A northern hardware company is studying a plan to open a new distribution center in southeast. The company plans to rent a warehouse and an adjacent office, and distribute its main products to the local dealers. The company has decided to initially start with four of its main products: Pressure washers, Go karts, Generators, and Water pumps. The table below describes how much each of the products will cost the company (including transportation costs):

Item

Cost (in Dollars)[LD1]

Pressure Washer

330

Go-kart

370

Generator

410

Case of 5 Water Pumps

635

The company has set aside a purchasing monthly budget of $170,000 [LD2]for the new location. The selling prices (per unit) for each item are given in the table below:

Item

Cost (in Dollars)[LD3]

Pressure Washer

499.99

Go-kart

729.99

Generator

699.99

Case of 5 Water Pumps

269.99

Other than the budget, another of the company's concern is the available space in the warehouse. The warehouse has 82 shelves and each shelf is 30 ft long and 5 ft wide. Pressure washers and generators each are stored on 5 ft by 5 ft pallets whereas each Go Kart is stored on an 8 ft by 5 ft pallet. Furthermore, a 5 ft by 5 ft pallet is used to store four cases of water pumps[LD4]

For promoting its brand products, the company's marketing department has decided to allocate at least 30% of its inventory to pressure washers and Go Karts, and sell at least twice as many generators as water pumps[LD5].

As a consultant, your task is perform a monthly analysis using a linear programming model to maximize the company's net profit. Complete the following in a Word document and in R. Submit both the Word document and the R file as attachments

1.In a Word document, write the mathematical formulation of the problem.

2.Set up the linear programming formulation in R.

3.Use the LpsolveAPI in R to solve the problem, and generate a report.

4.Describe the optimal solutions obtained in the Word document. These will consist of the inventory level for all four products and the optimal monthly profit.

5.One of the decision variables has an optimal value of zero. Use the R report to determine the smallest selling price for that item so that this optimal zero solution value changes to a non-zero value.

6.In the word document explain whether, in addition to the $170,000 [LD6]allocated to the purchasing budget during the first month, the company should allocate additional money. If yes, calculate the amount of additional investment do you recommend, and how much should the company expect its net monthly profit to increase as a consequence of this increase?

7.In the word document, explain whether you recommend that the company should rent a smaller or a larger warehouse. In any case, indicate the ideal size of your recommended warehouse in square feet, and indicate how much this change in the size of the warehouse will contribute to the monthly profit.