1. Production-related Decision Making in Large Corporations

2. Production-related Decision Making in Large Corporations (borrowed from Heizer and Render)

3. Product and Process Design, Sourcing, Equipment Selection and Capacity Planning

4. Major Topics Product and Process Design Documenting Product and Process Design Sourcing Decisions: –A simple “Make or Buy” model –Decision Trees: A scenario-based approach Equipment Selection and Capacity Planning

5. Product Selection and Development Stages (borrowed from Heizer & Render)

6. Quality Function Deployment (DFD) and the House of Quality QFD: The process of –Determining what are the customer “requirements” / “wants”, and –Translating those desires into the target product design. House of quality: A graphic, yet systematic technique for defining the relationship between customer desires and the developed product (or service)

7. House of Quality Example (borrowed from Heizer & Render)

8. The “House of Quality” Chain (borrowed from Heizer & Render)

9. Concurrent Engineering: The current approach for organizing the product and process development The traditional US approach (department-based): Research & Development => Engineering => Manufacturing => Production Clear-cut responsibilities but lack of communication and “forward thinking”! The currently prevailing approach (cross-functional team-based): Product development (or design for manufacturability, or value engineering) teams: Include representatives from: –Marketing –Manufacturing –Purchasing –Quality assurance –Field service –(even from) vendors Concurrent engineering: Less costly and more expedient product development

10. The time factor: Time-based competition Some advantages of getting first a new product to the market: –Setting the “standard” (higher market control) –Larger market share –Higher prices and profit margins Currently, product life cycles get shorter and product technological sophistication increases => more money is funneled to the product development and the relative risks become higher. The pressures resulting from time-based competition have led to higher levels of integrations through strategic partnerships, but also through mergers and acquisitions.

11. Additional concerns in contemporary product and process design –promote robust design practices Robustness: the insensitivity of the product performance to small variations in the production or assembly process => ability to support product quality more reliably and cost-effectively. –Control the product complexity –Improve the product maintainability / serviceability –(further) standardize the employed components Modularity: the structuring of the end product through easily segmented components that can also be easily interchanged or replaced => ability to support flexible production and product customization;increased product serviceability. –Improve job design and job safety –Environmental friendliness: safe and environmentally sound products, minimizing waste of raw materials and energy, complying with environmental regulations, ability for reuse, being recognized as good corporate citizen.

12. Documenting Product Designs Engineering Drawing: a drawing that shows the dimensions, tolerances, materials and finishes of a component. (Fig. 5.9) Bill of Material (BOM): A listing of the components, their description and the quantity of each required to make a unit of a given product. (Fig. 5.10) Assembly drawing: An exploded view of the product, usually via a three-dimensional or isometric drawing. (Fig. 5.12) Assembly chart: A graphic means of identifying how components flow into subassemblies and ultimately into the final product. (Fig. 5.12) Route sheet: A listing of the operations necessary to produce the component with the material specified in the bill of materials. Engineering change notice (ECN): a correction or modification of an engineering drawing or BOM. Configuration Management: A system by which a product’s planned and changing components are accurately identified and for which control of accountability of change are maintained

13. Documenting Product Designs (cont.) Work order: An instruction to make a given quantity (known as production lot or batch) of a particular item, usually to a given schedule. Group technology: A product and component coding system that specifies the type of processing and the involved parameters, allowing thus the identification of processing similarities and the systematic grouping/classification of similar products. Some efficiencies associated with group technology are: –Improved design (since the focus can be placed on a few critical components –Reduced raw material and purchases –Improved layout, routing and machine loading –Reduced tooling setup time, work-in-process and production time –Simplified production planning and control

14. Engineering Drawing Example (borrowed from Heizer & Render)

15. Bill of Material (BOM) Example (borrowed from Heizer & Render)

16. Assembly Drawing & Chart Examples (borrowed from Heizer & Render)

17. Operation Process Chart Example (borrowed from Francis et. al.)

18. Route Sheet Example (borrowed from Francis et. al.)

19. “Make-or-buy” decisions Deciding whether to produce a product component “in- house”, or purchase/procure it from an outside source. Issues to be considered while making this decision: –Quality of the externally procured part –Reliability of the supplier in terms of both item quality and delivery times –Criticality of the considered component for the performance/quality of the entire product –Potential for development of new core competencies of strategic significance to the company –Existing patents on this item –Costs of deploying and operating the necessary infrastructure

20. A simple economic trade-off model for the “Make or Buy” problem Model parameters: c1 ($/unit): cost per unit when item is outsourced (item price, ordering and receiving costs) C ($): required capital investment in order to support internal production c2 ($/unit): variable production cost for internal production (materials, labor,variable overhead charges) Assume that c2 < c1 X: total quantity of the item to be outsourced or produced internally X Total cost as a function of X C C+c2*X c1*X X0 = C / (c1-c2)

21. Example: Introducing a new (stabilizing) bracket for an existing product Machine capacity available Required “infrastructure” for in-house production –new tooling: $12,500 –Hiring and training an additional worker: $1,000 Internal variable production (raw material + labor) cost: $1.12 / unit Vendor-quoted price: $1.55 / unit Forecasted demand: 10,000 units/year for next 2 years  X0 = (12,500+1,000)/(1.55-1.12) = 31,395 > 20,000  Buy!

22. Evaluating Alternatives through Decision Trees Decision Trees: A mechanism for systematically pricing all options / alternatives under consideration, while taking into account various uncertainties underlying the considered operational context. Example –An engineering consulting company (ECC) has been offered the design of a new product.The price offered by the customer is $60,000. –If the design is done in-house, some new software must be purchased at the price of $20,000, and two new engineers must be trained for this effort at the cost of $15,000 per engineer. –Alternatively, this task can be outsourced to an engineering service provider (ESP) for the cost of $40,000. However, there is a 20% chance that this ESP will fail to meet the due date requested by the customer, in which case, the ECC will experience a penalty of $15,000. The ESP offers also the possibility of sharing the above penalty at an extra cost of $5,000 for the ECC. –Find the option that maximizes the expected profit for the ECC.

23. Decision Trees: Example 1 2 3 0.8 0.2 0.8 0.2 60K-20K-2*15K=10K 10K 60K-40K=20K 60K-40K-15K=5K 17K 60K-45K=15K 60K-45K-7.5K=7.5K 13.5K 17K

24. Technology selection The selected technology must be able to support the quality standards set by the corporate / manufacturing strategy This decision must take into consideration future expansion plans of the company in terms of –production capacity (i.e., support volume flexibility) –product portfolio (i.e., support product flexibility) It must also consider the overall technological trends in the industry, as well as additional issues (e.g., environmental and other legal concerns, operational safety etc.) that might affect the viability of certain choices For the candidates satisfying the above concerns, the final objective is the minimization of the total (i.e., deployment plus operational) cost

25. Production Capacity Design capacity: the “theoretical” maximum output of a system, typically stated as a rate, i.e., product units / unit time. Effective capacity: The percentage of the design capacity that the system can actually achieve under the given operational constraints, e.g., running product mix, quality requirements, employee availability, scheduling methods, etc. Plant utilization = actual prod. rate / design capacity Plant efficiency = actual prod. rate / (effective capacity x design capacity) Notice that actual prod. rate = (design capacity) x (utilization) = (design capacity) x (effective capacity) x (efficiency)

26. Capacity Planning Capacity planning seeks to determine –the number of units of the selected technology that needs to be deployed in order to match the plant (effective) capacity with the forecasted demand, and if necessary, –a capacity expansion plan that will indicate the time-phased deployment of additional modules / units, in order to support a growing product demand, or more general expansion plans of the company (e.g., undertaking the production of a new product in the considered product family). Frequently, technology selection and capacity planning are addressed simultaneously, since the required capacity affects the economic viability of a certain technological option, while the operational characteristics of a given technology define the production rate per unit deployed and aspects like the possibility of modular deployment.

27. Quantitative Approaches to Technology Selection and Capacity Planning All these approaches try to select a technology (mix) and determine the capacity to be deployed in a way that it maximizes the expected profit over the entire life-span of the considered product (family). Expected profit is defined as expected revenues minus deployment and operational costs. Typically, the above calculations are based on net present values (NPV’s) of the expected costs and revenues, which take into consideration the cost of money: NPV = (Expense or Revenue) / (1+i) N where i is the applying interest rate and N the time period of the considered expense. Possible methods used include: –Break-even analysis, similar to that applied to the “make or buy” problem, that seeks to minimizes the total (fixed + variable) cost. –Decision trees which allow the modeling of problem uncertainties like uncertain market behavior, etc., and can determine a strategy as a reaction to these unknown factors. –Mathematical Programming formulations which allow the optimized selection of technology mixes.

28. Selecting the Process Layout

29. Operation Process Chart Example for discrete part manufacturing (borrowed from Francis et. al.)

30. Major Layout Types (borrowed from Francis et. al.)

31. Advantages and Limitations of the various layout types (borrowed from Francis et. al.)

32. Advantages and Limitations of the various layout types (cont. - borrowed from Francis et. al.)

33. Selecting an appropriate layout (borrowed from Francis et. al.)

34. The product-process matrix Jumbled flow (job Shop) Disconnected line flow (batch) Connected line flow (assembly Line) Continuous flow (chemical plants) Process type Production volume & mix Low volume, low standardi- zation Multiple products, low volume Few major products, high volume High volume, high standardization, commodities Commercial printer Heavy Equipment Auto assembly Sugar refinery Void

35. Cell formation in group technology: A clustering problem Partition the entire set of parts to be produced on the plant-floor into a set of part families, with parts in each family characterized by similar processing requirements, and therefore, supported by the same cell. Part-Machine Indicator Matrix

36. Clustering Algorithms for Cellular Manufacturing Row & Column Masking

37. Clustering Algorithms for Cellular Manufacturing: Similarity Coefficients - Motivation 1 1

38. Clustering Algorithms for Cellular Manufacturing: Similarity Coefficients - Definitions P(Mi) = set of parts supported by machine Mi |P(Mi)| = cardinality of P(Mi), i.e., the number of elements of this set SC(Mi,Mj) = |P(Mi)  P(Mj)| / |P(Mi)  P(Mj)| = |P(Mi)  P(Mj)| / (|P(Mi)|+|P(Mj)|-|P(Mi)  P(Mj)|) Notice that:0  SC(Mi,Mj)  1.0, and the closer this value is to 1.0 the greater the similarity among the part sets supported by machines Mi and Mj. By picking a desired threshold, one can cluster together all machines that have a similarity coefficient greater than or equal to this threshold.

39. A typical (logical) Organization of the Production Activity in Repetitive Manufacturing Raw Material & Comp. Inventory Finished Item Inventory S1,2 S1,1S1,n S2,1S2,2S2,m Assembly Line 1: Product Family 1 Assembly Line 2: Product Family 2 Fabrication (or Backend Operations) Dept. 1Dept. 2Dept. k S1,i S2,i Dept. j

40. Synchronous Transfer Lines: Examples (Pictures borrowed from Heragu)

41. Flow Patterns for Product-focused Layouts (borrowed from Francis et. al.)

42. Discrete vs. Continuous Flow and Repetitive Manufacturing Systems (Figures borrowed from Heizer and Render)

43. Production Planning and Scheduling

44. Dealing with the Problem Complexity through Decomposition Aggregate Planning Master Production Scheduling Materials Requirement Planning Aggregate Unit Demand End Item (SKU) Demand Corporate Strategy Capacity and Aggregate Production Plans SKU-level Production Plans Manufacturing and Procurement lead times Component Production lots and due dates Part process plans (Plan. Hor.: 1 year, Time Unit: 1 month) (Plan. Hor.: a few months, Time Unit: 1 week) Shop floor-level Production Control (Plan. Hor.: a day or a shift, Time Unit: real-time)

45. Aggregate Planning

46. Product Aggregation Schemes Items (or Stock Keeping Units - SKU’s): The final products delivered to the (downstream) customers Families: Group of items that share a common manufacturing setup cost; i.e., they have similar production requirements. Aggregate Unit: A fictitious item representing an entire product family. Aggregate Unit Production Requirements: The amount of (labor) time required for the production of one aggregate unit. This is computed by appropriately averaging the (labor) time requirements over the entire set of items represented by the aggregate unit. Aggregate Unit Demand: The cumulative demand for the entire set of items represented by the aggregate unit. Remark: Being the cumulate of a number of independent demand series, the demand for the aggregate unit is a more robust estimate than its constituent components.

47. Computing the Aggregate Unit Production Requirements Aggregate unit labor time = (.32)(4.2)+(.21)(4.9)+(.17)(5.1)+(.14)(5.2)+ (.10)(5.4)+(.06)(5.8) = 4.856 hrs

48. Aggregate Planning Problem Aggregate Planning Aggregate Unit Demand Aggregate Unit Availability (Current Inventory Position) Aggregate Production Plan Required Production Capacity Aggr. Unit Production Reqs Corporate Strategy Aggregate Production Plan: Aggregate Production levels Aggregate Inventory levels Aggregate Backorder levels Production Capacity Plan: Workforce level(s) Overtime level(s) Subcontracted Quantities

49. Pure Aggregate Planning Strategies 1. Demand Chasing: Vary the Workforce Level D(t)P(t) = D(t) W(t) PCWCHCFC D(t): Aggregate demand series P(t): Aggregate production levels W(t): Required Workforce levels Costs Involved: PC: Production Costs fixed (setup, overhead) variable (materials, consumables, etc.) WC: Regular labor costs HC: Hiring costs: e.g., advertising, interviewing, training FC: Firing costs: e.g., compensation, social cost

50. Pure Aggregate Planning Strategies 2. Varying Production Capacity with Constant Workforce: D(t)P(t) O(t) PCWCOCUC U(t) S(t) SC W = constant S(t): Subcontracted quantities O(t): Overtime levels U(t): Undertime levels Costs involved: PC, WC: as before SC: subcontracting costs: e.g., purchasing, transport, quality, etc. OC: overtime costs: incremental cost of producing one unit in overtime (UC: undertime costs: this is hidden in WC)

51. Pure Aggregate Planning Strategies 3. Accumulating (Seasonal) Inventories: D(t)P(t) I(t)PCWCIC W(t), O(t), U(t), S(t) = constant I(t): Accumulated Inventory levels Costs involved: PC, WC: as before IC: inventory holding costs: e.g., interest lost, storage space, pilferage, obsolescence, etc.

52. Pure Aggregate Planning Strategies 4. Backlogging: D(t)P(t) B(t) PCWCBC W(t), O(t), U(t), S(t) = constant B(t): Accumulated Backlog levels Costs involved: PC, WC: as before BC: backlog (handling) costs: e.g., expediting costs, penalties, lost sales (eventually), customer dissatisfaction

53. Typical Aggregate Planning Strategy A “mixture” of the previously discussed pure options: D PCWCHCFCOCUCSCICBC P W H F O U S I B + Additional constraints arising from the company strategy; e.g., maximal allowed subcontracting maximal allowed workforce variation in two consecutive periods maximal allowed overtime safety stocks etc. Io Wo

54. Solution Approaches Graphical Approaches: Spreadsheet-based simulation Analytical Approaches: Mathematical (mainly linear programming) Programming formulations

55. A prototype problem Forecasted demand: Jan: 1280 Feb: 640 Mar: 900 Apr: 1200 May:2000 Jun: 1400 On-hand Inventory: 500 Required on-hand Inventory at end of June: 600 Current Workforce Level: 300 Worker prod.capacity: 0.14653 units/day Working days per month Jan: 20 Feb: 24 Mar: 18 Apr: 26 May: 22 Jun: 15 Cost structure: Inv. holding cost: $80/unit x month Hiring cost: $500/worker Firing cost: $1000/worker

56. A prototype problem (cont.) Net predicted demand: Jan: 780 Feb: 640 Mar: 900 Apr: 1200 May: 2000 Jun: 2000 Forecasted demand: Jan: 1280 Feb: 640 Mar: 900 Apr: 1200 May:2000 Jun: 1400 On-hand Inventory: 500 Required on-hand Inventory at end of June: 600

57. An LP formulation for the prototype problem Problem Parameters D t = Forecasted demand for period t d t = working days at period t c = daily worker capacity W 0 =Initial workforce level I 0 = Current on-hand inventory C H = Hiring cost per worker C F = Firing cost per worker C I = Inventory holding cost per unit per period Problem Decision Variables H t = Workers hired at period t F t = Workers fired at period t W t = Workforce level at period t P t = Level of production at period t I t = Inventory at the end of period t

58. An LP formulation for the prototype problem s.t.

59. Optimal Plan for the considered example Fire 27 workers in January Hire 465 workers in May Produce at full (labor) capacity every month Resulting total cost: $379320.900

60. Analytical Approach: A Linear Programming Formulation min TC =  t ( PC t *P t +WC t *W t +OC t *O t +HC t *H t +FC t *F t + SC t *S t +IC t *I t +BC t *B t ) s.t.  t, P t +I t-1 +S t = (D t -B t )+B t-1 +I t  t, W t = W t-1 +H t -F t  t, (u_l_r)*P t  s_d)  w_d) t *W t +O t  t, P t, W t, O t, H t, F t, S t, I t, B t  0 ( )Any additional policy constraints Prod. Capacity: Material Balance: Workforce Balance: Var. sign restrictions: Time unit: month / unit_labor_req. /shift_duration (in hours) / (working_days) for month t

61. Demand (vs. Capacity) Options or Proactive Approaches to Aggregate Planning Influencing demand variation so that it aligns to available production capacity: –advertising –promotional plans –pricing (e.g., airline and hotel weekend discounts, telecommunication companies’ weekend rates) “Counter-seasonal” product (and service) mixing: Develop a product mix with antithetic (seasonal) trends that level the cumulative required production capacity. –(e.g., lawn mowers and snow blowers) => The outcome of this type of planning is communicated to the overall aggregate planning procedure as (expected) changes in the demand forecast.

62. Disaggregation and Master Production Scheduling (MPS)

63. The (Master) Production Scheduling Problem MPS Placed Orders Forecasted Demand Current and Planned Availability, eg., Initial Inventory, Initiated Production, Subcontracted quantities Master Production Schedule: When & How Much to produce for each product Capacity Consts. Company Policies Economic Considerations Product Charact. Planning Horizon Time unit Capacity Planning

64. MPS Example: Company Operations Mashing (1 mashing tun) Boiling (1 brew kettle) Fermentation (3 40-barrel ferm. tanks) Filtering (1 filter tank) Bottling (1 bottling station) Grain cracking (1 milling machine) Fermentation Times:

65. Example: Implementing the Empirical Approach in Excel

66. Computing Inventory Positions and Net Requirements Net Requirement: NR i = abs(min{0, IP i }) Inventory Position: IP i = max{IP i-1,0}+ SR i +BNR i -D i (Material Balance Equation) i DiDi IP i (IP i-1 ) + SR i +BNR i

67. Problem Decision Variables: Scheduled Releases

68. Testing the Schedule Feasibility

69. Fixing the Original Schedule

70. Infeasible Production Requirements

71. A feasible schedule with spoilage effects

72. Computing Spoilage and Modified Inventory Position Spoilage: SP i = max{0, IP i-1 -  SR i-1 +SR i-2 +…+SR i-sl+1 ) -  BNR i-1 +BNR i-2 +…+BNR i-sl+1 )} Inventory Position: IP i = max{IP i-1,0}+ SR i +BNR i -D i -SP i (Material Balance Equation) i DiDi IP i (IP i-1 ) + SR i +BNR i SP i

73. The Driving Logic behind the Empirical Approach DemandAvailability: Initial Inventory Position Scheduled Receipts due to initiated production or subcontracting Future inventories Net Requirements Lot Sizing Scheduled Releases Resource (Fermentor) Occupancy Product i Feasibility Testing Master Production Schedule Schedule Infeasibilities Revise Prod. Reqs Compute Future Inventory Positions

74. Materials Requirements Planning (MRP)

75. The “MRP Explosion” Calculus BOM MRP MPS Current Availabilities Planned Order Releases Priority Planning Lead Times Lot Sizing Policies

76. Example: The (complete) MRP Explosion Calculus Item BOM: Alpha C(2)D(2) B(1)C(1) E(1) F(1) Item Levels: Level 0: Alpha Level 1: B Level 2: C, D Level 3: E, F

77. (borrowed from Heizer and Render)

78. The “MRP Explosion” Calculus Level 0 Level 1 Level 2 Level N Initial Inventories Scheduled Receipts External Demand Capacity Planning Planned Order Releases Gross Requirements

79. Computing the item Scheduled Releases Synthesizing item demand series Projecting Inv. Positions and Net Reqs. Lot Sizing Time- Phasing Parent Sched. Rel. Item External Demand Gross Reqs Scheduled Receipts Initial Inventory Safety Stock Requirements Net Reqs Lot Sizing Policy Planned Order Receipts Lead Time Planned Order Releases

80. Some Lot Sizing Heuristics Economic Order Quantity (EOQ): Compute a lot size using the EOQ formula with the demand rate D set equal to the average of the demand values observed over the considered planning horizon. Periodic Order Quantity (POQ): Compute T = round(EOQ/D), and every time you schedule a new lot, size it to cover the net requirements for the subsequent T periods. Silver-Meal (SM): Every time you start a new lot, keep adding the net requirements of the subsequent periods, as long as the average (setup plus holding) cost per period decreases. Least Unit Cost (LUC): Every time you start a new lot, keep adding the net requirements of the subsequent periods, as long as the average (setup plus holding) cost per unit decreases. Part Period Balancing (PPB): Every time you start a new lot, add a number of subsequent periods such that the total holding cost matches the lot set up cost as much as possible.

81. Capacity Planning (Example) Available labor hours Periods123 4567 8 50 100 150

82. (borrowed from Heizer and Render) Pegging and Bottom-up Replanning

83. Shop floor-level Production Control / Scheduling

84. General Problem Definition Determine the timing of –the releases of the various production lots on the shop-floor and –the allocation to them of the system resources required for the execution of their various operations so that the production plans decided at the tactical planning - i.e., MPS & MRP - level are observed as close as possible.

85. Example W_q W_2W_i W_M W_1 J_1 J_2 J_N

86. A modeling abstraction M: number of machine types / workstations. N: number of jobs to be scheduled. Job routing: an ordered list / sequence of machines that a job needs to visit in order to be completed. Operation: a single processing step executed during the job visit to a machine. P_j: the set of operations in the routing of job j. t_kj: the processing time for the k-th operation of job j. d_j: due date for job j. r_j: the release date of job j, i.e., the date at which the material required for starting the job processing will be available.

87. Example

88. A feasible schedule and its Gantt Chart 1 2 3 4 5 5101520 Machine Time Job 1Job 2Job 3Job 4Job 5

89. Performance-related job and schedule attributes job completion time: C_j schedule makespan: max_j C_j job lateness: L_j = C_j - d_j (notice that, by definition, job lateness can be either positive or negative - in which case that the job is finished earlier than its due date) job tardiness: T_j = max (0, L_j) = [L_j]+ job flow time: F_j =C_j - r_j (i.e., the amount of time the job spends on the shop-floor) job tardy index: TI_j = 1 if job is tardy; 0 otherwise. Number of tardy jobs: NT job importance weight: w_j (the higher the weight, the more important the job)

90. Performance Criteria

91. Schedule Performance Evaluation

92. Problem variations Based on job routing: –job shop: each job has an arbitrary route –flow shop: all jobs have the same route, but different operational processing times –re-entrant flow shop: some machine(s) is visited more than once by the same job –flexible job shop / flow shop: each operation has a number of machine alternatives for its execution Based on the operational processing times: –deterministic: the various processing times are known exactly –stochastic: the processing times are known only in distribution Based on the possibility of pre-emption: –pre-emptive: the execution of a job on a machine can be interrupted upon the arrival of a new job –non-preemptive: each machine must complete its currently running job before switching to another one. Based on the considered performance objective(s)

93. Solution Approaches Analytical (Mixed Integer Programming) formulations: Notoriously difficult to solve even for relatively small configurations Heuristics: In the scheduling literature, the applied heuristics are known as dispatching rules, and they determine the sequencing of the various jobs waiting upon the different machines, based upon job attributes like –the required processing times –due dates –priority weights –slack times, defined as d_j - (current time + total remaining processing time for job j) –Critical ratios, defined as (d_j-current time)/rem. proc. time for job j

94. Assembly Line Balancing

95. Synchronous Transfer Lines: Examples (Pictures borrowed from Heragu)

96. Balancing Synchronous Transfer Lines Given: –a set of m tasks, each requiring a certain (nominal) processing time t_i, and –a set of precedence constraints regarding the execution of these m tasks, assign these tasks to a sequence of k workstations, in a way that –the total amount of work assigned to each workstation does not exceed a pre-defined cycle time c, (constraint I) –the precedence constraints are observed, (constraint II) –while the number of the employed workstations k is minimized. (objective) Remark: The problem is hard to solve optimally, and quite often it is addressed through heuristics.

97. Heuristics for Assembly Line Balancing Developed in class – c.f. your class notes!