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0 | C H 16 A P T E R Aggregate and Workforce Planning And I remember misinformation followed us like a plague, Nobody knew from time to time if the plans were changed. Paul Simon 16.1 Introduction A variety of manufacturing management decisions require information about what a plant will produce over the next ye... |
1 | 554 Part III Principles in Practice The module in which we address the important question of what will be produced and when it will be produced over the long range is the aggregate planning (AP) module. As Figure 13.9 illustrated, the AP module occupies a central position in the production planning and control (PPC)... |
2 | 16.2 Basic Aggregate Planning We start with a discussion of simple aggregate planning situations and work our way up to more complex cases. Throughout the chapter, we assume that we have a demand forecast available to us. This forecast is generated by the forecasting module and gives estimates of periodic demand over ... |
3 | t¯ r St â h It (16.1) t=1 Subject to: St ⤠dt t = 1, . . . , t¯ (16.2) 556 Part III Principles in Practice X t ⤠ct t = 1, . . . , t¯ (16.3) It = Itâ1 + X t â St t = 1, . . . , t¯ (16.4) X t , St , It ⥠0 t = 1, . . . , t¯ (16.5) The objective function computes net proï¬t by mult... |
4 | Chapter 16 Figure 16.1 Input spreadsheet for linear programming example. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 557 Aggregate and Workforce Planning B C D E F G H Constants: r h I_O t c_t d_t 10 1 0 1 100 80 2 100 100 3 100 120 4 120 140 5... |
5 | (16.16) X 5 ⤠120 (16.17) X 6 ⤠120 (16.18) Inventory balance constraints I1 â X 1 + S1 = 0 (16.19) I2 â I1 â X 2 + S2 = 0 (16.20) I3 â I2 â X 3 + S3 = 0 (16.21) I4 â I3 â X 4 + S4 = 0 (16.22) I5 â I4 â X 5 + S5 = 0 (16.23) I6 â I5 â X 6 + S6 = 0 (16.24) Non-negativity constra... |
6 | (16.26) I1 , I2 , I3 , I4 , I5 , I6 ⥠0 (16.27) 558 Part III Principles in Practice Some linear programming packages allow entry of a problem formulation in a format almost identical to (16.6) to (16.27) via a text editor. While this is certainly convenient for very small problems, it can become prohibitively ... |
7 | Figure 16.2 Speciï¬cation of objectives and constraints in Excel. Solver Parameters Set Target Cell: $B$16 Equal To: Max By Changing Cells: Min $B$11:$G$13 Solve Value of: 0 Close Guess Subject to the Constraints: $B$11:$G$13 >= 0 $B$19:$B$30 <= $D$19:$D$30 $B$31:$B$36 = 0 Options Add Change Reset All Delete... |
8 | 560 Part III Principles in Practice us to limit the time the model will run and to specify certain tolerances. If the model does not converge to an answer, the most likely reason is an error in one of the constraints. However, sometimes increasing the search time or reducing tolerances will ï¬x the problem when the... |
9 | Chapter 16 Figure 16.5 Output spreadsheet for LP example. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 561 Aggregate and Workforce Planning B C D E F G H Constants: r h I_0 t c_t d_t 10 1 0 1 100 80 2 100 100 3 100 120 4 120 140 5 120 90 6 120 ... |
10 | 80 100 120 120 90 140 100 100 100 120 110 120 0 0 0 0 0 0 r*(S_1+S_2+S_3+S_4+S_5+S_6) - h*(I_1+I_2+I_3+I_4+I_5+I_6) <= <= <= <= <= <= <= <= <= <= <= <= = = = = = = 80 100 120 140 90 140 100 100 100 120 120 120 0 0 0 0 0 0 d_1 d_2 d_3 d_4 d_5 d_6 c_1 c_2 c_3 c_4 c_5 c_6 Note: X_t, S_t and I_t must be >= 0 Sensitiv... |
11 | Microsoft Excel 12.0 Answer Report Worksheet: [BasicCap.xls]Figure 16.7 Report Created: 8/29/2007 3:11:48 PM Target Cell (Max) Cell Name $B$16 Net_Profit Original Value Final Value $0.00 $6,440.00 Adjustable Cells Cell Name $B$11 X_1 $C$11 X_2 $D$11 X_3 $E$11 X_4 $F$11 X_5 $G$11 X_6 $B$12 S_1 $C$12 S_2 $D$12 S_3 $E$... |
12 | Original Value Final Value $0.00 $6,440.00 Adjustable Cells Cell Name $B$11 X_1 $C$11 X_2 $D$11 X_3 $E$11 X_4 $F$11 X_5 $G$11 X_6 $B$12 S_1 $C$12 S_2 $D$12 S_3 $E$12 S_4 $F$12 S_5 $G$12 S_6 $B$13 I_1 $C$13 I_2 $D$13 I_3 $E$13 I_4 $F$13 I_5 $G$13 I_6 Original Value Final Value 0 100 0 100 0 100 0 120 0 110 0 120 0 80 ... |
13 | Status Not Binding Not Binding Not Binding Not Binding Not Binding Not Binding Binding Binding Binding Not Binding Binding Binding Binding Binding Binding Binding Not Binding Binding Not Binding Not Binding Not Binding Not Binding Not Binding Not Binding Not Binding Not Binding Not Binding Not Binding Not Binding Not B... |
14 | 564 Part III Principles in Practice To see how these data are interpreted, consider the information in Figure 16.8 on the seventh line of the constraint section for the capacity constraint X 1 ⤠100. The shadow price is $7, which means that if the constraint is changed to X 1 ⤠101, net proï¬t will increase by ... |
15 | 16.3 Product Mix Planning Now that we have set up the basic framework for formulating and solving aggregate planning problems, we can examine some commonly encountered situations. The ï¬rst realistic aggregate planning issue we will consider is that of product mix planning. To do this, we need to extend the model of ... |
16 | i equal to the raw materials cost of product i times a one-period interest rate to represent the opportunity cost of the money tied up in inventory; but it may make sense to use higher values to penalize inventory that causes long, uncompetitive cycle times. 566 Part III Principles in Practice We can use LP (16.28... |
17 | Input Data for Single-Period AP Example Product 1 2 Selling price Raw material cost Maximum weekly sales Minutes per unit on workstation A Minutes per unit on workstation B Minutes per unit on workstation C Minutes per unit on workstation D $90 $45 100 15 15 15 15 $100 $40 50 10 30 5 5 Chapter 16 Aggregate and... |
18 | 568 Part III Principles in Practice seeing which machine is most overloaded.7 This yields 15(100) + 10(50) = 2,000 minutes on workstation A 15(100) + 30(50) = 3,000 minutes on workstation B 15(100) + 5(50) = 1,750 minutes on workstation C 15(100) + 5(50) = 1,750 minutes on workstation D Only workstation B requires m... |
19 | Chapter 16 569 Aggregate and Workforce Planning Table 16.2 Input Data for Modiï¬ed Single-Period AP Example Product 1 2 Selling price Raw material cost Maximum weekly sales Minutes per unit on workstation A Minutes per unit on workstation B Minutes per unit on workstation C Minutes per unit on workstation D $... |
20 | 45X 1 + 60X 2 â 5,000 (16.33) Subject to: X 1 ⤠100 (16.34) X 2 ⤠50 (16.35) 15X 1 + 10X 2 ⤠2,400 (16.36) 15X 1 + 35X 2 ⤠2,400 (16.37) 15X 1 + 5X 2 ⤠2,400 (16.38) 25X 1 + 14X 2 ⤠2,400 (16.39) Chapter 16 571 Aggregate and Workforce Planning Problem (16.33)â16.39) is trivial for any ... |
21 | 16.3.3 Extensions to the Basic Model A host of variations on the basic problem given in formulation (16.28)â(16.32) are possible. We discuss a few of these next; the reader is asked to think of others in the problems at chapterâs end. Other Resource Constraints. Formulation (16.28)â(16.32) contains capacity cons... |
22 | Chapter 16 573 Aggregate and Workforce Planning q of full capacity, then we can replace constraints (16.30) in LP (16.28)â(16.32) by m ai j X it ⤠qc jt for all j, t i=t The result will be that a binding capacity constraint will occur whenever a workstation is loaded to 100q percent of capacity in a period... |
23 | 574 Part III Principles in Practice of product i? For that matter, why should the lateness penalty be linear in the number of periods late or the number of units that are late? Clearly, asking someone in the organization for these numbers is out of the question. Therefore, one should view this type of model as a too... |
24 | n t¯ {ri Sit â h i Iit+ â Ïi Iitâ â l j O jt } t=1 (16.49) j=1 Subject to: d it ⤠Sit ⤠dÌit for all i, t (16.50) ai j X it ⤠c jt + O jt for all j, t (16.51) Iit = Iitâ1 + X it â Sit for all i, t (16.52) Iit = Iit+ â Iitâ for all i, t (16.53) X it , Sit , Iit+ , Iitâ O jt... |
25 | (16.55) units of i into station j. These values can easily be computed in the manner used for the example in Figure 16.9 and updated in a spreadsheet or database as a function of the estimated yield loss at each station. Using equation (16.55) to adjust the production amounts X it in the manner illustrated in Figure 1... |
26 | 16.4.1 An LP Model To illustrate how an LP model can help address the workforce-resizing and overtime allocation questions, we will consider a simple single-product model. In systems where product routings and processing times are either almost identical, so that products can be aggregated into a single product, or en... |
27 | 578 Part III Principles in Practice with an LP model, keeping in mind that we are capturing general effects rather than elaborate details. Given that the AP and WP modules are used for long-term general planning purposes and rely on speculative forecasted data (e.g., of future demand), this is probably a reasonable ... |
28 | $2,980,600.00 = -d_1 0.00 -200 = -d_2 0.00 -220 = -d_3 0.00 -230 = -d_4 0.00 -300 = -d_5 0.00 -400 = -d_6 0.00 -450 = -d_7 0.00 -320 = -d_8 0.00 -180 = -d_9 0.00 -170 = -d_10 0.00 -170 = -d_11 0.00 -160 = -d_12 0.00 -180 = -2520.00 0 = 0.00 0 = 0.00 0 = 0.00 0 = 0.00 0 = 0.00 0 = 0.00 0 = 0.00 0 = 0.00 0 = 0.00 0 = 0.... |
29 | 1,000(d1 + · · · + d12 ) â 10(I1 + · · · + I12 ) â35(W1 + · · · + W12 ) â 52.5(O1 + · · · + O12 ) â15(H1 + · · · + H12 ) â 9(F1 + · · · + F12 ) (16.68) Subject to: I1 â I0 â X 1 = âd1 (16.69) I2 â I1 â X 2 = âd2 (16.70) I3 â I2 â X 3 = âd3 (16.71) I4 â I3 â X 4 ... |
30 | (16.102) 12X 11 â W11 â O11 ⤠0 (16.103) 12X 12 â W12 â O12 ⤠0 (16.104) X t , It , Ot , Wt , Ht , Ft ⥠0 t = 1, . . . , 12 (16.105) Objective (16.68) is identical to objective (16.61), except that the St variables have been replaced with dt constants.8 Constraints (16.69)â(16.80) are the usual ... |
31 | variables and could be left out without affecting the solution. We have kept it in so that our model reports a sensible proï¬t function. 582 Part III Principles in Practice left-hand side of the equality and constants (dt ) are on the right-hand side. This is often a convenient modeling convention, as we will see... |
32 | 3 230.00 2520.00 0.00 0.00 0.00 0.00 4 300.00 2520.00 0.00 0.00 0.00 0.00 5 400.00 2520.00 0.00 0.00 0.00 0.00 6 450.00 2520.00 0.00 0.00 0.00 0.00 7 320.00 2520.00 0.00 0.00 0.00 0.00 8 180.00 2520.00 0.00 0.00 0.00 0.00 9 170.00 2520.00 0.00 0.00 0.00 0.00 10 170.00 2520.00 0.00 0.00 0.00 0.00 11 160.00 2520.... |
33 | 3 302.86 3634.29 0.00 0.00 258.57 0.00 4 302.86 3634.29 0.00 0.00 261.43 0.00 5 302.86 3634.29 0.00 0.00 164.29 0.00 6 302.86 3634.29 0.00 0.00 17.14 0.00 7 302.86 3634.29 0.00 0.00 0.00 0.00 8 180.00 2160.00 0.00 1474.29 0.00 0.00 9 170.00 2040.00 0.00 120.00 0.00 0.00 10 170.00 2040.00 0.00 0.00 0.00 0.00 11 ... |
34 | t = 1, . . . , 12 Rerunning the model with these additional constraints produces the spreadsheet in Figure 16.13. As we expect, this solution does not include any layoffs. Somewhat surprising, however, is the fact that it does not involve any new hires either (that is, Ht = 0 for every period). Instead of increasing t... |
35 | 3 230.00 2520.00 0.00 0.00 0.00 240.00 4 300.00 2520.00 0.00 0.00 0.00 1080.00 5 400.00 2520.00 0.00 0.00 0.00 2280.00 6 450.00 2520.00 0.00 0.00 0.00 2880.00 7 320.00 2520.00 0.00 0.00 0.00 1320.00 8 180.00 2520.00 0.00 0.00 0.00 0.00 9 170.00 2520.00 0.00 0.00 0.00 0.00 10 170.00 2520.00 0.00 0.00 0.00 0.00 1... |
36 | 3 302.86 3028.57 0.00 0.00 258.57 605.71 4 302.86 3028.57 0.00 0.00 261.43 605.71 5 302.86 3028.57 0.00 0.00 164.29 605.71 6 302.86 3028.57 0.00 0.00 17.14 605.71 7 302.86 3028.57 0.00 0.00 0.00 605.71 8 180.00 3028.57 0.00 0.00 0.00 0.00 9 170.00 3028.57 0.00 0.00 0.00 0.00 10 170.00 3028.57 0.00 0.00 0.00 0.00... |
37 | 16.4.3 Modeling Insights In addition to providing a detailed example of a workforce formulation in LP (16.61)â (16.67), we hope that our discussion has helped the reader appreciate the following aspects of using an optimization model as the basis for an AP or WP module. 1. Multiple modeling approaches. There are oft... |
38 | Appendix 16A Linear Programming Linear programming is a powerful mathematical tool for solving constrained optimization problems. The name derives from the fact that LP was ï¬rst applied to ï¬nd optimal schedules or âprogramsâ of resource allocation. Hence, although LP generally does involve using a computer prog... |
39 | 45X 1 + 60X 2 Finally, we need to specify constraints. If we could produce as much of products 1 and 2 as we wanted, we could drive the above objective function, and hence weekly proï¬t, to inï¬nity. This is not possible because of limitations on demand and capacity. 9 Note that we are neglecting labor and overhead ... |
40 | 592 Part III Principles in Practice the user speciï¬es otherwise. In other packages, the user must include the non-negativity constraints explicitly. This is something to beware of when using LP software. Solution To get a general idea of how an LP package works, let us consider the above formulation from a mathem... |
41 | X2 100.00 80.00 60.00 X2 = 50 40.00 Feasible region 20.00 0.00 0 50 100 X1 150 Chapter 16 Figure 16.16 593 Aggregate and Workforce Planning 140.00 Solution to LP example. 120.00 Z = 5,557.94 X2 100.00 Z = 7,000 80.00 Optimal solution (75.79, 36.09) 60.00 40.00 Feasible region 20.00 0.00 0 Z = 3... |
42 | 594 Part III Figure 16.17 Principles in Practice 140.00 Effect of changing objective coefï¬cients in LP example. 120.00 45X1 + 60X2 = 5,557.94 X2 100.00 60X1 + 60X2 = 6,712.80 80.00 60.00 40.00 Feasible region 20.00 0.00 0 50 100 150 X1 equation (16.117) to 60 Z Z X2 = â X1 + = âX 1 + 60 60 60... |
43 | Chapter 16 595 Aggregate and Workforce Planning side will increase from 2,400 to 2,500. Since this is something we might want to consider, we would like to be able to determine its effect. We do this differently for two types of constraints: a. Slack constraints are constraints that do not deï¬ne the optimal extrem... |
44 | Figure 16.18 25X1 + 14X2 = 2,400 140.00 X1 = 100 15X1 + 35X2 = 2,770 120.00 15X1 + 35X2 = 2,400 100.00 X2 Feasible region when RHS of constraint of workstation B is increased to 2,770. 80.00 60.00 X2 = 50 40.00 Feasible region 20.00 0.00 0 50 100 X1 150 596 Part III Principles in Practice increase ... |
45 | 2. Show how to modify LP (16.49)-(16.54) to represent the case where overtime on all the workstations must be scheduled simultaneously (i.e., if one resource runs overtime, all resources run overtime). Describe how you would handle the case where, in general, different workstations can have different amounts of overtim... |
46 | 598 Part III Principles in Practice Vit = units of family i purchased from vendor in month t and available to meet demand in month t Iit = ï¬nished goods inventory of family i at end of month t dit = units of family i demanded (and shipped) during month t c jt = hours available on work center j( j = 1, . . . , 10) ... |
47 | Selling price Labor required Bottleneck machine time required Raw material required Bookcase 1 Bookcase 2 $15 0.75 hour/unit 1.5 hours/unit 2 bf/unit $8 0.5 hour/unit 0.8 hour/unit 1 bf/unit P1 = units of bookcase 1 produced per week P2 = units of bookcase 2 produced per week OT = hours of overtime used per week R... |
48 | DUAL PRICES .100000 .066667 3.866666 6.000000 4.500000 .000000 .000000 NO. ITERATIONS = 5 RANGES IN WHICH THE BASIS IS UNCHANGED: VARIABLE P1 P2 OT RM A1 A2 ROW 2 3 4 5 6 7 8 CURRENT COEF 15.000000 8.000000 -6.000000 -1.500000 -1.000000 -1.000000 OBJ COEFFICIENT RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE .9666... |
49 | j i 1 2 3 4 ri A B C 2.4 2.0 0.9 1.1 2.2 0.9 0.8 1.2 1.0 3.0 2.1 2.5 $50 $65 $70 The number of hours available (c jt ) and the upper and lower limits on demand (dÌit and d it ) for each product over the next four quarters are as follows: t 1 2 3 4 c1t c2t c3t c4t 640 640 1,920 1,280 640 640 1,920 1... |
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