1. Corso MAE Metodi Quantitativi per il Management Quantitative methods for Management Roma, 18 settembre - 24 ottobre 2003 Prof. Gianni Di Pillo Prof. Laura Palagi Dipartimento di Informatica e Sistemistica Universita` di Roma “La Sapienza” EXERCISE Each student must prepare its own exercise

2. Exercise 1 Blending problem Multi plant problem Given the problems:

3. Mathematical model for blending Max 40 x 1 + 30 x 2 + 20 x 3 + 40 x 4 + 35 x 5 x 1 + x 2 <= 200 x 3 + x 4 + x 5 <= 250 8.8 x 1 + 6.1 x 2 + 2 x 3 + 4.2 x 4 + 5 x 5 <= 6(x 1 + x 2 + x 3 + x 4 + x 5 ) 8.8 x 1 + 6.1 x 2 + 2 x 3 + 4.2 x 4 + 5 x 5 >= 3(x 1 + x 2 + x 3 + x 4 + x 5 ) x 1, x 2, x 3, x 4, x 5 >= 0 VEG1 = x 1, VEG2 = x 2, OIL1 = x 3, OIL2 = x 4, OIL3 = x 5

4. Excel table for blending

5. Mathematical model for the multi plant max 10 x 1 + 15 x 2 + 10 x 3 + 15 x 4 4 x 1 + 2 x 2 <= 80 5 x 3 + 3 x 4 <= 60 2 x 1 + 5 x 2 <= 60 5 x 3 + 6 x 4 <= 75 4 x 1 + 4 x 2 +4 x 3 + 4 x 4 <= 120 x 1, x 2, x 3, x 4 >= 0 More than two variables: we can solve it with the Solver

6. Excel table for the multiplant

7. Analyze the Excel’s report 1.Which constraints are binding 2.Analyze the reduced costs (for each variables) 3.Analyze the shadow prices (for each constraints) 4.Propose profitable changes to the coefficients 5.Verify (using Excel) the validity of the proposal For each problem:

8. Exercise 2 Given the multiperiod production problem A company produces footballs and it must decide how many footballs to produce each month. It has decide to use a six-month planning horizon.

9. A multiperiod production problem The forecasted demand for the next six months are 10.000, 15.000, 30.000, 35.000, 25.000 and 10.000. The company wants to meet these demands on time, knowing that it currently has 5000 footballs in inventory and it can use a given month’s production to help meet the demand for that month. During each month there is enough production capacity to produce up to 30.000 footballs, and there is enough storage capacity to store up to 10.000 footballs at the end of the month.

10. A multiperiod production problem The production costs per footballs for the next six months are EURO 12.50, 12.55, 12.70, 12.80, 12.85 and 12.95 respectively. The holding cost per football held in inventory at the end of any month is figured at 5% of the production cost for that month. The company wants to determine the production schedule that minimizes the overall costs

11. Answer the following 1.Write the Excel table with data of the problem 2.Write the mathematical model: defining the objective function and the constraints 3.Write the model in Excel 4.Solve the problem using Excel’s Solver 5.Produce the Excel’s reports 6.Make some sensitivity analysis using Excel solver and/or the reports (it is up to you !)