ANSWERS AND HINTS TO PROBLEMS

Transcription

1 APPENDIX 1 ANSWERS AND HINTS TO PROBLEMS 2.1 Period II Demand a) b) Month I a MAD Month I b Forecast for month 5 after month 2: ' 0.24 == a5 == 234, h5 == 10.4, MADs == 33.5, a == 41.9, Months 6-8: mean == 764.4, standard deviation == a) Define We obtain

2 186 ANSWERS AND runts It follows that both fl and gl will approach zero. b) Define ht =at -c-d (t+i). We obtain h t = (I-a)ht-I -d = (I-a)ht-I +a(-dl a), and from our results on exponential smoothing it follows that hi will approach -dla in the long run. 3.1 a) b) QA = 1000, Q8 = 939. The relative cost increase is given by ~ =!(.J125/200 +.J2001l25) = C 2 For product A the correct cost is C =.J = $15179 so the cost increase is $ ~ =!(.JM +.J1I0.6) = C C Q. Q. Q. Q. d(s S) =-1 VI +-1 V2 +-1 v3 +-1 v ' Q 3.4 C Q. Q. Q. d(s S) =-lvi+-\v2+-\v ' Q Q= 24d(SI+S2) i(vi +3v2 +8v3) 3.5 a) Period t I <it k=t k=t k=t Deliveries 65 in period I, 80 in period 3, and 90 in period 5. Cost $380.

3 ANSWERS AND HINTS 187 b) Two periods ( )/2 = 70 ~ 100, Three periods ( )/3 = 73.3 > 70, delivery in period 3. Two periods ( )/2 = 70 ~ 100, Three periods ( )/3 = 106.7> 70, delivery in period 5, i.e., same solution. c) Two periods 40 ~ 100, Three periods > 100, delivery in period 3. Two periods 40 ~ 100, Three periods > 100, delivery in period 5, i.e., same solution. 3.6 a) Period t d. k=t k=t+l k=t Deliveries 50 in period 1, 165 in period 3, and 170 in period 5. Cost $380. b) Two periods ( )/2 = 60 ~ 100, Three periods ( )/3 = > 60, delivery in period 3. Two periods ( )/2 = 67.5 ~ 100, Three periods ( )/3 = 85> 67.5, delivery in period 5, Two periods ( )/2 = 62.5 ~ 100, i.e., same solution. 3.7 a) Period t d k=t k=t k=t k=t k=t k=t k=t Deliveries 10 in period 1, 36 in period 2, and 19 in period 4. Cost $436. b) Two periods ( )/2 = 98 ~ 100, Three periods ( )/3 = 97.3 ~ 98, Four periods ( )/4 = 94 ~ 97.3, Five periods ( )/5 = 91.2 ~ 94, Six periods ( )/6 = 89.3 ~ 91.2,

4 188 ANSWERS AND HINTS Seven periods ( )n = 86.9 S 89.3, i.e., just one delivery in period 1. Cost $608. Note the large difference due to the special demand structure. 3.8 We obtain Il' = 200 and a' = 2 7 ~1t/2 40 = a) b) 3.9 a) b) c) 3.10 a) b) SS = :::: 104, and R = 304. S2 = 92 percent, S2 = 96 percent, and S2 = 99.7 percent. 68, 104, and , 98.3, and 99.2 percent. 77.3, and 99.9 percent. SS = = 35. SI = cp(0.7) = 75.8 percent. S2 = 96.4 percent. Let the lead-time demand be uniform on (100 - a, a). We obtain i.e., a = ~7500 = S( = ( ) = 70.2 percent. The average backordered quantity per cycle is E(B) = 181"6 (u -135) du = 7.686, ( and S2 = /200 = 96.2 percent a) Initially Il' = 800 and a' = 100. We get S2 = 96.7 percent. After reducing the leadtime Il' = 400 and a' = 50J2 = We get S2 > percent. We obtain the average stock on hand as The stock on hand is increasing from to 750. b) R can be reduced to 419. The stock on hand is a) Applying (4.12) and (4.13) we obtain T( = , T2 = , T3 = , and the costs C( = , C2 = , C 3 = The sum of these costs C = is a lower bound for the total costs. b) Assume that the independent solution is feasible and that we start production of product 3 at times t', t' , t' + 2' etc. Assume that the first production of product 2 after time t' starts at time t'+a(. Clearly a(2! a3 = But the next time the time difference is a2 = a( + T2 - T3 = a( The time after that it is a3 = a , etc. After some time we will obtain for some n, 0 < an < a3 which means that the solution is infeasible.

5 ANSWERS AND HINTS 189 c) Applying (4.17)-(4.19) we obtain Tmin = 7.292, and t = , i.e., Top! = d) t = The corresponding total costs per unit of time are C = Starting with W = we obtain n. = 1, n2 = n3 = 2. Next we get the final solution W = with the same multipliers and the total costs C = The solution is feasible if we produce items 2 and 3 in different basic periods. In basic periods where product 1 and 2 are produced, the time required is cr. + cr2 = = < In basic periods where product 1 and 3 are produced, the time required is cr. + cr3 = = < a) Change first to the equivalent R~ = 2 such that IP~o - R~ is a multiple of Q. 5.2 Next we use (5.4) to obtain R ~ = 8 and R ~ = 20. b) For both policies we have: Installation I order at times 7, 17,27,..., and installation 2 at times 7, 27, 47,... c) At times 6, 26, 46,... No equivalent installation stock policy exists, since the echelon stock policy is not nested. "tern A Period !Lead-time = 1 Gross requirements larder quantity = 30 Scheduled receipts Projected inventory Planned orders temb Period !Lead-time = 1 Gross requirements Iorder quantity = 30 Scheduled receipts Projected inventory Planned orders temc Period !Lead-time = 1 Gross requirements larder quantity = 90 Scheduled receipts Projected inventory Planned orders

6 190 ANSWERS AND HINTS 5.3 a) Item A Period I Lead-time = I Gross requirements Order quantity = 10 Scheduled receipts Safety stock = 10 Projected inventory II II Planned orders ltemb Period I [Lead-time = I Gross requirements Prder quantity = 10 Scheduled receipts ~afety stock = 10 Projected inventory Planned orders Iteme Period I ~ad-time = I Gross requirements Order quantity = 20 Scheduled receipts Safety stock = 20 Projected inventory Planned orders No delayed orders. b) tern A Period I Lead-time = I Gross requirements Order quantity = 10 Scheduled receipts Safety time = I Projected inventory I I Planned orders ~temb Period I !Lead-time = I Gross requirements prder quantity = 10 Scheduled receipts ~afety time = I Projected inventory Planned orders teme Period I [Lead-time = I Gross requirements Order quantity = 20 Scheduled receipts Safety time = I Projected inventory Planned orders The order for item A in period I is delayed one period.

7 ANSWERS AND HINTS a) We have e, = e2 = 1. Since A2/e2 > A,/e" the constraint Q2 ~ Q, will be satisfied automatically if we optimize the items separately in the relaxed problem. We obtain Q; = ~2A,dl e, = 20J5 == 44.72, and Q; = ~2A2d I ez = 100. A lower bound for the costs is ~2A,de, +~2Azde2 == Without any lack of generality we can assume that 20J51.,fi < q ::; 20J5.,fi. This means that Q, = q. Furthermore we should choose Qz = 2q for q ~ 50 I.,fi and Qz = 4q otherwise. We obtain C(q) = 5~+ 2~0, 20.J51.,fi < q < 501.,fi, We obtain the optimal q == 48.30, Q, = q, and Qz =2q. The optimal costs are C == , i.e., only 0.13 percent above the lower bound. b) From (5.17) we obtain k =.J5. Since.J5 12 $. 31.J5 it is optimal to have k = 2. Inserting in (5.14) and (5.15) we get the same solution as in a). Le., b) p(2)=3,p(l)=2. We obtain A3 =A3, h; =e3,and

8 192 ANSWERS AND HINTS and 5.6 Note first that 112 = 02 = 0 and that S; does not change. Consequently (5.40) degenerates to Since the costs for S2 > S; are increasing with S2, the optimal S2 must occur for S2 ~ S;. From (5.37) we obtain This is the cost for a single-echelon system, since there is no stock at installation 2, and since the possible order-up-to level at installation 1 is bounded by S2' We obtain the optimal S2 from the condition J S; ~~i') = _b_, _ =..!E... = 0.870, ~l 0, h,+b, 1l.5. S " " 3 (;' Th A '.e., 2=1l,+1.l3 0,= ,,6=73.8. ecostsarec2( S 2)= a) Using (5.48) we obtain P(lLo = 1) = , P(lLo = 2) = , P(ILo = 3) = The holding costs are determined from (5.49). For Poisson demand the fill rate is equivalent to the ready rate, i.e., P(lLo > 0) = = 42.3 percent. b) From (5.51) and (5.52) we obtain E(lLo) = and E(Wo) =0.672/3 = Consequently, II = I2 = For retailer 1 we obtain P(lLI = I) = , P(lLI = 2) = , P(lLI = 3) = The holding costs are and the backorder costs The fill rate is 87.4 percent. For retailer 2 we get P(I~ = 1) = , P(I~ = 2) = , P(I~ = 3) = , P(I~ = 4) = , P(l~ = 5) = The holding costs are and the backorder costs The fill rate is 89.8 percent.

9 APPENDIX 2 NORMAL DISTRIBUTION TABLES x <p(x) = r;::-:: e 2, <p( -X) = <p(x). ",,21t x ci>(x) = J <p(v)dv, ci>(-x) = 1- ci>(x). ~ G(X) = J (v - x)<p(v)dv = <p(x) - x(l- ci>(x», G(-X) = G(X) + x. x ~ 1 [ ] 1 H(x)= JG(v)dv=- (x 2 +1)(l-cI>(x»-x<p(x),H(-x)=-H(x)+-(x 2 +1). x 2 2 X <p(x) ci>(x) G(x) H(x) X <p(x) ci>(x) G(x) H(x)

10 194 NORMAL DISTRffiUTION x <p(x) CI>(x) G(x) H(x) x <p(x) CI>(x) G(x) H(x)

11 NORMAL DISTRIBUTION 195 x <p(x) <I>(x) G(x) H(x) x <p(x) <I>(x) G(x) H(x) Ql

12 196 NORMAL DISTRIBUTION x <p(x) <l>(x) G(x) H(x) x <p(x) <l>(x) G(x) H(x)

13 INDEX ABC inventory analysis, 179 Adaptive forecasting, 21 Afentakis, P., 147, 170 Aggregation, 177, 179 Andersson, J., 167, 170 Arborescent system, 116 Archibald, B. C., 74, 86 Assembly system, 117 Atkins, D. R., 111 AutoRegressive Moving Average (ARMA),16 Axsater, S., 30, 47, 86, 100, 103, 111, 123, 125, 126, 132, 139, 148, 149, 165, 167, 169, 170, 172, 184 Backlog, see Backorders Backorder costs, see Shortage costs Backorders, Baganha, M. P., 135, 170 Baker, K. R., 47,86,133, 170 Balintfy, J. L., 111 Base stock policy, 30 Basic period, 91-95, ,144 Batch quantities, 30-49, 77-86, , , see also Order quantities Benton, W. C., 36, 86 Berry, W. L., 4, 113, 172, 184 Bertrand, J. W. M., 103, 112 Beyer, D., 74, 82, 86 Billington, P. J., 95, 112 Bill of material (BOM), , 127 Blackburn, J. D., 49, 86, 146, 147, 170 Bomberger, E. A., 97, 98, 99, 100, 101, 112 Bomberger's problem, 97 Bowman, R. A., 102, 112 Box, G. E. P., 16,22 Box-Jenkins approach, Browne, S., 51, 86 Brown, R. G., 3 Buchanen, D. J., 74, 87 Buffer stocks, see Safety stocks Bullwhip effect, Cachon, G. P., 167, 170 Candea, D., 3 Can-order policy, 111 Capacity constraints, , Capacity Requirements Planning (CRP), Capital costs, see Holding costs Carrying costs, see Holding costs Chen, F., 65, 87, 135, 157, 165, 171 Chikful, A., 3 Clark, A. J., 151, 157, 159, 171 Clark-Scarf model, Cohen,M.A., 135, 169, 170,171 Compound Poisson demand, Constant demand model, 6-7 Continuous review, 27-28, 51-68, 72-86, , Costs, Croston, J. D., 16, 22 Customer service, see Service levels Cycle counting, 177 Cycle time, Cyclic schedule, , Dannenbring, D. G., 21, 22 Dekker, R., 113 De Kok, A. G., 157, 172 Del Vecchio, A., 135, 172 Demand forecasts, see Forecasting Demand models, 6-8

14 198 INDEX Demand processes, De Matteis, J. J., 47, 87 Dependent demand, see Material Requirements Planning Deuermeyer, B., 163, 171 Distribution Requirements Planning (DRP), 132, see also Material Requirements Planning Distribution system, Dobson, G., 102, 112 Doll, C. L., 100, 112, 113 Dynamic programming, 43-45, Echelon holding costs, 138, 147 Echelon stock, Echelon stock policy, Economic Lot Scheduling Problem (ELSP), Economic Order Quantity (EOQ), see Order quantities Education, Electronic Data Interchange (EDI), 115 Elmaghraby, S. E., 100, 102, 112 Enterprise Resource Planning (ERP), 133 EOQ, see Order quantities Eppen, G. D., 95, 112, 157, 171 Erkip, N. K., 150, 171 Exponential smoothing, 8-11 Exponential smoothing with trend, Federgruen, A., 45, 69, 77, 79, 87, 88, 102,111,112,151,156,157,171 Feller, W., 49, 87 Fill rate, 56-57, Forecast errors, Forecasting, 5-23 adaptive forecasting, 21 Box-Jenkins approach, correlated stochastic deviations, demand models, 6-8 exponential smoothing, 8-11 exponential smoothing with trend, li- B forecast errors, initial forecast, 10, 12, 18 manual forecasts, Mean Absolute Deviation (MAD), monitoring forecasts, moving average, 8 sporadic demand, 16 Winters' trend-seasonal method, Forrester, J. W., 135, 171 Forsberg, R., 167, 171 Frenk, J. B. G., 113 Gallego, G., 102, 112 Gardiner, J. S., 22 Gardner, E. S., 21, 22 Gavish, B., 170 General system, 118 Goyal, S. K., 107, 112 Graves, S. C., 3, 87, 112, 113, 148, 164, 170, 171, 172 Groenevelt, H., 112 Groff, G., 87 Gross requirements, Grouping of items, Grubbstrijm, R. W., 26, 87 G(x),54, Hadley, G., 3, 74, 87 Harris, F. W., 31, 87 Hausman, W. H., 150, 171 Hax, A., 3 Heyman, D. P., 87 Hill, R. M., 74, 87 Holding costs, 25-26, see also Echelon holding costs Holt, C. C., Hopp, D. L Hsu, W., 102, 112 H(x),64, Hyndman, R. J., 22 Iglehart, D. 82, 86, 87 Implementation, 82-86, ABC inventory analysis, 179 cycle counting, 177 education, grouping of items, inventory records, performance evaluation, periodic counting, selective inventory control, 179 short-run consequences of adjustments, simulation. 182 step-by-step implementation, 181 Inderfurth, K., 172 Initial forecast. 10, Installation stock policy, 120 Intermittent demand, see Sporadic demand Inventory costs, Inventory level, 27

15 INDEX 199 Inventory position, 27 Inventory position distribution, 52, 66-70, 167 Inventory records, Inventory turnover, 177 Iyogun, P., III Jackson, P., 110, 112 Jenkins, G. M., 16,22 Johansen, S. G., 74, 87 Johnson, L. A., 3, 22 Joint optimization of order quantity and reorder point, Joint replenishments, Juntti, L., 125, 126, 170 KANBAN policy, 29, 96,120,126,131, Karmarkar, U. S., 103, 112, 170 Katalan, Z., 102, 112 Kleindorfer, P. R., 171 Kolen, A., 88 Kunreuther, H., 45,87 Lagrangean optimization, , Lateral transshipments, 169 Lead-times, 27, Least Unit Cost (LUC), 47 Lee, H. L., 87,135, 163, 165, 169, 171, 172 Little's formula, 161 Loss function, 54, Lost sales, Lot sizing, see Order quantities Love, R. F., 74, 87 Love, S. F., 3 Lundin, R., 45,87 MAD,17-19 Makridakis, S., 22 Manne, A. S., 95, 112 Manual forecasts, Manufacturing Resource Planning (MRP), 132 Marklund, J., 167, 172 Martin, R. K., 95, 112 Master Production Schedule (MPS), 127 Material Requirements Planning, bill of material, , 127 Capacity Requirements Planning (CRP), Distribution Requirements Planning (DRP),132 Enterprise Resource Planning (ERP), 133 gross requirements, Manufacturing Resource Planning (MRP),132 Master Production Schedule (MPS), 127 MRP II,132 nervousness, 132 net requirements, 128 pegging requirements, 131 planned orders, projected inventory, 128 Rough Cut Capacity Planning (RCCP), safety time, 130, 150 scheduled receipts, 128 time horizon, 127 Maxwell, W., 112 McClain, J. 0., 3, 112 Meal, H. C., 45, 47, 48,88,89 Mean Absolute Deviation (MAD), Mendoza, A. G., 47,87 METRIC approach, Millen, R. A., 49, 86, 146, 147, 170 MOD-METRIC, Moinzadeh, K., 163, 165, 171, 172 Monitoring forecasts, Montgomery, D. C., 3, 22 Moon, I., 102, 112 Morton, T., 45, 87 Moving average, 8 MRP, see Material Requirements Planning MRP II, 132 Muckstadt, J. A., 102, 110, 112, 140, 150, 163,172 Multi-echelon systems, 34-36, Multi-stage systems, 34-36, Naddor, E., 3 Nahmias, S., 3, 22, 87, 169, 172 Nervousness, 132 Nested policy, 123 Net requirements, 128 Net stock, see Inventory level Newsboy model Normal distribution. 51, Nuttle, H. L. W., 139, 148, 170

16 200 INDEX Optimality of ordering policies, 82 Ordering costs, 26 Ordering policies, 27-30, , see also Ordering systems Ordering systems, 27-30, 1I9-135 comparison of installation stock and echelon stock policies, continuous or periodic review, different ordering policies, echelon stock reorder point policies, installation stock reorder point policies, 120 inventory position, 27 KANBAN policy, 29, 96, 120, 126, 131, Material Requirements Planning, , see also Material Requirements Planning Ordering system dynamics, Order point, see Reorder point Order quantities, coordinated replenishments, joint replenishments, powers-of-two policies, production smoothing, Roundy's 98 percent approximation, 107-1I0 Order quantities, multi-echelon systems, constant demand, Roundy's 98 percent approximation, time-varying demand, Order quantities, single-echelon systems, 30-49,77-86 backorders allowed, classical economic order quantity, finite production rate, joint optimization of order quantity and reorder point, Least Unit Cost (LUC), 47 quantity discounts, sensitivity analysis, several stages, Silver-Meal heuristic, 45-47, time-varying demand, updating in practice, Wagner-Whitin algorithm, 43-45, Order-up-to-S policy, 30 Orlicky, J., 3, 133, 172 Overage cost, Padmanabhan,P., 171, 172 Palm, C., 76, 87 Park, S., 36, 86 Pegging requirements, 131 Performance evaluation, Periodic counting, Periodic review, 27-28, 68-70, Peterson, R., 3, 22, 88, 1I3, 172, 184 Planche, R., III, II2 Planning horizon, 45 Plossl, G. W., 3 Poisson demand, Poisson distribution, 49 Porteus, E. L., 87,178,184 Powers-of-two policies, 91-95, , J07-110, Probabilistic demand, see Safety stocks Production smoothing, Product structure, Projected inventory, 128 Pull system, Push system, Pyke, D., 3, 22, 88, II3, 172, 184 (Q, R) policy, see (R, Q) policy Quantity discounts, Ready rate, 57, Reinsell, G. C., 22 Renberg, B., III, 112 Reorder point, 28-30, see also Safety stocks Repairable items, 169 Replenishment quantities, see Order quantities Review period, 27 Rinnooy Kan, A., 3 Rolling horizon, 45, 48-49, 127 Rosling, K., 65, 72, 74, 80, 82, 87,123, 126, 132, 147, 157, 170, 172 Rough Cut Capacity Planning (RCCP), Roundy, R., 2, 95, 102, 105, 107, 109, HO, 1I2, 140, 141, , 174 Roundy's 98 percent approximation, 107- HO, (R, Q) policy, Safety capacity, 150 Safety factor, 58

17 INDEX 201 Safety stocks, multi-echelon systems, distribution systems, exact solution, distribution systems, METRIC approach, optimization, safety stocks and safety times, 150 serial systems, Clark-Scarf model, serial systems, (R, Q) policies, Safety stocks, single-echelon systems, 49-82, continuous review (R, Q) policy, inventory level distribution, continuous review (s, S) policy, inventory level distribution, demand processes, determining reorder points for given service levels, determining reorder points for given shortage costs, joint optimization of order quantity and reorder point, lost sales, newsboy model, optimality of ordering policies, 82 periodic review, safety factor, 58 service levels, shortage costs, stochastic lead-times, updating in practice, Safety time, 130, 150, see also Safety stocks Samroengraja, R., 135, 171 Satir, A. T., 107, 112 Scarf, H., 151, 157, 159, 171 Scheduled receipts, 128 Schrage, L., 157, 171 Schwarz, L. B., 163, 171, 172 Seasonal index, 7,13-15 Selective inventory control, 179 Sensitivity analysis, 31-33, see also Powers-of-two policies Serial system, 116 Service constraints, 26, 56-57, Service levels 26, 56-57, Sethi, S. P., 82, 86 Setup costs. 26 Shapiro. J. F Sherbrooke. C. C., Shortage costs , Silver. E. A , , Ill Silver-Meal heuristic, Simulation, 182 Single period problem, see Newsboy model Smoothing constant. 8-10,12, 14 Smoothing production Sobel. M Song, J. S., Spearman,M.L., 120, 172 Spence, A. M S policy, 30 Sporadic demand, 16 (S - 1. S) policy. 30 (s. S) policy Standard deviation. 17 Step-by-step implementation, 181 Sterman Stochastic lead-times, Stockout costs, see Shortage costs Supply Chain Management, Svoronos, A. 163, 172 Tersine, R. J., 3, 184 Thomas, L. J.. 3, Thorstenson, A., 74, 87 Tijms, H., 112 Time-varying demand , Towill, D., Trend model, 7 Trend-seasonal model. 7 Tzur, M., 45, 87 Underage cost, Van Hoesel. S. 88 Van Houtum, G. J., 156, 157, 172 Variance, 17 Veatch. M. H Veinott, A., Verrijdt, J. H. C. M. 157, 172 Viswanathan, S., Ill, 113 Vollman. T. E Wagelmans. A Wagner. H. M., ,145,146 Wagner-Whitin algorithm, 43-45, Waiting time, 39

18 202 INDEX Wein, L. M., 120, 172 Whang, S., 171, 172 Wheelwright, S. C., 22 Whitin, T. M., 3, 43, 45, 47, 48, 49, 74, 87,88,89,145,146 Whybark, D. C., 4,100,112,113,172, 184 Wight, O. W., 3 Wildeman, R. E., 107, 113 Williams, 1. F., 140, 172 Wilson, R. H., 31, 88 Winters, P. R., 13, 22 Winters' trend-seasonal method, Woodruff, D. L., 172 Zangwill, W. I., 45, 88 Zhang, W. F., 167, 170 Zheng, Y. S., 30, 65, 69, 77, 79, 82, 87, 88, 157, 165, 171 Zijrn, W. H. M., 156, 172 Zipkin, P. H., 3, 4, 51, 77, 86, 88, 102, 103,113,151,156,157,163,171,172