Anja Wolter, Stefan Helber

Transcription

1 Generation of the Parameters for the Test Instances for the Paper entitled Simultaneous Production and Maintenance Planning for a Single Capacitated Resource facing both a Dynamic Demand and Intensive Wear and Tear Anja Wolter, Stefan Helber Leibniz Universität Hannover, Institute of Production Management Königsworther Platz 1, D Hannover, Germany anja.wolter@prod.uni-hannover.de, stefan.helber@prod.uni-hannover.de, August 9,

2 Demand is only given at the end of each macroperiod τ, i.e., the last microperiod of each macroperiod, and otherwise zero. The demand for Problem Classes A and B is given in Tables 1 and 2 respectively. Table 1: Given demand d kt for Problem Class A k\t Table 2: Given demand d kt for Problem Class B k\t The average demand d k of product k per macroperiod τ is given by d k = t T M d kt P k K. (1) The average production time tp per macroperiod is tp = tp k d k. (2) The setup times are determined as a fraction ts rel of the average production time per product and macroperiod. The setup times of the products depend on the average production time per macroperiod of the respective product. The setup times for product k are then calculated as follows: ts k = ts rel tp k d k k K. (3) The average setup time ts per macroperiod over all products is then ts = ts rel tp k d k = ts rel tp. (4) Two different wear and tear cases are assumed. In the first case all products cause the same wear and tear a on the resource. In the second case all products have a different 2

3 wear and tear on the resource in an interval [0.5 a; 1.5 a]. The wear and tear parameters for the respective cases are then (i) a k = a k K, (5) (ii) a k = a 2 + a (k 1) k K. (6) K 1 The average rate of wear and tear awr per macroperiod over all products is awr = (d k a k ). (7) Inserting equations (5) and (6) respectively into equation (7) gives then: (i) awr = a (ii) awr = a d k, (8) ( 1 2 d k + ) d k k 1. (9) K 1 Given the time between two maintenance activities TBM and the maximal state of the resource s max, the following equation also holds for the average wear and tear rate: awr = smax T BM. (10) Inserting equ. (8) into (10) and solving for a gives then for case (i): (i) a = s max T BM k K d. (11) k Inserting equation (11) then into equation (5) gives the wear and tear of the products in the case that all products cause the same wear and tear on the resource. Inserting equ. (9) into (10) and solving for a as well gives then for case (ii): (ii) a = T BM s max ( 1 K 2 d k + ). (12) K d k k 1 K 1 Inserting equation (12) into (6) gives the wear and tear parameters of the products in the case that all products have a different wear and tear. In the case of partial maintenance (β = 1) the average time for maintenance in a macroperiod is 3

4 tm = tm pm d k a k (13) with tm as the time required to restore one unit of stock of wear and tear. pm The average maintenance time per macroperiod tm is assumed to be a fraction tm rel of the average setup time ts: tm = tm rel ts. (14) Inserting equations (4) and (13) into (14) and solving for tm gives then the maintenance time per partial maintenance unit tm = tmrel ts rel pm K tp k d k K d k a k. (15) In order to create comparable problem instances, we assume that the time needed to increase the state of the resource by one unit is equal for the cases of partial and complete maintenance. Hence the following relationship holds: tm pm = tv s max. (16) Inserting equ. (15) in (16) and solving for tv gives the corresponding maintenance time for one complete maintenance tv = tmrel ts rel s max K tp k d k K d k a k. (17) The average capacity needed per macroperiod is the sum over the average production time tp, the average setup time ts and the average maintenance time tm per macroperiod and is calculated as follows c = tp + ts + tm. (18) Inserting equations (4) and (14) gives c = tp + ts rel tp + tm rel (ts rel tp) = tp (1 + ts rel + tm rel ts rel ). (19) Dividing the average capacity needed c by the assumed utilization gives the assumed capacity per macroperiod τ c τ = c Util = tp (1 + tsrel + tm rel ts rel ) Util τ P. (20) The setup costs cs k for product k are given taken from the formula for the time between orders (TBO) from the economic order quantity mode and solving for the setup/order costs: 4

5 cs k = d k T BO 2 ci k 2 k K. (21) The sum of average setup and inventory costs per time unit over all products is determined by summing over the products: csi = k K 2 d k cs k ci k, (22) Inserting equ. (21) into (22) gives then csi = k K(d k T BO ci k ). (23) The average maintenance costs per macroperiod τ are cm = (d k a k ) cm pm. (24) k K The average maintenance costs cm are assumed to be a fraction cm rel of the average setup and inventory costs csi: cm = cm rel csi. (25) Inserting equ. (25) into (24) and solving for cm gives the maintenance costs of one unit of partial maintenance cm = pm cm rel csi k K (d k a k ). (26) The same relation as explained above for the maintenance time (equation (16)) is also assumed for the maintenance costs for the two cases partial and complete maintenance, cv = s max cm pm. (27) 5