We have briefly discussed the meaning of models, various types of models; we are particularly more interested in the mathematical models.
Linear Programming
– Problem formulation [Mathematical Modeling]
We have briefly discussed the meaning of models, various types of
models; we are particularly more interested in the mathematical models. Let us
consider the following situation pertaining to a furniture manufacturing firm.
A furniture company
manufactures desks and chairs. There are four departments namely carpentry,
upholstery, painting and varnishing with capacities as given below:
Assuming
that raw materials are available in adequate quantities and the manufacturer
wishes to know how many desks and chairs he should produce. He enjoys a good
market share. The contribution from a desk is Rs.40/- and that from a chair is Rs.25/- Let us develop the mathematical model of the given
situation as follows: Let us say, we want to produce X1 units of Desks
and X2 Units of Chairs. By producing one unit of desk, the manufacturer
gets Rs. 40/- as profit and for chairs, he gets Rs. 25/- Therefore, the profits from desks | = 40 * X1 |
Profits from chairs | = 25 * X2 |
Therefore, the total profits = 40X1 + 25X2 Obviously, the objective of the firm, therefore, is
Maximize the total profits, which is expressed as a mathematical expression
(function) as Max Z = 40X1 + 25X2 However, in achieving the objective, the firm faces
several constraints. The first constraint is from the carpentry section; to
produce, one unit of desk, it needs 4 hours from this section and for producing
chairs, and this section needs to spend 2 hours. However, this section, in
total, available, only for 120 hours in the week. Thus, the first constraint is formulated as
follows: 4X1 + 2X2 <= 120 …………………..
(1) Upholstery section’s service is needed only for
producing chairs and to produce one unit of chair, this section has to spend 3
hours and its maximum availability is 240 hours per week. Thus, the second constraint
is derived as follows: 3X2 <= 240 …………………..
(2) Similarly, the painting section’s availability per
week is limited to 90 hours. To produce a desk or chair, this section has to
spend 2 hours for each unit. Hence, the constraint for painting section is, 2X1 + 2X2 <= 90 …………………..
(3) Finally, the varnishing section’s services are
needed for both the products, and it is limited to 100 hours per week. To
produce a desk, it needs to spend 1 hour and to produce a chair; it has to
spend 2 hours. Thus, the constraint for varnishing section is, X1 + 2X2 <= 100 …………………..
(4) Obviously, the production quantities like number of
desks (X1) and number of chairs (X2) cannot be negative; we add two more
constraints to this situation. They are X1 ≥ 0 and X2 ≥0 Therefore, the mathematical formulation of the
given situation is Max Z = 40X1 + 25X2 Subject to 4X1 + 2X2
<= 120 3X2 <=
240 2X1 + 2X2
<= 90
X1 + 2X2 <= 100 X1 ≥ 0 and X2 ≥0 Therefore by reading, Max Z = 40X1 + 25X2 Subject
to 4X1 + 2X2
<= 120 3X2 <=
240 2X1 + 2X2
<= 90 X1 + 2X2
<= 100 X1 ≥ 0 and X2 ≥0 The
reader can understood that the firm has to decide the quantities to be produced
in desks and chairs, so as to make the maximize profit / contribution and
achieving this objective is subject to 4 constrains – availability of raw
materials from its 4 departments. Thus, the mathematical model summarizes the
information provided in the context / situations in terms of mathematical
symbols and notations. We will take few more examples. Example1:
Distillers
Production Schedule M/S. RK Distillers Ltd (RKDL) has two bottling plants, one located at Pondicherry and the
other at Chennai. Each plant produces three brands of liquor products,
Challenge, Royal and Salute under the job order contract to the leading liquor
baron MB Distillers and Bottlers Ltd (MBDL).
The number of cases produced per day is as follows:
As per the sales forecast given by the marketing team, MBDL expects a
minimum of 30000 cases of Challenge, 40000 cases of Royal and 44000 cases of
Salute from RKDL for the next fortnight. The operating costs per day for plants
at Chennai and Pondicherry are 600 and 450 thousands per day. How many days
each plant should run to fulfill the orders for the next fortnight? [You can
assume that the factory runs all the 7 days in a week-since there is a shift
system to take care of weekly off] Solution Let X1 be the number of days Chennai plant to be operated and X2 be the
number of days Pondicherry plant should run. The firm aims to reduce the overall operating cost arising out
operations subject to fulfilling the market demand. Therefore, the objective is to minimize the total operating cost; to
operate the Chennai plant for X1 days, the firm has to incur 600X1+ 450X2 Min Z = 600X1 + 450X2 Subject to 1500X1 + 1500X2 >= 30,000 3000X1 + 1000X2 >= 40,000 2000X1 + 5000X2 >= 44000 Where, X1, X2 >= 0 Example2: Product
Mix Decision A company producing a
standard line and a deluxe line of dish washers has the following time
requirements (in minutes) in departments where either model can be processed.
The standard models contribute Rs. 20 each and the
deluxe Rs. 30 each to profits. Because the company produces other items that
share resources used to make the dishwashers, the stamping machine is available
only 30 minutes per hour, on average. The motor installation production line
has 60 minutes available each hour. There are two lines for wiring, so the time
availability is 90 minutes per hour. Write the formulation for this linear program Solution Let X =
number of standard dishwashers produced per hour Y = number of deluxe
dishwashers produced per hour Therefore, the objective is to maximize the total
contribution from these two products.
Thus, it is written as Max Z= 20X + 30 Y This
contribution realization is subject to the following constraints; 3X + 6Y ≤ 30 -------------- | (Stamping
Machine constraint) |
10X + 10Y ≤ 60
-------------- | (Motor
installation constraint) |
10X + 15Y ≤ 90
-------------- | (Wiring
machine constraint) |
And
obviously, X & Y cannot be negative quantities, hence, X>=0 &
Y>=0 Example3: Product Mix Decision @ Whoppy
Soft Drinks The production manager for the Whoppy soft drink
company is considering the production of 2 kinds of soft drinks: regular (R)
and diet (D). The company operates one “8 hour” shift per day. Therefore, the
production time is 480 minutes per day. During the production process, one of
the main ingredients, syrup is limited to maximum production capacity of 675
gallons per day. Production of a regular case requires 2 minutes and 5 gallons
of syrup, while production of a diet case needs 4 minutes and 3 gallons of
syrup. Profits for regular soft drink are Rs.3.00 per case and profits for diet
soft drink are Rs.2.00 per case. Write the formulation for this linear program. Solution Let R =
number of regular drinks produced per days D = number of diet drinks produced
per days Therefore, the objective is to
maximize the total contribution from these two products. Thus, it is written as Max Z= 3R + 2D This
contribution realization is subject to the following constraints; 2R + 4D ≤
480 -------------- (Production time constraint) 5R + 3D ≤
675 -------------- (Syrup availability constraint) And obviously, R & D cannot be negative
quantities, hence, R>=0 & D>=0 Example4:
Sales Mix
Decision for computer retail sales A computer retail store sells two types of flat
screen monitors: 17 inches and 19 inches, with a profit contribution of Rs. 300
and Rs. 250, respectively. The monitors are ordered each week from an outside
supplier. As an added feature, the retail store installs on each monitor a
privacy filter that narrows the viewing angle so that only persons sitting
directly in front of the monitor are able to see on-screen data. Each 19”
monitor consumes about 30 minutes of installation time, while each 17” monitor
requires about 10 minutes of installation time. The retail store has
approximately 40 hours of labor time available each week. The total combined
demand for both monitors is at least 40 monitors each week. How many units of
each monitor should the retail store order each week to maximize its weekly
profits and meet its weekly demand?
Solution Let X1 = number of 17 inches monitor
to be ordered per week X2 = number of 19 inches monitor
to be ordered per week Therefore, the objective of the retail service firm
is to maximize the total contribution from these computer monitor sales. The
retail firm gets Rs. 300 and Rs. 250 per monitor for 17 and 19 inches
respectively, therefore, the objective function arrived as follows: Max Z= 300 X1 + 250 X2 This
contribution realization is subject to the following constraints; 10X1 + 30X2 ≤ 2400 | -------------- (Labor time constraint) |
X1 + X2 ≥ 40 -------------- | (Market demand for the computer |
monitor per week constraint) And
obviously, X1 & X2 cannot be negative quantities, hence, X1>=0 &
X2>=0 Exercises [Try on
your own] 1. A
furniture store produces beds and desks for college students. The production
process requires assembly and painting. Each bed requires 6 hours of assembly
and 4 hours of painting. Each desk requires 4 hours of assembly and 8 hours of
painting. There are 40 hours of assembly time and 45 hours of painting time
available each week. Each bed generates $35 of profit and each desk generates
$45 of profit. As a result of a labor strike, the furniture store is limited
to producing at most 8 beds each week. Formulate the situation as a linear
programming problem, which can determine number beds and desks should be
produced each week to maximize weekly profits. 2. A bank is
attempting to determine where its assets should be allocated in order to
maximize its annual return. At present, $750,000 is available for investment in
three types of mutual funds: A, B, and C. The annual rate of return on each
type of fund is as follows: fund A, 15%; fund B, 12%; fund C; 13%. The bank’s
manager has placed the following restrictions on the bank’s portfolio: 1. No more
than 20% of the total amount invested may be in fund A 2. The
amount invested in fund B cannot exceed the amount invested in fund C Determine the optimal
allocation that maximizes the bank’s annual return.
Tags : Operations Management - Introduction to Operations Research
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