1. Chapter 2:
Linear Programming
Dr. Alaa Sagheer

2. Chapter Outline Part I: Introduction The Linear Programming Model
Examples of Linear Programming Problems
Developing Linear Programming Models
Graphical Solution to LP Problems

3. Introduction
Mathematical programming is used to find the best or optimal solution to a problem that requires a decision or set of decisions about how best to use a set of limited resources to achieve a state goal of objectives.
Steps involved in mathematical programming
Conversion of stated problem into a mathematical model that abstracts all the essential elements of the problem.
Exploration of different solutions of the problem.
Finding out the most suitable or optimum solution.
Linear programming requires that all the mathematical functions in the model be linear functions.

4. Introduction Example:
The Burroughs garment company manufactures men's shirts and women’s blouses for Walmark Discount stores. Walmark will accept all the production supplied by Burroughs. The production process includes cutting, sewing and packaging. Burroughs employs 25 workers in the cutting department, 35 in the sewing department and 5 in the packaging department. The factory works one 8-hour shift, 5 days a week. The following table gives the time requirements and the profits per unit for the two garments:

5. Minutes per unit Introduction
Garment
Cutting
Sewing
Packaging
Unit profit($)
Shirts 20 70 12
8.00
Blouses 60 4
12.00
Determine the optimal weekly production schedule for Burroughs.

6. Introduction Profit got = 8 x1 + 12 x2
Solution: Assume that Burroughs produces x1 shirts and x2 blouses per week:
Profit got =
8 x x2
Time spent on cutting =
20 x x2 mts
Time spent on sewing =
70 x x2 mts
Time spent on packaging =
12 x1 + 4 x2 mts

7. Introduction
The objective is to find x1, x2 so as to maximize the profit
z = 8 x x2
satisfying the constraints:
20 x x2 ≤ 25  40  60
70 x x2 ≤ 35  40  60
12 x x2 ≤ 5  40  60
x1, x2 ≥ 0, integers

8. Introduction This is a typical optimization problem
Any values of x1, x2 that satisfy all the constraints of the model is called a feasible solution. We are interested in finding the optimum feasible solution that gives the maximum profit while satisfying all the constraints.

9. The Linear Programming Model
Let x1, x2, x3, ………, xn are decision variables and
Z = Objective function or linear function
Requirement: Maximization of the linear function Z
Z = c1x1 + c2x2 + c3x3 + ………+ cnxn
subject to the following constraints:
where aij, bi, and cj are given constants.

10. The Linear Programming Model
The linear programming model can be written in more efficient notation as:
The decision variables, xI, x2, ..., xn, represent levels of n competing activities

11. The Linear Programming Model
An LPP satisfies the three basic properties:
1. Proportionality: It means, the contributions of each decision variable in the objective function and its requirements in the constraints are directly proportional to the value of the variable
2. Additivity: This property requires the contribution of all the variables in the objective function and its constraints to be the direct sum of individual contributions of each variables.
3. Certainty: All the objective and constraint coefficient of the LP model are deterministic. This mean that they are known constants- a rare occurrence in real life.

12. Examples of LP Problems
1. A Product Mix Problem
A manufacturer has fixed amounts of different resources such as raw material, labor, and equipment.
These resources can be combined to produce any one of several different products.
The quantity of the ith resource required to produce one unit of the jth product is known.
The decision maker wishes to produce the combination of products that will maximize total income.

13. Examples of LP Problems
2. A Blending Problem
Blending problems refer to situations in which a number of components (or commodities) are mixed together to yield one or more products.
Typically, different commodities are to be purchased. Each commodity has known characteristics and costs.
The problem is to determine how much of each commodity should be purchased and blended with the rest so that the characteristics of the mixture lie within specified bounds and the total cost is minimized.

14. Examples of LP Problems
3. A Production Scheduling Problem
A manufacturer knows that he must supply a given number of items of a certain product each month for the next n months.
They can be produced either in regular time, subject to a maximum each month, or in overtime. The cost of producing an item during overtime is greater than during regular time. A storage cost is associated with each item not sold at the end of the month.
The problem is to determine the production schedule that minimizes the sum of production and storage costs.

15. Examples of LP Problems
4. A Transportation Problem
A product is to be shipped in the amounts al, a2, ..., am from m shipping origins and received in amounts bl, b2, ..., bn at each of n shipping destinations.
The cost of shipping a unit from the ith origin to the jth destination is known for all combinations of origins and destinations.
The problem is to determine the amount to be shipped from each origin to each destination such that the total cost of transportation is a minimum.

16. Examples of LP Problems
5. A Flow Capacity Problem
One or more commodities (e.g., traffic, water, information, cash, etc.) are flowing from one point to another through a network whose branches have various constraints and flow capacities.
The direction of flow in each branch and the capacity of each branch are known.
The problem is to determine the maximum flow, or capacity of the network.

17. Developing LP Model
The variety of situations to which linear programming has been applied ranges from agriculture to zinc melting.
Steps Involved:
Determine the objective of the problem and describe it by a criterion function in terms of the decision variables.
Find out the constraints.
Do the analysis which should lead to the selection of values for the decision variables that optimize the criterion function while satisfying all the constraints imposed on the problem.

18. Developing LP Model + Example: Product Mix Problem
Graphical Solution
+ Example: Product Mix Problem
The N. Dustrious Company produces two products: I and II. The raw material requirements, space needed for storage, production rates, and selling prices for these products are given in Table I.
The total amount of raw material available per day for both products is 15751b. The total storage space for all products is 1500 ft2, and a maximum of 7 hours per day can be used for production.

19. Developing LP Model Example: Product Mix Problem Solution
All products manufactured are shipped out of the storage area at the end of the day. Therefore, the two products must share the total raw material, storage space, and production time. The company wants to determine how many units of each product to produce per day to maximize its total income
Solution
The company has decided that it wants to maximize its sale income, which depends on the number of units of product I and II that it produces.
Therefore, the decision variables, x1 and x2 can be the number of units of products I and II, respectively, produced per day.

20. Developing LP Model The object is to maximize the equation:
Z = 13x x2
subject to the constraints on storage space, raw materials, and production time.
Each unit of product I requires 4 ft2 of storage space and each unit of product II requires 5 ft2. Thus a total of 4x1 + 5x2 ft2 of storage space is needed each day. This space must be less than or equal to the available storage space, which is 1500 ft2. Therefore,
4X1 + 5X2  1500
Similarly, each unit of product I and II produced requires 5 and 3 1bs, respectively, of raw material. Hence a total of 5xl + 3x2 Ib of raw material is used.

21. Developing LP Model
This must be less than or equal to the total amount of raw material available, which is 1575 Ib. Therefore,
5x x2  1575
Prouct I can be produced at the rate of 60 units per hour. Therefore, it must take I minute or 1/60 of an hour to produce I unit. Similarly, it requires 1/30 of an hour to produce 1 unit of product II. Hence a total of x1/60 + x2/30 hours is required for the daily production. This quantity must be less than or equal to the total production time available each day. Therefore,
x1 / 60 + x2 / 30  7
or x x2  420
Finally, the company cannot produce a negative quantity of any product, therefore x1 and x2 must each be greater than or equal to zero.

22. Developing LP Model
Graphical Solution
The linear programming model for this example can be summarized as:

23. Developing LP Model + Example (2):
Graphical Solution
+ Example (2):
Reddy Mikks produce both interior and exterior paints from two raw materials, M1 and M2.
The following table provides the basic data of the problem:

24. Developing LP Model Example Problem Solution
A market survey indicates that the daily demand for the interior paint cannot exceed that for extirior paint by more than one ton. Also, the maximam daily demand for the interior paint is 2 tons. Reddy Mikks wants to determine the optimum(best) product mix of the interior and exterior paints that maximizes the total daily profit.
Solution
Reddy Mikks has decided that it wants to maximize its total daily profit, which depends on the product mix of the interior and exterior paints.
Therefore, the decision variables, x1 and x2 can be the ton of exterior and interior paints, respectively, produced per day.

25. Developing LP Model The object is to maximize the equation:
Z = 5x x2
The daily raw material M1 is 6tons per ton of exterior paint and 4 tons per ton of interior paint must be equal the daily avaliability of raw material M1 (24 ton) , Therefore,
6x1+ 4x2  24
Similary, The daily usage of raw material M2 is 1ton per ton of exterior paint and 2 tons per ton of interior paint must be equal the daily aviliability or raw material M2 (6 tons), Therefore,
x1+ 2x2  6

26. Developing LP Model
The first demand restriction stipulates that the excess the daily production of interior over exterior paint should not exceed 1 ton, Therefore ,
x2 - x1  1
The second demand restrection stipulates that the maximum daily demand of interior paint is limited to 2 tons. Therefore,
x2  2

27. Developing LP Model
Graphical Solution
The linear programming model for this example can be summarized as:

28. Developing LP Model Example (3):
Wild West produces two types of cowboy hats. Type I hat requires twice as much labor as a Type II. If all the available labor time is dedicated to Type II alone, the company can produce a total of 400 Type II hats a day. The respective market limits for the two types of hats are 150 and 200 hats per day. The profit is $8 per Type I hat and $5 per Type II hat. Formulate the problem as an LPP so as to maximize the profit.

29. Developing LP Model Solution:
Assume that Wild West produces x1 Type I hats and x2 Type II hats per day.
8 x1 + 5 x2
Per day Profit got =
Assume the time spent in producing one type II hat is c minutes.
Labour Time spent is
(2 x x2) c minutes

30. maximize the profit z = 8 x1 + 5 x2
Developing LP Model
The objective is to find x1, x2 so as to
maximize the profit z = 8 x1 + 5 x2
satisfying the constraints:
(2 x1 + x2 ) c ≤ 400 c
x ≤ 150
x ≤ 200
x1, x2 ≥ 0, integers

31. Developing LP Model Day 1 2 3 4 5 Napkins Needed 80 50 100 150
Example (4):
BITS wants to host a Seminar for five days. For the delegates there is an arrangement of dinner every day. The requirement of napkins during the 5 days is as follows:
Day 1 2 3 4 5
Napkins Needed 80 50
100
150

32. Developing LP Model
Institute does not have any napkins in the beginning. After 5 days, the Institute has no more use of napkins. A new napkin costs Rs The washing charges for a used one are Rs A napkin given for washing after dinner is returned the third day before dinner. The Institute decides to accumulate the used napkins and send them for washing just in time to be used when they return. How shall the Institute meet the requirements so that the total cost is minimized ? Formulate as a LPP.

33. Developing LP Model Solution:
Let xj be the number of napkins purchased on day j, j=1,2,..,5
Let yj be the number of napkins given for washing after dinner on day j, j=1,2,3
Thus we must have
x1 = 80, x2 = 50, x3 + y1 = 100, x4 + y2 = 80
x5 + y3 = 150
Also we have
y1 ≤ 80, y2 ≤ (80 – y1) + 50
y3 ≤ (80 – y1) + (50 – y2) + 100

34. z = 2(x1+x2+x3+x4+x5)+0.5(y1+y2+y3)
Developing LP Model
Thus we have to Minimize
z = 2(x1+x2+x3+x4+x5)+0.5(y1+y2+y3)
Subject to
x1 = 80, x2 = 50, x3 +y1 =100,
x4 + y2 = 80, x5 + y3 = 150,
y1 ≤ 80, y1+y2 ≤ 130, y1+y2+y3 ≤ 230,
all variables ≥ 0, integers

35. Developing LP Model Example (5):
A Post Office requires different number of full-time employees on different days of the week. The number of employees required on each day is given in the table below. Union rules say that each full-time employee must receive two days off after working for five consecutive days. The Post Office wants to meet its requirements using only full-time employees. Formulate the above problem as a LPP so as to minimize the number of full-time employees hired.

36. No. of full-time employees required
Developing LP Model
Requirements of full-time employees day-wise
Day
No. of full-time employees required
1 - Monday 10
2 - Tuesday 6
3 - Wednesday 8
4 - Thursday 12
5 - Friday 7
6 - Saturday 9
7 - Sunday 4

37. Developing LP Model Solution: Minimize Subject to
Let xi be the number of full-time employees employed at the beginning of day i (i = 1, 2, …, 7). Thus our problem is to find xi so as to
Minimize
Subject to

38. The Graphical Solution
The determination of the solution space that defines the feasible solutions that satisfy all the constrains of the model.
The determination of the optimum solution from among all the points in the feasible solution space.

39. Graphical LP Solution Model
Example:
Electra produces two types of electric motors, each on a separate assembly line. The respective daily capacities of the two lines are 150 and 200 motors. Type I motor uses 2 units of a certain electronic component, and type II motor uses only 1 unit. The supplier of the component can provide 400 pieces a day.The profits per motor of types I and II are $8 and $5 respectively. Formulate the problem as a LPP and find the optimal daily production.

40. Graphical LP Solution Model
Let the company produce x1 type I motors and x2 type II motors per day.
The objective is to find x1 and x2 so as to
Maximize the profit
Subject to the constraints

41. Graphical LP Solution Model
Step 1 Determination of the Feasible solution space
The non-negativity restrictions tell that the solution space is in the first quadrant. Then we replace each inequality constraint by an equality and then graph the resulting line (noting that two points will determine a line uniquely).
Step 2 Determination of the optimal solution
The determination of the optimal solution requires the direction in which the objective function will increase (decrease) in the case of a maximization (minimization) problem. We find this by assigning two increasing (decreasing) values for z and then drawing the graphs of the objective function for these two values. The optimum solution occurs at a point beyond which any further increase (decrease) of z will make us leave the feasible space.

42. Graphical solution Maximize z=8x1+5x2 Subject to the constraints
Optimum =1800 at
2x1+x2  400
x  150
x2  200
x1,x2 0
(100,200)
(0,200)
z=1200
(150,100)
z=1800
z=1000 x1
z=400
(150,0)
z=1700

43. Realted to Reddy Mikks problem
Graphical LP Solution Model
Realted to Reddy Mikks problem
Problem

44. Graphical Solution to LP Problems
Related to Product Mix Problem
Problem

45. Feed Mix Problem Graphical LP Solution Model Minimize z = 2x1 + 3x2
Subject to x1 + 3 x2  15
2 x1 + 2 x2  20
3 x1 + 2 x2  24
x1, x2  0

46. Graphical Solution of Feed Mix Problem
(0,12)
z = 26
z = 30
z= 36
z = 39
(4,6)
z = 42
z = 22.5
Minimum at
(7.5,2.5)
(15,0)

47. Optimum = 60 at (20, 20) Maximize z = 2x1 + x2 Subject to x1 + x2 ≤ 40
Graphical LP Solution Model (5)
Maximize z = 2x1 + x2
Subject to x1 + x2 ≤ 40
4 x1 + x2 ≤ 100
x1, x2 ≥ 0
Optimum = 60 at (20, 20)

48. (0,40)
(20,20)
(25,0)
z maximum at
Graphical solution

49. The Graphical Solution
Ozark farms uses at least 800 Ib of special feed is a mixture of corn and soybean meal with the following compositions:
The dietary requirement of the special feed are at least 30% protein and at most 5% fiber.
Ozark Farms wishes to determine the daily minimm- cost feed mix.

50. The Graphical Solution
We can consider that,
And the objective function seeks to minimize the total daily cost (in dollars) of the feed mix and is thus expressed as
Subject to constrains:

51. The Graphical Solution
Unlike previous examples, the second and the third constraints pass through the origin. To plot associated straight lines, we need one additional point, which can be obtained by assigning value to one of the variables and then solving for other variables. For example, in the second constraint, x1=200, then x2=140

52. The Graphical Solution
Problem