best solution for minimizing

Project Description:

a company is planning production for the next 4 quarters. they want to minimize the cost of production. the production cost, demand, and production capacity vary from quarter to quarter, as shown in the table below. the maximum amount of inventory which can be held is 100 units and management wants to keep at least 50 units on hand. quarterly inventory holding cost is 4% of the cost of production. there are currently 50 units in inventory. the company wants to produce at no less than one half of its maximum capacity in any quarter.
quarter
1 2 3 4
unit production cost $55 $50 $50 $45
units demanded 100 150 180 120
maximum production 150 150 160 130
holding cost $2.20 $2.00 $2.00 $1.80
let pi be the number of units produced in quarter i (i = 1,2,3,4), let bi be the beginning inventory for quarter i (i = 2,3,4,5).
a. formulate the lp objective function for this problem.
b. formulate the constraints that bound the values of the pi variables.
c. formulate the constraints that bound the values of the bi variables.
d. formulate the lp constraints that relate the beginning inventory in quarter i to the beginning inventory and production level in the previous quarter.
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