraints 5x₁+20x₂≤400, 10x₁+15x₂≤450, x₁≤400, x₁≤700

Non Negative Restrictions x₁, x_2, ≥0

Z=2200 X₁=24, X₂=14