There exist some other heuristic algorithms for lot sizing purposes. Two examples are Least Unit Cost and Part Period Balancing. Since the main purpose of this course was to show the existence of different ways of determining lot sizes, we will not cover these other heuristics. Have you realized that, so far, we have assumed no capacity constraints in producing or ordering the requirements? So, in the next topic, we will consider this situation. 4. LOT SIZING WITH CAPACITY CONSTRAINTS As mentioned above, in real life companies usually have capacity constraints. Therefore, assume that in addition to known requirements (r1,,rn) in each period, there are also production capacities (c1,,cn). Hence, we now wish to find the optimal production quantities (y1,,yn) subject to the constraints yici, for 1in. Consider the same example, r=(52,87,23,56), but now suppose that the production capacity in each period is c=(60,60,60,60 ). First, we must determine if the problem is feasible. That is, whether at least one solution exists. On the surface of the problem looks solvable, as the total requirements over the four periods is 218 and the total capacity is 240 . But this problem is infeasible because the most that can be produced in the first two periods is 120 , but the requirements for those periods sum to 139. Therefore, before even trying to solve the problem, we need to check the feasibility condition of the problem. This is checked as follows: i=1jcii=ijriforj=1,,n EXAMPLE r=(20,40,100,35,80,75,25)c=(60,60,60,60,60,60,60) The feasibility condition test is satisfied (make sure you know how to perform this check). We now present: (1) an approximate lot-shifting technique to obtain a feasible solution. LOT SHIFTING TECHNIQUE PROCEDURE The idea of this method is to back-shift demand from periods in which demand exceeds the capacity to prior periods in which there is excess capacity. This process is repeated for each period in which demand exceeds capacity until we construct a new requirements schedule in which lot-for-lot is feasible. In the example, the first period in which demand exceeds capacity is period 3 . We replace r3 with c3 (which is the capacity available). The difference of 40 units must now be redistributed back to periods 1 and 2 . We consider the first prior period, which is period 2. There are 20 units of excess capacity in period 2, which we absorb. We still have 20 units of demand from period 3 that are not yet accounted for; this is added to the requirements for period 1. Out of the 40 units in excess in period 3, 20 units were redistributed to period 2 and 20 units to period 1. Summarizing these results up until now, we have, Notice that since the r vector has changed, we have renamed it as r. The next period in which demand exceeds capacity is period 5 . The excess demand of 20 units can be back-shifted to period 4 . Finally, the 15 units of excess demand in period 6 can be backshifted to periods 4 (5 units) and 1 (10 units). The feasibility condition guarantees that this process leads to a feasible condition. This leads to, Hence, the modified requirements schedule obtained is, r=(50,60,60,60,60,60,25) Do you think this is the best schedule you can get? Why or why not? Homework Exercise \#5 Consider the Valve Casing Assembly from the Harmon Music Company example. a. Suppose that the production capacity in any week is 100 valve casings. Using the algorithm presented in this section, determine the planned order release for the valve casing assemblies. b. What gross requirements schedule for the valves does the lot sizing you obtained in part (a) give? c. Suppose that the production capacity for the valves is 200 valves per week. Is the gross requirements schedule from part (b) feasible? If not, suggest a modification in the planned order release computed in part (a) that would result in a feasible gross requirements schedule for the valves. There exist some other heuristic algorithms for lot sizing purposes. Two examples are Least Unit Cost and Part Period Balancing. Since the main purpose of this course was to show the existence of different ways of determining lot sizes, we will not cover these other heuristics. Have you realized that, so far, we have assumed no capacity constraints in producing or ordering the requirements? So, in the next topic, we will consider this situation. 4. LOT SIZING WITH CAPACITY CONSTRAINTS As mentioned above, in real life companies usually have capacity constraints. Therefore, assume that in addition to known requirements (r1,,rn) in each period, there are also production capacities (c1,,cn). Hence, we now wish to find the optimal production quantities (y1,,yn) subject to the constraints yici, for 1in. Consider the same example, r=(52,87,23,56), but now suppose that the production capacity in each period is c=(60,60,60,60 ). First, we must determine if the problem is feasible. That is, whether at least one solution exists. On the surface of the problem looks solvable, as the total requirements over the four periods is 218 and the total capacity is 240 . But this problem is infeasible because the most that can be produced in the first two periods is 120 , but the requirements for those periods sum to 139. Therefore, before even trying to solve the problem, we need to check the feasibility condition of the problem. This is checked as follows: i=1jcii=ijriforj=1,,n EXAMPLE r=(20,40,100,35,80,75,25)c=(60,60,60,60,60,60,60) The feasibility condition test is satisfied (make sure you know how to perform this check). We now present: (1) an approximate lot-shifting technique to obtain a feasible solution. LOT SHIFTING TECHNIQUE PROCEDURE The idea of this method is to back-shift demand from periods in which demand exceeds the capacity to prior periods in which there is excess capacity. This process is repeated for each period in which demand exceeds capacity until we construct a new requirements schedule in which lot-for-lot is feasible. In the example, the first period in which demand exceeds capacity is period 3 . We replace r3 with c3 (which is the capacity available). The difference of 40 units must now be redistributed back to periods 1 and 2 . We consider the first prior period, which is period 2. There are 20 units of excess capacity in period 2, which we absorb. We still have 20 units of demand from period 3 that are not yet accounted for; this is added to the requirements for period 1. Out of the 40 units in excess in period 3, 20 units were redistributed to period 2 and 20 units to period 1. Summarizing these results up until now, we have, Notice that since the r vector has changed, we have renamed it as r. The next period in which demand exceeds capacity is period 5 . The excess demand of 20 units can be back-shifted to period 4 . Finally, the 15 units of excess demand in period 6 can be backshifted to periods 4 (5 units) and 1 (10 units). The feasibility condition guarantees that this process leads to a feasible condition. This leads to, Hence, the modified requirements schedule obtained is, r=(50,60,60,60,60,60,25) Do you think this is the best schedule you can get? Why or why not? Homework Exercise \#5 Consider the Valve Casing Assembly from the Harmon Music Company example. a. Suppose that the production capacity in any week is 100 valve casings. Using the algorithm presented in this section, determine the planned order release for the valve casing assemblies. b. What gross requirements schedule for the valves does the lot sizing you obtained in part (a) give? c. Suppose that the production capacity for the valves is 200 valves per week. Is the gross requirements schedule from part (b) feasible? If not, suggest a modification in the planned order release computed in part (a) that would result in a feasible gross requirements schedule for the valves