A company has three factories, a distribution center, and five stores. They produce their products in the factory, ship them to the distribution center for sorting, and then the products are shipped from the distribution center to the stores to stock up the shelves. Each factory is responsible for producing one product only. Factory 1 can produce a maximum of 10K products, factory 2 can produce a maximum of 20K products, and factory 3 can produce a maximum of 15K products in a cycle. The distribution center's total capacity is 40K products total. The shipping costs from factories to the distribution center are per item shipped, and Table 1 summarizes the costs.

Table 1: Shipping cost from factory to the distribution center, per product
Factory 1               Factory 2                 Factory 3
      $6                           $7                             $10
Each store requires a mix of the three products with a minimum required number of products to be received from the factories every cycle (Table 2). The shipping cost from the distribution center to the stores is a constant cost, and Table 3 summarizes the costs. However, if the store is receiving more than a total of 8K of products, they incur a constant cost of $1,000 for the extra shipping trucks. Finally, because where the stores are located, and supply and demand forces, each item sells differently in different stores. The selling price is detailed in Table 4. The company wants to maximize their net profit.

Table 2: Minimum order from each product for each store
                              Store 1             Store 2           Store 3          Store 4            Store 5
Product                1 2000               1000               3000               1000               1000
Product                2 4000               3000               4000               2000               2000
Product                3 2000               2000               3000               1000               1000

Table 3: Transportation cost from distribution center to store
Store 1               Store 2               Store 3               Store 4               Store 5
  $800                  $1000                   $700                    $600                   $900

Table 4: Selling price of each product in each store, $
                            Store 1        Store 2          Store 3        Store 4        Store 5
Product 1               170               175               180               165               160
Product 2               160               155               165               150               155
Product 3               140               130               135               145               130

Model the problem as a linear program. Fully define any variable or parameter that you use.
Solve the problem using software to obtain an optimal solution.

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Model the problem as a linear program Let x1 x2 and x3 represent the number of products from Factory 1 Factory 2 and Factory 3 respectively shipped to ... View the full answer