OR topics in MRP-II. Mads Jepsens. OR topics in MRP-II p.1/25

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1 OR topics in MRP-II Mads Jepsens OR topics in MRP-II p.1/25

2 Overview Why bother? OR topics in MRP-II p.2/25

3 Why bother? Push and Pull systems Overview OR topics in MRP-II p.2/25

4 Overview Why bother? Push and Pull systems Manufacturing Resource Planning (MRP-II) OR topics in MRP-II p.2/25

5 Overview Why bother? Push and Pull systems Manufacturing Resource Planning (MRP-II) Rough-Cut Capacity Planning(RCCP) OR topics in MRP-II p.2/25

6 Overview Why bother? Push and Pull systems Manufacturing Resource Planning (MRP-II) Rough-Cut Capacity Planning(RCCP) Work center scheduling OR topics in MRP-II p.2/25

7 Why bother? MRP-II is still used in many production planning systems including Dynamics AX OR topics in MRP-II p.3/25

8 Why bother? MRP-II is still used in many production planning systems including Dynamics AX Room for improvements, many constraints are tightened to obtain feasible solutions. OR topics in MRP-II p.3/25

9 Why bother? MRP-II is still used in many production planning systems including Dynamics AX Room for improvements, many constraints are tightened to obtain feasible solutions. Academically challenging. Problems are typically NP-hard in the strong sense. OR topics in MRP-II p.3/25

10 Why bother? MRP-II is still used in many production planning systems including Dynamics AX Room for improvements, many constraints are tightened to obtain feasible solutions. Academically challenging. Problems are typically NP-hard in the strong sense. Closely related to other interesting problems OR topics in MRP-II p.3/25

11 Production Planning Problem A set of resources (Machines, personal etc). OR topics in MRP-II p.4/25

12 Production Planning Problem A set of resources (Machines, personal etc). Each resource have different types of constraints. OR topics in MRP-II p.4/25

13 Production Planning Problem A set of resources (Machines, personal etc). Each resource have different types of constraints. Each product (item) is defined by its bill of material (BOM) and a routing OR topics in MRP-II p.4/25

14 Production Planning Problem A set of resources (Machines, personal etc). Each resource have different types of constraints. Each product (item) is defined by its bill of material (BOM) and a routing The BOM describes how many and which components should be used to produce the item OR topics in MRP-II p.4/25

15 Production Planning Problem A set of resources (Machines, personal etc). Each resource have different types of constraints. Each product (item) is defined by its bill of material (BOM) and a routing The BOM describes how many and which components should be used to produce the item The routing describes which work processes are needed. Each Bom may have several different routings. OR topics in MRP-II p.4/25

16 Production Planning Problem A set of resources (Machines, personal etc). Each resource have different types of constraints. Each product (item) is defined by its bill of material (BOM) and a routing The BOM describes how many and which components should be used to produce the item The routing describes which work processes are needed. Each Bom may have several different routings. For a finite time period and a finite set of orders the Production Planning Problem is to determine when and where to produce a given item OR topics in MRP-II p.4/25

17 Solution Models EOQ inventory model. Limited to single product with constant lead time. OR topics in MRP-II p.5/25

18 Solution Models EOQ inventory model. Limited to single product with constant lead time. Material Requirement Planning(MRP), does not consider capacity on resources. OR topics in MRP-II p.5/25

19 Solution Models EOQ inventory model. Limited to single product with constant lead time. Material Requirement Planning(MRP), does not consider capacity on resources. Manufacturing Resource Planning (MRP-II). Considers capacity on critical resources. OR topics in MRP-II p.5/25

20 Solution Models continued Kanban/lean manufacturing. Can require huge changes in factory layout, in the philosophy at the company and has some issues handling less frequent ordered items. OR topics in MRP-II p.6/25

21 Solution Models continued Kanban/lean manufacturing. Can require huge changes in factory layout, in the philosophy at the company and has some issues handling less frequent ordered items. Enterprise Resource Planning. Is a comprehensive system and much more than just planning. Introduces supply chain optimization. OR topics in MRP-II p.6/25

22 Solution Models continued Kanban/lean manufacturing. Can require huge changes in factory layout, in the philosophy at the company and has some issues handling less frequent ordered items. Enterprise Resource Planning. Is a comprehensive system and much more than just planning. Introduces supply chain optimization. Advanced Planning Systems. Specialized plug-in based systems to handle specific groups of companies. OR topics in MRP-II p.6/25

23 Push and Pull systems A example of a pull systems is the kanban system OR topics in MRP-II p.7/25

24 Push and Pull systems A example of a pull systems is the kanban system In a push system items are pushed towards the customer. MRP-II is a example of a push system. OR topics in MRP-II p.7/25

25 Push and Pull systems A example of a pull systems is the kanban system In a push system items are pushed towards the customer. MRP-II is a example of a push system. In a Pull/Kanban system the amount of wip is reduced. OR topics in MRP-II p.7/25

26 Push and Pull systems A example of a pull systems is the kanban system In a push system items are pushed towards the customer. MRP-II is a example of a push system. In a Pull/Kanban system the amount of wip is reduced. In a Pull system bottlenecks are visible since machines after the bottleneck have no kanban cards. OR topics in MRP-II p.7/25

27 Push and Pull systems A example of a pull systems is the kanban system In a push system items are pushed towards the customer. MRP-II is a example of a push system. In a Pull/Kanban system the amount of wip is reduced. In a Pull system bottlenecks are visible since machines after the bottleneck have no kanban cards. Bottlenecks are less visible in a push system since items pile up several places. OR topics in MRP-II p.7/25

28 Push and Pull systems A example of a pull systems is the kanban system In a push system items are pushed towards the customer. MRP-II is a example of a push system. In a Pull/Kanban system the amount of wip is reduced. In a Pull system bottlenecks are visible since machines after the bottleneck have no kanban cards. Bottlenecks are less visible in a push system since items pile up several places. Less frequent ordered items is easier handled in a push system. OR topics in MRP-II p.7/25

29 Material Requirement Planning(MRP) OR topics in MRP-II p.8/25

30 Material Requirement Planning(MRP) Originally system was based on reorder point. OR topics in MRP-II p.8/25

31 Material Requirement Planning(MRP) Originally system was based on reorder point. Reorder point is suited for independent demand. OR topics in MRP-II p.8/25

32 Material Requirement Planning(MRP) Originally system was based on reorder point. Reorder point is suited for independent demand. But not for dependent demand. OR topics in MRP-II p.8/25

33 Material Requirement Planning(MRP) Originally system was based on reorder point. Reorder point is suited for independent demand. But not for dependent demand. MRP works backwards from independent demand to derive a schedule. OR topics in MRP-II p.8/25

34 Material Requirement Planning(MRP) Originally system was based on reorder point. Reorder point is suited for independent demand. But not for dependent demand. MRP works backwards from independent demand to derive a schedule. MRP is called a push system since it pushes items in the production chain. OR topics in MRP-II p.8/25

35 Schematic of MRP OR topics in MRP-II p.9/25

36 MRP Inputs and Outputs OR topics in MRP-II p.10/25

37 MRP Inputs and Outputs Master Production Schedule: OR topics in MRP-II p.10/25

38 MRP Inputs and Outputs Master Production Schedule: Item, Quantity and due dates. OR topics in MRP-II p.10/25

39 MRP Inputs and Outputs Master Production Schedule: Item, Quantity and due dates. Erp Database: OR topics in MRP-II p.10/25

40 MRP Inputs and Outputs Master Production Schedule: Item, Quantity and due dates. Erp Database: BOM, Routing, lot-sizing rule(lsr), lead time(plt) and On-Hand Inventory. OR topics in MRP-II p.10/25

41 MRP Inputs and Outputs Master Production Schedule: Item, Quantity and due dates. Erp Database: BOM, Routing, lot-sizing rule(lsr), lead time(plt) and On-Hand Inventory. Scheduled Receipts: Out standing orders and Jobs. Work in process. OR topics in MRP-II p.10/25

42 MRP Inputs and Outputs Master Production Schedule: Item, Quantity and due dates. Erp Database: BOM, Routing, lot-sizing rule(lsr), lead time(plt) and On-Hand Inventory. Scheduled Receipts: Out standing orders and Jobs. Work in process. MRP outputs: Planned order release, Change notices and Exception reports. OR topics in MRP-II p.10/25

43 MRP Procedure continued OR topics in MRP-II p.11/25

44 MRP Procedure continued Lot sizing: OR topics in MRP-II p.11/25

45 MRP Procedure continued Lot sizing: Wagner Whitin OR topics in MRP-II p.11/25

46 MRP Procedure continued Lot sizing: Wagner Whitin lot for lot OR topics in MRP-II p.11/25

47 MRP Procedure continued Lot sizing: Wagner Whitin lot for lot fixed order period OR topics in MRP-II p.11/25

48 MRP Procedure continued Lot sizing: Wagner Whitin lot for lot fixed order period Fixed order Quantity and EOQ. OR topics in MRP-II p.11/25

49 MRP Procedure continued Lot sizing: Wagner Whitin lot for lot fixed order period Fixed order Quantity and EOQ. Part-Period Balancing. Balancing inventory cost and Setup Cost. OR topics in MRP-II p.11/25

50 MRP Procedure continued Lot sizing: Wagner Whitin lot for lot fixed order period Fixed order Quantity and EOQ. Part-Period Balancing. Balancing inventory cost and Setup Cost. Time fasing. All lead times are considered for items, not for status on floor OR topics in MRP-II p.11/25

51 MRP Procedure continued Lot sizing: Wagner Whitin lot for lot fixed order period Fixed order Quantity and EOQ. Part-Period Balancing. Balancing inventory cost and Setup Cost. Time fasing. All lead times are considered for items, not for status on floor Bom Explosion. Netting and lot sizing is done for each sub item. OR topics in MRP-II p.11/25

52 Issues with MRP OR topics in MRP-II p.12/25

53 Issues with MRP Assume constant lead times. To take care of variation Safety Stock and Safety Lead time is used. OR topics in MRP-II p.12/25

54 Issues with MRP Assume constant lead times. To take care of variation Safety Stock and Safety Lead time is used. Capacity Infeasibility, there is no capacity check. OR topics in MRP-II p.12/25

55 Issues with MRP Assume constant lead times. To take care of variation Safety Stock and Safety Lead time is used. Capacity Infeasibility, there is no capacity check. Long Planned Lead time due to variation in delivery time. OR topics in MRP-II p.12/25

56 Issues with MRP Assume constant lead times. To take care of variation Safety Stock and Safety Lead time is used. Capacity Infeasibility, there is no capacity check. Long Planned Lead time due to variation in delivery time. System Nervousness. Plans that are feasible can become infeasible. OR topics in MRP-II p.12/25

57 MRP-II Long range planning Long term forecast Resource planning Aggregate production planning Firm orders Short term forecast Rough cut capacity planning Master production scheduling Demand management Bills of material On hand & scheduled reciepts Material requirement planning Job pool Capacity requirement planning Intermediate range planning Short term control Job release Routing data Job dispatching I/O control OR topics in MRP-II p.13/25

58 Rough-Cut Capacity Planning(RCCP) OR topics in MRP-II p.14/25

59 Rough-Cut Capacity Planning(RCCP) Considers aggregated work OR topics in MRP-II p.14/25

60 Rough-Cut Capacity Planning(RCCP) Considers aggregated work Allocates work in time buckets OR topics in MRP-II p.14/25

61 Rough-Cut Capacity Planning(RCCP) Considers aggregated work Allocates work in time buckets Determines resources in order to reach due dates OR topics in MRP-II p.14/25

62 Rough-Cut Capacity Planning(RCCP) Considers aggregated work Allocates work in time buckets Determines resources in order to reach due dates Both regular and nonregular (outsourcing over time etc.) is considered OR topics in MRP-II p.14/25

63 Rough-Cut Capacity Planning(RCCP) Considers aggregated work Allocates work in time buckets Determines resources in order to reach due dates Both regular and nonregular (outsourcing over time etc.) is considered Time driven RCCP is when project dates must be meet. OR topics in MRP-II p.14/25

64 Rough-Cut Capacity Planning(RCCP) Considers aggregated work Allocates work in time buckets Determines resources in order to reach due dates Both regular and nonregular (outsourcing over time etc.) is considered Time driven RCCP is when project dates must be meet. In resource-driven RCCP only regular capacity can be used OR topics in MRP-II p.14/25

65 Rough-Cut Capacity Planning(RCCP) Considers aggregated work Allocates work in time buckets Determines resources in order to reach due dates Both regular and nonregular (outsourcing over time etc.) is considered Time driven RCCP is when project dates must be meet. In resource-driven RCCP only regular capacity can be used This session will consider the time driven OR topics in MRP-II p.14/25

66 Some notation OR topics in MRP-II p.15/25

67 Some notation n jobs J 1,J 2,,J n OR topics in MRP-II p.15/25

68 Some notation n jobs J 1,J 2,,J n k resources R 1,,R k OR topics in MRP-II p.15/25

69 Some notation n jobs J 1,J 2,,J n k resources R 1,,R k T time buckets OR topics in MRP-II p.15/25

70 Some notation n jobs J 1,J 2,,J n k resources R 1,,R k T time buckets Q kt is the regular capacity for resource k in period t OR topics in MRP-II p.15/25

71 Some notation n jobs J 1,J 2,,J n k resources R 1,,R k T time buckets Q kt is the regular capacity for resource k in period t Job J j requires q jk units of resource k OR topics in MRP-II p.15/25

72 Some notation n jobs J 1,J 2,,J n k resources R 1,,R k T time buckets Q kt is the regular capacity for resource k in period t Job J j requires q jk units of resource k x kt denotes the fraction of job J j performed in period t OR topics in MRP-II p.15/25

73 Some notation n jobs J 1,J 2,,J n k resources R 1,,R k T time buckets Q kt is the regular capacity for resource k in period t Job J j requires q jk units of resource k x kt denotes the fraction of job J j performed in period t J j must be performed in time window [r j,d j ] OR topics in MRP-II p.15/25

74 Some notation n jobs J 1,J 2,,J n k resources R 1,,R k T time buckets Q kt is the regular capacity for resource k in period t Job J j requires q jk units of resource k x kt denotes the fraction of job J j performed in period t J j must be performed in time window [r j,d j ] p j is the minimum number of periods job J j can use. OR topics in MRP-II p.15/25

75 Some notation n jobs J 1,J 2,,J n k resources R 1,,R k T time buckets Q kt is the regular capacity for resource k in period t Job J j requires q jk units of resource k x kt denotes the fraction of job J j performed in period t J j must be performed in time window [r j,d j ] p j is the minimum number of periods job J j can use. 1 p j is the maximum fraction of a job that can be completed in single time bucket. OR topics in MRP-II p.15/25

76 Precedence constraints If job J i must finish before J j there is a precedence relation. For a period τ this can be modelled as: OR topics in MRP-II p.16/25

77 Precedence constraints If job J i must finish before J j there is a precedence relation. For a period τ this can be modelled as: x jτ > 0 τ 1 t=d t x it = 1 OR topics in MRP-II p.16/25

78 Precedence constraints If job J i must finish before J j there is a precedence relation. For a period τ this can be modelled as: x jτ > 0 τ 1 t=d t x it = 1 There will be d j r j constraints per precedence relation. OR topics in MRP-II p.16/25

79 Precedence constraints If job J i must finish before J j there is a precedence relation. For a period τ this can be modelled as: x jτ > 0 τ 1 t=d t x it = 1 There will be d j r j constraints per precedence relation. Time windows can in some cases be tightened, due to precedence constraints. OR topics in MRP-II p.16/25

80 Nonregular capacity Let Q kt denote the nonregular capacity for resource k in period t. Then for each resource and time period: OR topics in MRP-II p.17/25

81 Nonregular capacity Let Q kt denote the nonregular capacity for resource k in period t. Then for each resource and time period: U kt = max{0, n j=1 q jk x jt Q kt } OR topics in MRP-II p.17/25

82 Nonregular capacity Let Q kt denote the nonregular capacity for resource k in period t. Then for each resource and time period: U kt = max{0, n j=1 q jk x jt Q kt } The cost of using nonregular capacity for resource k in time period t is c kt OR topics in MRP-II p.17/25

83 Nonregular capacity Let Q kt denote the nonregular capacity for resource k in period t. Then for each resource and time period: U kt = max{0, n j=1 q jk x jt Q kt } The cost of using nonregular capacity for resource k in time period t is c kt It is assumed that there is no limit on the nonregular resources. OR topics in MRP-II p.17/25

84 Mathematical Model min T t=1 K k=1 c ktu kt subject to dj t=r j x jt = 1 1 j n x jt 1 p j 1 j n, 1 t T n j=1 q jkx jt U kt 0 1 j n, 1 t T Precedencerelations x jt,u kt 0 1 j n, 1 t T OR topics in MRP-II p.18/25

85 Controlling feasibility Allowed To Work window for job J j is defined as [S j,c j ]. OR topics in MRP-II p.19/25

86 Controlling feasibility Allowed To Work window for job J j is defined as [S j,c j ]. Job J j cannot start before S j or after C j OR topics in MRP-II p.19/25

87 Controlling feasibility Allowed To Work window for job J j is defined as [S j,c j ]. Job J j cannot start before S j or after C j A ATW for job J j is feasible if: OR topics in MRP-II p.19/25

88 Controlling feasibility Allowed To Work window for job J j is defined as [S j,c j ]. Job J j cannot start before S j or after C j A ATW for job J j is feasible if: 1. S j r j and C j d j 2. C j S j p j 1 OR topics in MRP-II p.19/25

89 Controlling feasibility Allowed To Work window for job J j is defined as [S j,c j ]. Job J j cannot start before S j or after C j A ATW for job J j is feasible if: 1. S j r j and C j d j 2. C j S j p j 1 A set S of ATW windows is feasible if: OR topics in MRP-II p.19/25

90 Controlling feasibility Allowed To Work window for job J j is defined as [S j,c j ]. Job J j cannot start before S j or after C j A ATW for job J j is feasible if: 1. S j r j and C j d j 2. C j S j p j 1 A set S of ATW windows is feasible if: 1. Every ATW window is feasible 2. S j > C j if J i J j OR topics in MRP-II p.19/25

91 Mathematical Model ATW windows s jt = { 1 S j t C j 0 otherwise (P S ) min T t=1 K k=1 c ktu kt subjectto dj t=r j x jt = 1 1 j n x jt s jt p j 1 j n, 1 t T n j=1 q jkx jt U kt 0 1 j n, 1 t T x jt,u kt 0 1 j n, 1 t T OR topics in MRP-II p.20/25

92 Constructive heuristics (H BASIC ) Construct a feasible set S of ATW windows. OR topics in MRP-II p.21/25

93 Constructive heuristics (H BASIC ) Construct a feasible set S of ATW windows. Solve problem P S OR topics in MRP-II p.21/25

94 Constructive heuristics (H BASIC ) Construct a feasible set S of ATW windows. Solve problem P S To obtain a feasible set of ATW windows construct them as follows: OR topics in MRP-II p.21/25

95 Constructive heuristics (H BASIC ) Construct a feasible set S of ATW windows. Solve problem P S To obtain a feasible set of ATW windows construct them as follows: Set S j = r j OR topics in MRP-II p.21/25

96 Constructive heuristics (H BASIC ) Construct a feasible set S of ATW windows. Solve problem P S To obtain a feasible set of ATW windows construct them as follows: Set S j = r j Set C j = min{d j, min {k Ji J k } r k 1} OR topics in MRP-II p.21/25

97 Constructive heuristics (H CPM ) Define the slack of job J j as L j = C j (S j + p j ) OR topics in MRP-II p.22/25

98 Constructive heuristics (H CPM ) Define the slack of job J j as L j = C j (S j + p j ) {J i1,j i2,,j ik } is a ordered set if: OR topics in MRP-II p.22/25

99 Constructive heuristics (H CPM ) Define the slack of job J j as L j = C j (S j + p j ) {J i1,j i2,,j ik } is a ordered set if: J i1 J i2,j i3 J i4,,j ik 1 J ik OR topics in MRP-II p.22/25

100 Constructive heuristics (H CPM ) Define the slack of job J j as L j = C j (S j + p j ) {J i1,j i2,,j ik } is a ordered set if: J i1 J i2,j i3 J i4,,j ik 1 J ik A critical path is a path where: OR topics in MRP-II p.22/25

101 Constructive heuristics (H CPM ) Define the slack of job J j as L j = C j (S j + p j ) {J i1,j i2,,j ik } is a ordered set if: J i1 J i2,j i3 J i4,,j ik 1 J ik A critical path is a path where: L j1 = L j2 = = min 1 j n L j OR topics in MRP-II p.22/25

102 Constructive heuristics (H CPM ) Define the slack of job J j as L j = C j (S j + p j ) {J i1,j i2,,j ik } is a ordered set if: J i1 J i2,j i3 J i4,,j ik 1 J ik A critical path is a path where: L j1 = L j2 = = min 1 j n L j S ji = S ji 1 + p i 1 for 2 i R OR topics in MRP-II p.22/25

103 Constructive heuristics (H CPM ) Define the slack of job J j as L j = C j (S j + p j ) {J i1,j i2,,j ik } is a ordered set if: J i1 J i2,j i3 J i4,,j ik 1 J ik A critical path is a path where: L j1 = L j2 = = min 1 j n L j S ji = S ji 1 + p i 1 for 2 i R C ji = C ji+1 p ji+1 for1 i R OR topics in MRP-II p.22/25

104 Constructive heuristics (H CPM ) Define the slack of job J j as L j = C j (S j + p j ) {J i1,j i2,,j ik } is a ordered set if: J i1 J i2,j i3 J i4,,j ik 1 J ik A critical path is a path where: L j1 = L j2 = = min 1 j n L j S ji = S ji 1 + p i 1 for 2 i R C ji = C ji+1 p ji+1 for1 i R A critical path is maximal if for all J l : OR topics in MRP-II p.22/25

105 Constructive heuristics (H CPM ) Define the slack of job J j as L j = C j (S j + p j ) {J i1,j i2,,j ik } is a ordered set if: J i1 J i2,j i3 J i4,,j ik 1 J ik A critical path is a path where: L j1 = L j2 = = min 1 j n L j S ji = S ji 1 + p i 1 for 2 i R C ji = C ji+1 p ji+1 for1 i R A critical path is maximal if for all J l : {J i1,j i2,,j ik,j il } is critical {J il,j i1,j i2,,j ik } is critical OR topics in MRP-II p.22/25

106 (H CPM ) continued Initialize S j = r j and C j = d j for all J j OR topics in MRP-II p.23/25

107 (H CPM ) continued Initialize S j = r j and C j = d j for all J j Compute the slack for all jobs OR topics in MRP-II p.23/25

108 (H CPM ) continued Initialize S j = r j and C j = d j for all J j Compute the slack for all jobs Find a maximal critical path {J i1,j i2,,j ik } OR topics in MRP-II p.23/25

109 (H CPM ) continued Initialize S j = r j and C j = d j for all J j Compute the slack for all jobs Find a maximal critical path {J i1,j i2,,j ik } Compute the total slack L = C jr (S j1 + R i=1 p j i ) OR topics in MRP-II p.23/25

110 (H CPM ) continued Initialize S j = r j and C j = d j for all J j Compute the slack for all jobs Find a maximal critical path {J i1,j i2,,j ik } Compute the total slack L = C jr (S j1 + R i=1 p j i ) Set S ji = S j1 + i 1 k=1 p i k + L i 1 k=1 p i k / R k=1 p i k OR topics in MRP-II p.23/25

111 (H CPM ) continued Initialize S j = r j and C j = d j for all J j Compute the slack for all jobs Find a maximal critical path {J i1,j i2,,j ik } Compute the total slack L = C jr (S j1 + R i=1 p j i ) Set S ji = S j1 + i 1 k=1 p i k + L i 1 k=1 p i k / R k=1 p i k Set C ik = S ik+1 1 OR topics in MRP-II p.23/25

112 H CPM example S C p OR topics in MRP-II p.24/25

113 H CPM example S C p L = C 4 (0 + 20) = 5 OR topics in MRP-II p.24/25

114 H CPM example S C p L = C 4 (0 + 20) = 5 Recall ˆ Sji = S ji + i 1 k=1 p j k + L i 1 k=1 / R k=1 p jk OR topics in MRP-II p.24/25

115 H CPM example S C p L = C 4 (0 + 20) = 5 Recall ˆ Sji = S ji + i 1 k=1 p j k + L i 1 k=1 / R k=1 p jk S 2 = = 6 OR topics in MRP-II p.24/25

116 H CPM example S C p L = C 4 (0 + 20) = 5 Recall ˆ Sji = S ji + i 1 k=1 p j k + L i 1 k=1 / R k=1 p jk S 2 = = 6 S 3 = = 12 OR topics in MRP-II p.24/25

117 H CPM example S C p L = C 4 (0 + 20) = 5 Recall ˆ Sji = S ji + i 1 k=1 p j k + L i 1 k=1 / R k=1 p jk S 2 = = 6 S 3 = = 12 S 4 = = 24 OR topics in MRP-II p.24/25

118 H CPM example S C p L = C 4 (0 + 20) = 5 Recall ˆ Sji = S ji + i 1 k=1 p j k + L i 1 k=1 / R k=1 p jk S 2 = = 6 S 3 = = 12 S 4 = = 24 C 1 = S 2 1 = 5, C 2 = 11, C 3 = 23 OR topics in MRP-II p.24/25

119 Questions? OR topics in MRP-II p.25/25

120 Questions? Thank you! OR topics in MRP-II p.25/25

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