Problem 6 FW Incorporated is a company that produces four different types of fur
ID: 387200 • Letter: P
Question
Problem 6
FW Incorporated is a company that produces four different types of furniture items. Each of these products passes through five production steps before being ready for distribution dlling, painting, assembling, cleaning, and packaging. FW has collected the information given in Table 1 about the total number of hours required for one unit of each product to go though each production step. For example, one unit of product 1 requires 0.03 hours of drilling (note that product 3 does not require painting) The total number of hours dedicated for each production step cannot exceed the bound given in Table 1. For example, the total number of hours used for drilling on all the products cannot be more than 400. Furthermore, during the assembly process, products 2 and 4 need a special glue that comes in the form of plastic sheets. Product 2 requires 2.0 ft2 of the special glue per unit produced, whereas Product 4 needs 1.4 ft2. The company has only 2000 ft of glue available for the assembly process. Production rate Ste Painting Cleaning uct 1 uct 2 0.15 0.12 0.10 0.20 0.06 uct 3 0.05 uct 4 Hours available 0.10 0.10 0.12 0.12 0.12 ps 0.06 0.05 0.04 0.02 0.05 0.03 0.02 400 400 500 450 400 Packaging Table 1: Production information The marketing team at FW used historical data to generate a detailed forecast of the demand of each product. The team observed that the demand always falls between a range as indicated in Table 2. In other words, according to the data, there is a minimum amount of product that FW is guaranteed to sell, but there is also a limit on the number of items FW expects to sell. Any production beyond that will most likely remain unsold and wasted. In addition to the forecast for the demand, Table 2 also presents both the selling price and production cost per unit for each product type Potential demand Product Sell price /unit Production cost /unit MinimumMaximum 10 6 1,000 0 500 100 6.000 500 3,000 1,000 16 20 14 Table 2: Marketing information FW wants to know how many units of each product must be produced guaranteeing that all the conditions are satisfied. Answer the following problems a) Provide an explicit formulation for this problem.Explanation / Answer
a) LP formulation for this problem is following:
Decision variables: Let X1, X2, X3, X4 be the number of units of each product that must be produced.
Objective: Max (10-6)X1+((25-15)X2+(16-11)X3+(20-14)X4
s.t. Constraints:
X1 >= 1000
X3 >= 500
X4 >= 100
X1 <= 6000
X2 <= 500
X3 <= 3000
X4 <= 1000
0.03X1+0.15X2+0.05X3+0.10X4 <= 400
0.06X1+0.12X2+0.00X3+0.10X4 <= 400
0.05X1+0.10X2+0.05X3+0.12X4 <= 500
0.04X1+0.20X2+0.03X3+0.12X4 <= 450
0.02X1+0.06X2+0.02X3+0.12X4 <= 400
2X2 + 1.4X4 <= 2000
X1, X2, X3, X4 >= 0