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1. A company produces bicycles and motorcycles. The required man- power and production cost to produce the bicycles and motorcycles are presented in table below: Table 1: Required manpower and producetion cost required manpower production cost bicycles 4 manpower per unit 1000 tl motorcycles 10 manpower per unit 8000 tl The pre-ordered demand quantities for the products for 2021 are given in table below: Table 2: Yearly demand Year 2021 bicycles 500 motorcycles 400 The company sells pre-ordered bicycles and motorcycles 3000 and 15000 tl, respectively. The available yearly manpower is 15000 and the max- imum yearly budget which can be used for production is 4000000 tl. The company can sell any number of surplus bicycles 2000 tl at the end of 2021. For example, if the company produces 550 bicycles in 2021, it can sell 500 bicycles 3000 tl and the remaining 50 surplus bicycles 2000 tl. Also, the company can sell any number of surplus motorcycles 11000 tl at the end of 2021. For example, if the company produces 600 motorcycles in 2021, it can sell 400 motorcycles 15000 tl and the remaining 200 surplus motorcycles 11000 tl. The production manager of the company needs a production plan which determines the amount of produced bicycles and motorcycles for 2021 and maxi- mizes the company's profit such that the pre-ordered demand and the above constraints are satisfied. Formulate this problem as a linear pro- gramming model. Define the decision variables, objective function, and provide a clear explanation of the constraints of your model.
Let x no of bicycles and y no of motorcycles be produced
Production cost per bicycle is 1000 and motorcycle is 8000tl
Hence total production cost = 1000x + 8000y
Since budget available is 4000000
1000x + 8000y <= 4000000
Further, 4 manpower per unit of bicycle and 10 manpower per unit of motorcycle is required with 15000 available manpower
Hence 4x + 10y <= 15000
Pre-ordered demand for bicycles and motorcycles are 500 and 400 respectively
500 bicycles are sold at 3000 tl and above this sold at 2000
400 motorcycles are sold at 15000 and above this sold at 11000
Since preordered demand is to be met x >= 500 and y >= 400
Further based on above price constraints
Total profit = 500X3000 + (x-500)*2000 + 400X15000 + (y-400)*11000
= 1500000 + 2000x – 1000000 + 6000000 + 11000y – 4400000
= 2000x + 11000y + 2100000
That needs to be maximised
Hence Objective function is maximize profit = 2000x + 11000y + 2100000
Decision variables are x and y
And constraints are:
1000x + 8000y <= 4000000
4x + 10y <= 15000
x >= 500 and y >= 400
