IMSE 361 Summer 2018 Dr. Majid Aldaihani Due: June 5, 2018 HW# 1 A manufacturing

Question

IMSE 361 Summer 2018 Dr. Majid Aldaihani Due: June 5, 2018 HW# 1 A manufacturing company produces two types of vending machines: machines A and B. The price of machine A is 7000 KD while the price of machine B is 6000 KD. Producing machine A costs the company a total of 6500 KD per year while machine B costs the company 5600 KD per year. The major resources of the company to manufacture the vending machines are labor hours and steel. The available labor hours and steel for the company are 120,000 hours and 15,000 tons per year respectively. For production, machine A requires 100 units of labor hours and 4 tons of steel, while machine B requires 80 units of labor hours and 3 tons of steel. The company must produce at least 20 machines of type A every year and at most 60 machines of type B. Use a linear programing (LP) formulation to help the company to maximize the annual profit. 1.

Explanation / Answer

Answer:

Given Conditions

·       There are Two Type of Machines to be manufactured with below details.

·       X = Machine A – 7000 KD - cost 6500 KD/year – 100 labor hours , 4 tons of steel – min 20 machines/year

·       Y = Machine B – 6000 KD - cost 5600 KD/year - 80 labor hours , 3 tons of steel, max 60 machine per year.

·       Available Labor hours = max 120000 hours

·       Available Steels = 15000 tons

Linear Programming Problem: The linear programming problem statement is the objective functions which maximize or minimize or optimize the function for the interested variable of the problem statement. The aim of objective function is to have the optimized solution with meeting the constraints of each of the variables. We can say that the linear programming problem objective optimize the function with compiling all the constraints equations.

Linear Objective Function: This is the main function of the problem, made up by including all relevant functional variables, which impacts the objective function significantly.

Machine A – 7000 KD & Machine B - 6500KD are the two variable that will deliver the objective of the organization.

Then the objective function to have optimized benefit for the organization is the combinations of quantities of each machine for the firm

Thus

Objective Function Max. Z = 7000 X + 6500 Y

Constraints Conditions: These are the conditions which makes the boundary conditions for the problem statement. Thus in the given problem statement, this conditions are must meet requirements.

So available hours and steel are the constraints for the manufacturing of machines as below

First Constraint Equation

Total available Labor hours = max 120000 hours.

Machine A (X) needs 100 labor hours and Machine B (Y) needs 80 labor hours

·       100 X + 80 Y <,= 120000 hours

Second Constraints equation

Total available Steels = 15000 tones.

Machine A (X) needs 4 tons of steel and Machine B (Y) needs 3 tomes of steel, then

·       4 X + 3Y <,=15000 tons of steel

Third Constraint equation

Machine A need to be minimum 20 nos.

·       X >,= 20

Machine B can be maximum 60 nos.

·       Y<,= 60

Fourth Constraint equation

Machine A cost to be 6500 KD/year

·       X<, = 6500

Machine B cost to be 5600KD/year

·       Y <,= 5600

Thus Summary of linear programming problem is as below

Objective Function

·       Max. Z = 7000 X + 6500 Y

Subjected to Constraints

·       100 X + 80 Y <,= 120000 hours

·       4 X + 3Y <,=15000 tons of steel

·       X >,= 20 , Y<,= 60

·       X<, = 6500 , Y <,= 5600

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