Problem 2

You are the owner of the new fancy restaurant in town. To operate the restaurant you use disposable napkins and tablecloths made from recyclable materials. Based on the number of customers you expect to have during the following two months, you have an estimation on number of napkins and tablecloths that you will need for the following 8 weeks (see Table 3).

To satisfy the demand of napkins and tablecloths for each week you have three choices. One possibility is buying them from a small local retailer who can sell you individual units of napkins and tablecloths for a price of $3 and $9 per unit, respectively. Unfortunately, being a small retailer, they can sell you at the most 2,000 napkins and 400 tablecloths per week. A second option is buying them from a large manufacturer. The problem with this option is that, as a large manufacturer, they do not sell individual units, but sell product by box. Each box they sell costs $1,500 and contains 400 napkins and 50 tablecloths. This manufacturer can sell you as many boxes as you want each week. A third option is to order the napkins and tablecloths online from a company located in China. They sell napkins and tablecloths at a price of $2 and $8 per unit, respectively. The problem with this option is that it takes two weeks for the order to arrive. For example, if you order 100 napkins and 50 tablecloths during week 1, those will become available upon arrival in week 3. You can order as many items as you want from this company.

For your inventory strategy, it is possible to buy more napkins and tablecloths from the manufacturers than the ones required for each week and save the excess and use them during the subsequent weeks. You can also buy fewer that the ones you need as long as you have enough napkins and tablecloths in your inventory to fulfill the given requirements. However, given the space limitations of your restaurant, you have available a small room with only 3,000 ft3 to store the napkins and tablecloths left at the end of each week. Assume that napkin and tablecloths have an individual size of 0.5 ft3 and 1 ft3 , respectively. Suppose that the beginning of week 1 you have 200 napkins and 80 tablecloths available. Also, assume that you want to leave at least 300 napkins and 100 tablecloths as your final inventory for future weeks.

Notice that the possibility of buying items from different manufacturers gives you many interesting options. For example buying in bulk from the large manufacturer is cheaper than buying from the local retailer, but because the product comes in a box with a predetermined number of items, you may end up with more items than the ones you need. Also, buying from the company in China seems to be a good option as well, but you must consider the 2-week delay. The main objective of this problem is to identify the inventory policy that minimizes your costs, while satisfying the requirements of the restaurant.

a. Provide a formulation for this problem specifying the sets, parameters, decision variables, constraints, and objective function. You MUST provide a general formulation.

b. Solve the problem using Gurobi.

c. Report your results using tables to present: 1) the number of items of each type that you buy (immediate delivery) or order (delayed delivery) from each provider each week, 2) the inventory of items left at the end of each week per type.

Now, consider the following variation of the problem. Suppose that the manufacturer that operates in China imposes the following additional condition. Since they have to ship the product from overseas, they decided not to sell you any product each week, unless the sum of the napkins and tablecloths that you order that week is more than 4,000 units. For example, with this new condition you can decide whether you order product from them during week t. If you do, the total amount of napkins and tablecloths you order from them that week must be at least 4,000 units.

d. Provide a formulation for this variation specifying the additional decision variables, constraints, and objective function. You MUST provide a general formulation.

e. Solve the problem using Gurobi.

f. Report your results using tables to present: 1) the number of items of each type that you buy (immediate delivery) or order (delayed delivery) from each provider each week, 2) the inventory of items left at the end of each week per type.