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Linear programming problem in Introduction to Operating research

A wholesale company has been growing by increasing product supply. They are interested into take in a product that needs to be stored at special conditions. They have their own warehouse, but need to change it to fulfil the storing condition for the new product. They want to start by take in the product for trial and if it is successful than the company is willing to do the changes. To start with they will rent a storage room for the product with a third party.  

They have contacted a storage company and have received the cost per pallet in storage. The cost is different after how many month they want to store the product, that is, the month payment lower according when more continuing months are bought. The problem is that forecasted demand after the product is not equal. To close a contract with the customer they have obliged them to own specific number of pallets in the inventory in the following month. For simplification you do not need to think about other cost.

In the table here below you can see stock count after month:

Month January February March April Estimated number of pallets 500 200 100 500

In the table here below you can see the total cost per pallet for different number of months. (If storage is rented in January for 4 months, then the total cost is 18 dollars for those 4 months)

Number of months Cost per pallet 1 13 16 18 4

a) Formulate a linear programming model that minimize the cost of the company and guarantees a space for appropriate number of pallets. Explain premises, variables, objective function and constrains.

b) Now a new demand comes from the company, they want to lower the storage cost in end of February. The reason is than the company is in settlements, and it would look better if the inventory where lowered. Because of that they do not want to rent more that 250 pallet spaces in February. Add this constrain to the linear program if it is possible.

c) The company gets information about new storage company they can use, they only have available space in April and can take maximum 150 pallets. The cost is than 5 usd/pallet/month. Add this constrain to the linear program.

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