Problems on Linear Programming

Problem 1: Rahim Factory manufactures two articles- share and towel. To manufacture share a certain machine has to be worked for 1.5 hours and in addition a craftsman has to work for two hours. To manufacture the towel the machine has to work for 2.5 hours, in addition the craftsman has to work for 1.5 hours. In a week, the factory can avail 80 hours of machine and 70 hours of craftsman. The profit on each share is Tk 50 that on each towel is TK 40.

If all the articles produced can be sold away, find how many of each kind i.e., share and towel should be produced to earn maximum profit per week. Formulate the model using linear programming model and solve it.

Problem 2: Mrs Saleha has learnt from a nutrition book that her family needs at least 330 gm of protein and 45 mg of iron per day. These nutrients can be obtained from meat and vegetables. Each pound of meat costs on an average of $1.6 and contains average of 150 gm. and 15 mg of iron, while each pound of vegetable costs 50 cents ($1/2) and has 10 gm. of protein and 5 mg of iron. Mrs. Saleha wants to determine the quantities of food that meet the nutritional requirements at least costs.

Mathematical Problem (Linear Programming)

Problem:

Zenith Inc. manufactures two types of kitchen utensils:

1.      Knives.

2.      Forks.

Both must be pressed and polished. The shop manager estimates that there will be a maximum 70 hours available next week in pressing machine center and 100 hours in polishing machine center. However each case of knives requires an estimated 12 minutes (.20 hour) of pressing and 30 minutes (.50 hour) of polishing while in case of forks requires 24 minutes (.40hour) of pressing and 15 minutes (.25 hour) of polishing. The company can sell as many knives as it produces at the prevailing market price of Tk 12 per case. Forks can be sold for Tk 9 per case. Cost of production per case knives is Tk 4 and forks Tk 3.

Zenith wants to determine how many cases of knives and forks the company should produce to maximize profit.

Solution (Equation type expression):

Zenith’s problem is to determine the quantity of knives and forks that will maximize profit and selected quantities cannot be used more than available pressing and polishing time.

Objective: To maximize profit.

Zenith’s total profit = The contribution from knives + the contribution from forks.

Since,

Each case of knives can be sold @ Tk 12

Each case of knives’ production cost@ Tk 4

So, contribution from each case of knives = Tk (12-4) = Tk 8

Since,

Each case of forks can be sold @ Tk 9

Each case of fork’s production cost@ Tk 3

So, contribution from each case of forks = Tk (9-3) = Tk 6

This Tk 8 and Tk 6 per case knives and per case forks contribution multiplied by number of cases gives total profit from knives and forks respectively.

By letting,

X= Number of knives zenith will produce in next week.

Y= Number of forks zenith will produce in next week.

Z= Zenith’s want of total profit.

Now we can represent Zenith’s total profit, 
                     Z= 8X + 6Y ------------- (I)

The company wants to choose the level of decision variables (X and Y) that maximizes total profit (Z). The objective can be expressed as,
                       Maximize Z = 8X + 6Y ---------(II)

Restrictions:

Available pressing and polishing capacity will limit knives and forks Zenith can produce. Since each case of knives uses .20 hour of pressing time i.e., .20X is the total time required to press X cases knives, similarly .40Y is the total time required to press Y cases of forks.

Consequently, .20X + .40Y gives the total time required to press X cases of knives and Y cases of forks.

Zenith can select any product combination doesn’t require more than 70 hours available pressing time. The mathematical representation of this condition will be-
                     .20X + .40Y ≤ 70 (hour) ----------- (III)

Another system constraint deals with polishing operation management of the business/ company knows that each case of knives uses .50 hour and each case of forks uses .25 hour polishing hour.

Since, there are only 100 hours of polishing time. So we can represent the above description as following:
                           .50X + .20Y ≤ 100 (hour) ------------ (IV)

It is physically impossible for zenith to produce negative number of knives and forks. Therefore, management must ensure that decision variables X and Y have values greater than or equal to 0 (zero).

                           Symbolically, X ≥ 0 and Y ≥ 0
                           In abbreviated form X, Y ≥ 0 ------------ (V)

Complete Formulation:

By collecting objective function (II), system constraints (III) and (IV) and non-negativity condition (V), zenith’s management can be represented the machine shop problem with the following mathematical equation/ function:

Maximize, Z = 8X + 6Y
Subject to,
.20X + .40Y ≤ 70 (System Constraints)
.50X + .25Y ≤ 100 (System Constraints)
 X, Y ≥ 0 (Non-negative function)

Solution (Numerical Expression):

Let us, first consider the inequality into equation we have, 
.20X + .40Y = 70 --------- (I)

And .50X + .25Y = 100 -------- (II)

For the equation number (I), when X=0 then

.20 x 0 + .40Y = 70

Or, Y = 175             [X, Y= 0, 175]

When Y=0 then

.20X   + .40 x 0 = 70

Or, X = 350            [X, Y= 350, 0]

For the equation number (II), when X=0 then

.50 x 0 + .25Y = 100

Or, Y = 400             [X, Y= 0, 400]

When Y=0 then

.50X   + .25 x 0 = 70

Or, X = 200            [X, Y= 200, 0]

Now equation number (I) multiplied by 5 and equation number (II) multiplied by 2. Then deduct (II) from (I), we get-

X + 2Y = 350

X + .50Y = 200

-------------------

1.50Y = 150

So, Y = 100

Putting the value of Y, in equation (I) we get,

.20X + .40 x 100= 70

Or, X = 150

So, X, Y = 150, 100

Now all the values of X and y are ---

For equation Number (I)

When X = 0,   then    X, Y = 0, 175

When Y = 0    then X, Y = 350, 0

For equation Number (II)

When X = 0,   then    X, Y = 0, 400

When Y = 0    then X, Y = 200, 0

And X, Y = 150, 100

 

Putting the above values on graph, we get---

Click on the picture to see the original size.

 

By putting values of variables on X and Y axis and using shadow, we get the crossing point of two lines at (150, 100). So it would be the target point where zenith may get highest profit or lowest profit.

Now, Z = 8X + 6Y

 = 8 x 1500 + 6x 100

 = 1800

Competitive analysis:

Corner point	Total profit
0, 175	Z = (8 x 0) + (6 x 175)= 1050
0, 0	Z = (8 x 0) + (6 x 0)= 0
200, 0	Z = (8 x 200) + (6 x 0)= 1600
150, 100	Z = (8 x 150) + (6 x 100)= 1800

Comment:

Under the above solution, we suggest the production manager to produce 150 cases of knives and 100 cases of forks to maximize profit or minimize costs.

Mathematical Problem

Problem:

Nabila is a university student. She produces playing tools for children with clay. She would like to earn as much as possible. She uses her spare time to produce these tools. She has a specified amount of time for this work. Nabila earns a given amount of profit per product. There is practically an unlimited demand of her work. Show the above description with mathematical expression.

Solution:

Since Nabila wants to earn as much as possible, her objective is to maximize profit. Total earnings are determined multiplying the profit per playing tool and number of playing tools produced.

By Letting,

P= total profit,

R= Profit per playing tool.

Q= Number of playing tools produced.

Nabil’s objective is to maximize profit can be stated as follows:

P=QR---------------------- (i)

This type of mathematical expression is called objective function or goal of the problem.

Here, Total profit is restricted by Nabila’s available time. The demand for her work will equal to the time utilized per playing tool multiplied by quantity of playing tools. This demand must not exceed her available time.

By Letting,

t= Time utilized per playing tool,

T= Nabila’s available time.

The relationship can be described with the following mathematical expression:

tQ, ≤ T ------------------ (ii)

The symbol less than equal to (≤) indicates that the total time required must be less than or equal to the available time period. This type of expression is known as constraints.

Another restriction is that Nabila cannot produce a negative number playing tool, i.e.,

Q ≥ 0 ---------------- (iii)

The above mathematical expression states that the quantity of pages must be greater than or equal to zero (0). This type of expression is known as non-negative function.

Nabila’s problem is to determine the quantity of pages (Q) that will maximize her profit (P) per period from the production service. This problem also recommended quantity must not require more than his available time.

By accumulating, the equation number (i), (ii), (iii), Jackson’s problem can be represented with the following mathematical model:

Maximize P= QR----------- (i)

Subject to,      tq ≤T--------------- (ii)

Q ≥ O ------------------- (iii)

Role of Quantitative Techniques (QT) in Industry and Business.

Quantitative technique especially operation research technique has gained increasing importance since World War II in the technology of business administration. This technique greatly helps in tackling the integrated and complex problems of the modern business and industry. Quantitative techniques for decision making are infecting examples of the use of scientific management. However the roles of quantitative techniques are explained below.

1.     They provide a tool for scientific analysis:

These techniques provide the executives with a more precise description of a cause. They replace the intuitive and subjective approach. The use of these techniques has transformed the conventional techniques of operational and investment problems in business and industry. Quantitative techniques thus encourage and enforce disciplined thinking about organization’s problems.

2.     They provide solutions for various business problems:

The quantitative techniques are being used in the field of production, procurement, marketing, and such other fields. Problems like- how best can the manager and executives allotted the available resources to various departments. So that in a given time the profit are maximized or costs are minimized planning decision business and industry largely governed by the picture of anticipated demands and quantitative techniques help to forecast about demand. So, quantitative techniques are very important.

3.     They enable proper deployment of resources:

Quantitative techniques render valuable help in proper deployment of resources. For example- Programming- Evaluation- Review- Technique (PERT) requires various related data to identify critical path. In the same way when it require supply data and determine the probability of completing an event or project itself by specified data.

4.     They help in minimizing waiting and servicing costs:

The waiting line and/ or queuing theory help the management in minimizing the total waiting and servicing costs. This technique also analyses the feasibility of adding facilities and thereby helps the business people to take the correct and profitable decision.

5.     They assist in choosing an optimum strategy:

Game theory is specially used to determine the optimum strategy in a competitive situation and enable the businessman to maximize profits or minimize losses by adopting optimum strategy.

6.     They help in resources allocation:

They render great help in optimum resource allocation by the help of linear programming. Linear programming techniques are popularly used by modern management in resource allocation and selecting production mix.

7.     They facilitates the process of decision making:

The decision theory enables the businessman to select the best courses of action when information is given in probabilistic form.

8.     Inventory problem:

These techniques enable the management to decide when to buy and how much to buy.

9.     Statistical techniques:

Statistical techniques are also of great help to business man in more than one way. Some of the statistical techniques are considerable importance in sales forecasting whereas other facilitates from comparison between the various phenomena. In statistics there are various techniques such as quality control technique, sampling theory to decision making, various significant tests to judge the reliability etc. Similarly regression analysis, variance analysis, time series analysis, index number etc are useful tools of statistical analysis from where business get a great help and right decision is being taken.

Limitation of Quantitative Techniques (QT)

Management Science Process

Quantitative decision making is not a substitute for competent management. Rather it is methodology that can significantly improve the executive decision making. So, every business students should learn about it. Followings are the graphical representation and description of management science process:

Management Science Process

1. Define the problem:

The First step of the management science process is to define the problem. Quantitative decision making approach is problem oriented. So we should define the managerial problem clearly and concisely. The problem must be stated precisely and that should be suitable for analysis. Many operation research studies were failed simply because the problem was poorly defined. So at first we should emphasize more on it.

2. Formulate a quantitative model:

In the second step, we need to formulate a model. In formulating model, we should consider controllable and uncontrollable inputs. A model is a simplified representation of real objects or situations. The representation inputs only essential, relevant and important features.

3. Gather relevant quantitative data:

Organization would gather data from past accounting records, sales, financial, inventory, production and engineering records and reports. Published documents such as government statistical summaries may be important sources of data. Managers and operating personnel can provide information about markets, financial conditions, productivity and other factors that are unavailable elsewhere.

4. Analyze and solve the quantitative model:

In this step we have to analyze collected data and solve the quantitative model. In most cases, there is tremendous volume of available data and a considerable amount of time is required to collect and organize the information. Furthermore, data are usually not in form of suitable form for decision making purposes. More effort then, is necessary for processing and analyzing the data. As a result many organizations have designed and implemented formal system for collecting, analyzing and reporting relevant and timely information. Such a structure id referred to MIS (management Information System).

5. Implement the solution:

It is the last phase of decision making. The quantitative technique analysis process is not complete, until the model’s solution information is reported to the decision maker and results are implemented. Such data constitute only one of the inputs considered by the manager when a final decision is being made. So it shows the success or failure of the process. For this reason it is called vital step among all.