LittleBoxes is a small-sized company that produces different types of presents packed in boxes. The typical products that they put in a box are: a box of filter coffee, a mug, a notebook, a box of white tea, vanilla candle, a cactus, and a frame. The sets that they sell are given in Table 1.

The boxes are produced (packed and assembled) in two shops: one in London, one in Paris. The labor requirements and packing times of each set are given in Table 2. The requirements do not differ with respect to the shop the sets are produced.

The firm gets orders from all over the country, and assigns the demand to the two shops. They require a two-week preparation time at most before the products are sent. The orders that should be fulfilled in the following two weeks are given in Table 3. The individual products that make up each set should be ordered from different companies. According to the agreements LittleBoxes made with the suppliers, there are upper/lower limits on the number of products that can be ordered weekly. The data is given in Table 4. They do not keep any inventory of the products since the products are delivered the same day they are ordered.

LittleBoxes is planning to determine its production quantities in each shop for the next two weeks. The product costs for the first week are given in Table 4 and these costs will increase by 8% for the second week. LittleBoxes has the option of producing more than a weeks requirement and keep those products as inventory for the next week. Currently, there is no inventory for any of the sets. Inventory costs are based on end-of-period inventory for each set, and will cost 5% of the selling prices given in Table 1. For the third week, LittleBoxes anticipate that demand will increase more. Therefore, they are planning to keep an inventory of 20 boxes of each set for that week. Your job is to determine a recommendation for the company that will minimize the total cost of production and inventory in the following two weeks. A recommendation must include a plan for production and inventory. You can assume that fractional production amounts are acceptable. In addition, you should also address the following sensitivity analysis issues in your recommendation (consider each issue independently):

a. In week 1, at which shop would you like to add extra labor minutes? How much would you be willing to pay per minute? How many extra minutes would you like? Repeat the same for the labor minutes in week 2, and extra packing minutes in weeks 1 and 2.

b. Marketing department is suggesting an advertising plan that increases the sales in each week by approximately 5% (you can assume fractional values for the sales). The additional sales revenue after the advertising campaign is estimated to be around $10,000. Should the company consider this advertising plan?

c. The company managers realized that the total labor minutes available in London shop is underestimated as 20,000 for the second week. If the true value causes a change in the basis, find a lower bound on the true total available labor minutes for Week 2. Find the objective function value of the new optimal solution using the lower bound.

Write the mathematical model and explain the variables, objective function and the constraints in detail,summarize your main findings (the value of the objective function, the production amounts in the two shops for the two weeks, etc.),for parts a-c, explain your decisions well. If you make any changes in the mathematical model, explain those changes both on the model and verbally and very shortly, mention what the case was about and what you have done. Highlight your most important results and discussions.

TABLES ARE SHOWN IN BELOW:

Set 1 2 3 4 5 6 Table 1. Sets of Presents Contents Price () Coffee, mug, frame 200 Tea, vanilla candle, frame 180 Coffee, tea, mug 170 Tea, notebook, cactus 150 Coffee, mug, cactus 160 Vanilla candle, frame, cactus 130 45 Product Boxes of coffee Boxes of white tea Notebook Mug Vanilla candles Cactus Frame Table 4. Product Data Cost for First Week (6) Policy 40 The total amount of tea and coffee boxes supplied can be at most 1300 weekly 30 At most 250 notebooks can be supplied weekly 22 At most 1000 mugs can be supplied weekly 15 At least 400 candles can be supplied weekly 15 At most 750 cactuses can be supplied weekly 35 Set Table 3. Demand for the Following Two Weeks Week 1 Week 2 Paris London Paris London 1 117 64 104 126 2 150 76 76 108 3 110 67 55 93 4 110 53 65 122 5 99 50 91 142 6 77 69 58 140 2 al 15 Table 2. Resource Requirements (minutes/set) Set Labor Packing 1 30 9 2 12 3 20 10 4 40 8 5 35 12 6 40 10 Total Available (minutes/week) 20,000 8,000 Set 1 2 3 4 5 6 Table 1. Sets of Presents Contents Price () Coffee, mug, frame 200 Tea, vanilla candle, frame 180 Coffee, tea, mug 170 Tea, notebook, cactus 150 Coffee, mug, cactus 160 Vanilla candle, frame, cactus 130 45 Product Boxes of coffee Boxes of white tea Notebook Mug Vanilla candles Cactus Frame Table 4. Product Data Cost for First Week (6) Policy 40 The total amount of tea and coffee boxes supplied can be at most 1300 weekly 30 At most 250 notebooks can be supplied weekly 22 At most 1000 mugs can be supplied weekly 15 At least 400 candles can be supplied weekly 15 At most 750 cactuses can be supplied weekly 35 Set Table 3. Demand for the Following Two Weeks Week 1 Week 2 Paris London Paris London 1 117 64 104 126 2 150 76 76 108 3 110 67 55 93 4 110 53 65 122 5 99 50 91 142 6 77 69 58 140 2 al 15 Table 2. Resource Requirements (minutes/set) Set Labor Packing 1 30 9 2 12 3 20 10 4 40 8 5 35 12 6 40 10 Total Available (minutes/week) 20,000 8,000