Question

Digital Imaging (DI) produces photo printers for both the professional and consumer markets. The DI consumer division recently introduced two photo printers that provide color prints rivaling those produced by a professional processing lab. The DI-910 model can produce a 4” × 6” borderless print in approximately 37 seconds. The more sophisticated and faster DI-950 can even produce a 13” × 19” borderless print. Financial projections show profit contributions of $42 for each DI-910 and $87 for each DI-950. The printers are assembled, tested, and packaged at DI’s plant located in New Bern, North Carolina. This plant is highly automated and uses two manufacturing lines to produce the printers. Line 1 performs the assembly operation with times of 3 minutes per DI-910 printer and 6 minutes per DI-950 printer. Line 2 performs both the testing and packaging operations. Times are 4 minutes per DI-910 printer and 2 minutes per DI-950 printer. The shorter time for the DI-950 printer is a result of its faster print speed. Both manufacturing lines are in operation one 8-hour shift per day.

Managerial Report

Perform an analysis for Digital Imaging in order to determine how many units of each printer to produce. Prepare a report to DI’s president presenting your findings and recommendations. Include (but do not limit your discussion to) a consideration of the following:

1. The recommended number of units of each printer to produce to maximize the total contribution to profit for an 8-hour shift. What reasons might management have for not implementing your recommendation?

2. Suppose that management also states that the number of DI-910 printers produced must be at least as great as the number of DI-950 units produced. Assuming that the objective is to maximize the total contribution to profit for an 8-hour shift, how many units of each printer should be produced?

3. Does the solution you developed in part (2) balance the total time spent on line 1 and the total time spent on line 2? Why might this balance or lack of it be a concern to management?

4. Management requested an expansion of the model in part (2) that would provide a better balance between the total time on line 1 and the total time on line 2. Management wants to limit the difference between the total time on line 1 and the total time on line 2 to 30 minutes or less. If the objective is still to maximize the total contribution to profit, how many units of each printer should be produced? What effect does this workload balancing have on total profit in part (2)?

5. Suppose that in part (1) management specified the objective of maximizing the total number of printers produced each shift rather than total profit contribution. With this objective, how many units of each printer should be produced per shift? What effect does this objective have on total profit and workload balancing?

For each solution that you develop, include a copy of your linear programming model and graphical solution in the appendix to your report.

Answer 1

1.         

 

            Capacity: 8 hours × 60 minutes/hour = 480 minutes per day

 

            Let       D1 = number of units of the DI-910 produced

                        D2 = number of units of the DI-950 produced

 

                    

 

Using The Management Scientist, the optimal solution is D1 = 0, D2 = 80. The value of the optimal solution is $6960.

 

Management would not implement this solution because no units of the DI-910 would be produced.

 

2.         Adding the constraint D1 ≥ D2 and resolving the linear program results in the optimal solution D1 = 53.333, D2 = 53.333. The value of the optimal solution is $6880.

 

3.         Time spent on Line 1: 3(53.333) + 6(53.333) = 480 minutes

 

            Time spent on Line 2: 4(53.333) + 2(53.333) = 320 minutes

 

Thus, the solution does not balance the total time spent on Line 1 and the total time spent on Line 2. This might be a concern to management if no other work assignments were available for the employees on Line 2.

 

4.         Let       T1 = total time spent on Line 1

                        T2 = total time spent on Line 2

 

            Whatever the value of T2 is, 

 

                        T1 ≤ T2 + 30

                        T1 ≤ T2 - 30

 

            Thus, with T1 = 3D1 + 6D2 and T2 = 4D1 + 2D2

 

                        3D1 + 6D2 ≤ 4D1 + 2D2 + 30

                        3D1 + 6D2 ≥ 4D1 + 2D2 - 30   

 

Hence,

                        -1D1+ 4D2 ≤ 30

                        -1D1 + 4D2 ≥ -30

 

            Rewriting the second constraint by multiplying both sides by -1, we obtain 

                        -1D1 + 4D2 ≤ 30

                         1D1 - 4D2 ≥ 30

 

Adding these two constraints to the linear program formulated in part (2) and resolving using The Management Scientist, we obtain the optimal solution D1 = 96.667, D2 = 31.667. The value of the optimal solution is $6815. Line 1 is scheduled for 480 minutes and Line 2 for 450 minutes. The effect of workload balancing is to reduce the total contribution to profit by $6880 - $6815 = $65 per shift.

 

5.         The optimal solution is D1 = 106.667, D2 = 26.667. The total profit contribution is 

                                    42(106.667) + 87(26.667) = $6800

 

Comparing the solutions to part (4) and part (5), maximizing the number of printers produced (106.667 + 26.667 = 133.33) has increased the production by 133.33 - (96.667 + 31.667) = 5 printers but has reduced profit contribution by $6815 - $6800 = $15. 

 

But, this solution results in perfect workload balancing because the total time spent on each line is 480 minutes.