The penalty points

The ten-point system separates defects by the warp and weft directions, which adds another layer of fabric grading. This system is challenging in everyday use. Penalty points are determined by the number of defects and length of each defect (see Table 5.4). If the penalty point total does not exceed the total yardage, the fabric is... 

CostiUo-Manzano, J.L, Castro-Nun5, M., and Pedregal, D.J. 2010. An econometric analysis of the penalty points system driver s licence in Spain. Accident Analysis and Prevention, 42(A), 1310-1319. 

If more than one candidate is left, compute the Penalty points for each candidate nj using the equation ... 

The weighting of the orders that contribute to the first component of the aggregated order state vector can be done in the following way. We define the penalty points of an order as the penalty costs for that order divided by p, so the penalty points may be 1,2,3 etc. Now we define r as the sum of the penalty points of those orders that have to be produced by the end of the period or in previous periods. The other elements of the order state vector, r2,..,rn, are defined as the sum of the number of orders with a particular residual lead time. For instance, r z is the sum over all priorities of the number of orders with a residual lead time of 2 periods. The description of the transition probabilities for this order state vector r will be slightly more complex, but on the other hand, there is a large state space reduction. 

Because of the conq>etition with the other types, there is not always capacity available for a type for which the nuniber of penalty points would justify a setup. Therefore, it will be wise to choose the penalty points and the level x,- in such a way that, if they were chosen the same in the uncapacitated situation, they would lead to a more frequent production. 

The elements sketched above will give an indication for the choice of the penalty points and the required minimum x,. Finding a good choice will now be a question of trial-and-error combined with some intuition. Now we will describe the extended (x,7>-rule for a given set of penalty points, with fj, i=l,..,M, j=l,..,N, the penalty points for an order of type i with a residual lead time of J periods. We also assume that the values for T,-, i=l,2,..,M are determined according to the uncapacitated (x,7>-nile according to the method described in 4.2.4.4. 

In the extended (x,7>-rule, we now choose mcl=6 and ecl=6. The matrix Q with the penalty points fij for type i and residual lead time J is chosen in more or less the some way as in Exanqile 5.1., where orders which cannot yet be produced, and also some of the orders with a small probability for production, are given no penalty point The best... 

In the Examples 5.1 and 5.2 the capacity restrictions were very tight. In order to see whether the differences in average costs also appear in the situation in which the capacity is rather loose, we will consider a third example. Therefore we take again Example 5.1, but now with an available capacity of 15 instead of 12. With one set-up every period, about 72 percent (10.75 out of 15) of the available capacity will be used. The cyclic rules w,- are not affected by the change in capacity, therefore (5.2.18) still holds and we can also use the same semi-fixed cycle. For the extended (x,7>-mle we used the same set of values for the pairs iXi,Ti) as in (5.2.20). Because it will not be very profitable to produce orders with a longer residual lead time, the penalty points for these orders have been decreased and after some trials we have found the following set of values for the penalty points ... 

The semi-fixed cycle we will use in this example is the same cycle as the one in Example 5.2 with 107 periods. The penalty points for the extended overtime (x,T)-nile and the pairs (xiTi) and the level of mcl and eel are also the same as in Example 5.2, not because this would be the best choice, but to illustrate the robusmess of this rule. The results for this example are given in Table 5.6. 

For this example, we will only consider an (x,7)-rule with a multi-type level L=1 and with no extra penalty points per period, C,=0, i = l,2,3,4. The choices for the penalty points for an order of type i and priority j are denoted by fij. We will consider three different choices for the matrices XF, which contains the values for the minimum level and the penalty points for the different types. 

We will consider one choice for the matrix XF for the multi-type level L = 1 and three choices for the matrices XFCV for L=2. The matrices XFCW contain the values for the tninimiitn level x, the penalty points for the different types and priorities q, the extra penalty points per period C and the maximum level v that allows another production. We will consider the following choices ... 

The extended (x.T)-nile for this situation contains a lot of well-known elements from the various (x,r)-rules described in the previous chapters. Again, the basis is a separate (x,7)-nde for every type which says that we can produce the orders which have to be delivered within T periods if there is a sufficient number of penalty points for the type. Due to the firm-initiated lead times it will often happen that the due date is the same for almost all the orders for a specific type. The first orders for the type obtained for instance a lead time of 4 periods, the orders arrived in the next period a lead time of 3 periods, then 2 periods and so on. This reduces the importance of the choice of T quite a lot, because usually most of the orders that are produced are orders that have to be delivered by the end of the period and the number of orders that will be produced will depend more on the capacity limitations than on the choice of T. The choice of the penalty points is important for the differences in delivery times for the various types and priorities and for the set-ups. This choice will be described more... 

The penalty points for a type will be the sum of the penalty points of all the orders for the type and an additional constant for every period after the last production period in which there has been orders for this type. This is similar to the production model in Subsection 6.2., where we have introduced the extra constant for type i as C,-. In the model in this chapter the extra points are lost as soon as the production of the type has been started, even if not all the orders for the type have been produced. In order to promote a further production of the type for which not all of the orders have been produced in the previous period one more element is added to the penalty points for a type. This element is the sum of the penalty points of all the orders for the type which have to be delivered by the end of the first period multiplied by a constant which we will denote by D. For this type the penalty points are given by... 

In order to make a decision about the types that will be produced in the following periods, we have to consider the expected number of penalty points during those periods. This expected amount will be described in the penalty point function, one for each product type, depending on the period t, containing the sum of the penalty points of all planned orders at time r, the constant C,- multiplied by a number of periods and the sum of the penalty points of the expected future orders until t. We will assume that the future orders will all obtain a lead time that is equal to r+1, so the penalty points for these orders will only be i, where ( i may have different values for different types and priorities. Once the type has been allocated to one of the machines, the only penalty points are due to the future orders that arrive after the preliminary production period. 

For this example we want to find good values for the parameters qi and 92 in the penalty point functions for the diffraent types and priorities. For the values of 91 we will start with one of the best choices for the penalty points in Example 6.2. In this example we will use only one value for 92 which is choosen arbitrarily 92=2 for all types and priorities. This means that starting fi om the due date period, the penalty points for an order are doubled every period. We start with an arbitrary value for the minimum number of penalty points that is necessary for the production x,=10 for every type i. The extra points per period are chosen to be C,=3 for every type. These choices have been placed in a matrix XQCi, where qij indicates the q value fOT orders for type i and priority J. The 92 values are not included in this matrix. 

In this example the lead times that the firm proposes are accepted with a certain probability i4. The values of these probabilities have been given in the formulae (7.3.1) and (7.3.2). The normal available capacity on each machine will be 9 units. If no orders would be withdrawn and if we have one set-up on each machine every period, the machines would be occupied more than 90 percent (25 out of 27 units). Due to the withdrawals the real occupancy rate will be around 85 percent. For this example we will first make an arbitrary choice for the penalty points and for the additional elements. We start with ci = 3, D = 3 and mcl = 3. The 2 value is chosen arbitrarily as 2 =2 for all types and priorities. The choices for the other parameters are given in the matrix XQC. ... 

We also have seen that an arbitrary choice for the penalty points and the additional elements yields a performance that is already quite well. Most of the changes that we have considered have been quite obvious. Therefore we expect that a similar approach can be used in a lot of practical situations in which a firm produces to order. 

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