report na ni
DESCRIPTION
Linear ProgrammingTRANSCRIPT
Linear Programming
Irene Lara M. Abad
WHAT IS LINEAR PROGRAMMING (LP)?
A mathematical modelling technique designed to optimize the usage of limited resources.
Formulation of LP Model
WHAT ARE THE BASIC ELEMENTS OF THE LP MODEL?
1) Decision variables
2) Objective
3) Constraints
SAMPLE PROBLEMReddy Mikks produces both interior and exterior paints from two raw materials, M1 and M2. The following table provides the basic data of the problem:
Tons of Raw Material per ton ofMaximum Daily
Availability (tons)Exterior Paint Interior Paint
Raw Material, M2 6 4 24
Raw Material, M1 1 2 6
Profit per ton (1000 dollars)
5 4A market survey restricts the maximum daily demand of interior paint to 2 tons. Additionally, the daily demand for interior paint cannot exceed that of exterior paint by more than 1 ton. Reddy Mikks wants to determine the optimum (best) product mix of interior and exterior paints that maximizes the total daily profit.
1) Decision Variables
Tons of Raw Material per ton ofMaximum Daily
Availability (tons)Exterior Paint Interior Paint
Raw Material, M2 6 4 24
Raw Material, M1 1 2 6
Profit per ton (1000 dollars)
5 4
FORMULATION OF LP MODEL
2) Objective
Tons of Raw Material per ton ofMaximum Daily
Availability (tons)Exterior Paint Interior Paint
Raw Material, M2 6 4 24
Raw Material, M1 1 2 6
Profit per ton (1000 dollars)
5 4
FORMULATION OF LP MODEL
3) Constraints (on raw materials usage)
Tons of Raw Material per ton ofMaximum Daily
Availability (tons)Exterior Paint Interior Paint
Raw Material, M2 6 4 24
Raw Material, M1 1 2 6
Profit per ton (1000 dollars)
5 4
FORMULATION OF LP MODEL
3) Constraints (on raw materials demand)
FORMULATION OF LP MODEL
A market survey restricts the maximum daily demand of interior paint to 2 tons. Additionally, the daily demand for interior paint cannot exceed that of exterior paint by more than 1 ton. Reddy Mikks wants to determine the optimum (best) product mix of interior and exterior paints that maximizes the total daily profit.
SAMPLE PROBLEM
FORMULATION OF LP MODEL
A manufacturing film produces two products X1 and X2 which undergo three production operations as presented in the ff. table. X1 is sold at a price of P140 per unit while X2 is sold at P80. while market demand for X1 is limited to 75 units per day, X2 should not exceed by 50 units per day operation. Machine hours required per unit are
Operation X1 X2
Cutting 0.50 0.30
Machining 1.50 1.00
Finishing 0.50 0.30
The available time in hours per day is 45 for cutting, 150 for machining and 75 for finishing. Production cost per hour is P30. Formulate the LP Model.
SAMPLE PROBLEM
FORMULATION OF LP MODEL
A poultry consumes not less than 50 lbs. of special feed daily. The feed is prepared as a mixture of corn and soybean meal with the following compositions:
The dietary requirements are
Pounds per pound of feedstuff
Feedstuff Calcium Protein Fiber Cost(dollar/lb)
Corn 0.001 0.09 0.02 0.20
Soybean 0.002 0.60 0.06 0.60
1) At most 1.5% of calcium
2) At least 20% but no more than 40% of protein
3) At most 4.5% fiber
Formulate the LP Model
SAMPLE PROBLEMFORMULATION OF LP MODEL
A company supplies nozzles to a corporation which assembles spray guns. The company producing the nozzles has 3 plants located in the localities. A, B, and C while the corporation that needs it for its spray guns has plants located in areas X, Y, and Z. Per contact entered into by both establishments, yearly requirement for nozzles in plants assembling the spray guns are as follows:
The company’s production capacities for nozzles are:
A 80000
B 50000
C 75000
X 50000
Y 70000
Z 60000
Cost of transporting nozzles in boxes, 1000 units per box, are as presented below
Source X Y Z
A 7 11 8
B 20 17 12
C 8 18 13
Formulate the LP model to optimize transport of nozzles
SPECIAL VARIABLES
Slack Variable
SPECIAL VARIABLES
Surplus Variable
SPECIAL VARIABLESUnrestricted Variable
- A variable which assumes a nonnegative valueMcBurger fastfood restaurant sells quarter-pounder and cheeseburger. A quarter-pounder uses a quarter pound of meat, and a cheeseburger uses only 0.2 lb. the restaurant starts the day with 200 lb of meat but may order more at an additional cost of 25 cents per pound to cover the delivery cost. Any surplus meat at the end of the day is donated to HotSoup Charity. McBurger’s profits are 20 cents from a quarter-pounder and 15 cents for a cheeseburger. All in all, McBurger does not expect to sell more than 900 sandwiches in any one day. How many of each sandwich should the McBurger make?
Objective Function: McBurger seeks to maximize the total profit, less any additional cost that may be incurred a result of ordering special delivery of additional pounds of meat
Solution of LP Model
3 WAYS IN SOLVING THE LP MODEL
1) Graphical Method
2) The Simplex Method
3) The M-Method