1. 1 Contents College 9 Chapter 9 additional (sheets): –workforce planning –resource loading

2. 2 Hierarchical capacity planning Strategic Aggregate/ strategic planning Resource loading Scheduling Tactical Operational Rough-cut process planning Engineering & process planning

3. 3 Workforce planning Strategic level: workforce planning –hiring/firing staff –training staff Tactical level: resource loading –planning overtime work –hiring temporary staff (e.g. Randstad) –subcontracting Operational level: workforce scheduling –staffing –shift scheduling –days-off scheduling Long term decisions Intermediate term decisions Short term decisions

4. 4 Capacity planning solution techniques (Strategic) workforce planning: –linear programming (Tactical) resource loading: –integer linear programming –stochastic techniques (Operational) workforce scheduling: –integer linear programming –cyclic staffing algorithms

5. 5 Workforce planning LP model (1 product, various machines) Demand forecast:  sales (S t ) restriction: Machine capacity restriction (machine j): a j = production time on machine j c jt = capacity machine j in period t X t = amount produced in period t 

6. 6 Inventory balance restrictions: I t = inventory in period t  Workforce balance restrictions (operators): W t = workforce in period t (initial workforce=w) H t = hired workforce in period t F t = fired workforce in period t  Workforce planning LP model (cont.)

7. 7 Operator capacity restriction (machine j): b = man-hours required to produce one item W t = Workforce in period t O t = Overtime in period t X t = amount produced in period t  objective: maximize net profit, including labor, overtime, inventory, hiring/firing costs

8. 8 Workforce planning LP model overview

9. 9 Resource loading example: furniture manufacturer Furniture manufacturer: Produces large quantities of furniture to order Mostly standard products (manufacture-to-order) 6 production activities: 1. Sawmilling (S) 2. Assembly (A) 3. Cleaning (C) 4. Painting (P) 5. Decoration (D) 6. Quality control (Q)

10. 10 Resource loading example (cont.) Order acceptance: Sales department negotiates prices & due dates (i.e. delivery dates) with customers Given are for each department: –available machine capacity (in hours per week) –operator capacity per week (regular & overtime)

11. 11 Resource loading example (cont.) example data SACPDQ Example customer order:

12. 12 Resource loading questions During order acceptance, for any given set of orders, these questions need to be answered: When should orders be released for production? Can the delivery dates be met? How much operator & machine capacity is required per department per week? Is irregular capacity (e.g. overtime work, subcontracting) required? This problem is the so-called resource loading problem

13. 13 Resource loading example (cont.) loading method of planner customer order consists of x jobs, each corresponding to a production activity lead time = 1 week per job operator and machine capacity check per department repair plan, if infeasible let each department solve their scheduling problem

14. Machine capacity check for D- department: Machine capacity check for Q- department:

15. Optimal loading of D-department: Machine capacity check for D- department:

16. Optimal loading of Q-department: Machine capacity check for Q- department:

17. 17 Repairing an infeasible plan Tardy orders may induce penalty costs Options to come to a feasible plan: Shift jobs in time; split jobs over 2 or more weeks Increase order lead time Expand operator capacity (overtime, hiring staff) Subcontracting jobs or entire orders

18. 18 Resource loading problems formal description Production system: job shop with operators and machine groups Demand: a predefined set of orders that consist of jobs Time horizon/time unit: T periods (=weeks) / hours Order constraints: release and due dates (weeks) Job constraints: –Pre-emption is allowed –Precedence relations (generic) –Machine and operator requirement (hours) –Optional: one-job-per-order-per-week production policy –Minimal duration

19. 19 Resource loading problems (cont.) formal description Resource capacity constraints: –Operator capacity: regular, overtime, hiring, subcontracting –Machine group capacity: regular Objective: assign jobs to machines and operators, and: minimize the cost of the use of non-regular capacity

20. 20 Resource loading Model formulation Order plan: indicates per job per order per week whether this job may be processed in that week  Input for the model as binary columns Order schedule: indicates per job per order per week the fraction that is assigned to that week  Output of the model; must match with order plan

21. 21 Resource loading Model formulation (cont.) Order plan represented by a binary column: 1 2 3

22. 22 Resource loading Model objective Select one order plan per order, and determine the corresponding order schedule. Minimize the use of non-regular capacity. PROBLEM: there are exponentially many feasible order plans SOLUTION: column generation approach

23. 23 Column generation algorithm Add columns to RLP Solve Restricted Linear Program (RLP) Solve pricing algorithm Columns exist with negative reduced costs? LP relaxation solved yes no

24. 24 Model formulation (cont.) MILP model for resource loading minimize select order plan match order schedule with order plan all work must be done machine capacity restriction operator capacity restriction objective function subject to: variable restrictions

25. 25 Workforce scheduling topics (chapter 9) Days-off scheduling –assigning employees to work-patterns –various assignment patterns over the cycle Shift scheduling –assigning employees to shift-patterns –each shift has its own cost –objective: minimize cost Cyclic staffing problem & extensions –assigning people to m-period cyclic schedule, so that requirement in each period is met Crew scheduling –transportation crew scheduling (airline industry)

26. 26 Days-off scheduling Find the minimum number of employees to cover a 7-day-a-week operation, so that the following constraints are satisfied: 1. The demand per day (n j ) is met (n 1 =Sunday) 2. Each employee is given each k 1 out of k 2 weekends off 3. Each employee works exactly 5 out of 7 days 4. Each employee works no more than 6 consecutive days

27. 27 Days-off scheduling (cont.) W = required workforce 3 lower bounds on W: 1. weekend constraint: 2. total demand constraint: 3. maximum daily demand constraint:

28. 28 Days-off scheduling (cont.) Define: n = max (n 1,n 7 ) = maximum weekend demand u j = surplus of employees =W - n j (if j=2,…,6) n - n j (if j=1,7) Assumption: the first day to be scheduled is a Saturday

29. 29 Days-off scheduling algorithm STEP 1: schedule the weekends off maximum demand in weekend is n  W-n empl. Free  Cyclically assign the weekends of to W-n employees STEP 2: Determine the additional off-day pairs Construct a list of n “off day”-pairs (k,l): –choose day k, such that u k =max (u 1,…,u 7 ) –choose day l (l  k) such that u l   0, if u l = 0 for all l, then set l = k. –decrease u k and u l with 1 Pairs (k,k) are non-distinctive pairs

30. 30 Days-off scheduling algorithm (cont.) STEP 3: Categorize employees in week i type T1 weekend i offweekend i+1 off 0 type T2 weekend i offweekend i+1 on 1 type T3 weekend i onweekend i+1 off 1 type T4 weekend i onweekend i+1 on 2 note: |T3|+|T4| = n, and |T2|+|T4| = n  |T2|=|T3|  pair each employee of T2 with an employee of T3 Off-days required during week i

31. 31 Days-off scheduling algorithm (cont.) STEP 4: Assign the n off-day pairs in week i First assign off-day pairs to the employees of T4 Then assign off-day pairs to the T2-T3 pairs of employees. Assign the earliest day to the T3-empl. Set i = i+1 and GO TO STEP 3.

32. 32 Days-off scheduling example Demand requirement: n = max(d 1,d 7 )=2 surplus: day-pairs: Sun-Mon & Mon-Mon

33. 33 Days-off scheduling example (cont.)

34. 34 Days-off scheduling example (cont.)

35. 35 Days-off scheduling example (cont.) Minimum 4-day, maximum 6-day work stretch:

36. 36 Days-off scheduling algorithm properties If all off-pairs are distinct  maximum work- stretch is 5 days Schedule always satisfies the constraints There exists an optimal cyclic schedule, that may be found by the algorithm

37. 37 Shift scheduling Generalization of days-off scheduling problem Cycle of periods is predetermined (e.g. 1 day cycle, with periods of hours) Several shift patterns, with associated costs c j. Shift pattern j is binary vector a(i,j). Personnel requirement b i per period solution method: linear programming

38. 38 Shift scheduling ILP model N patterns m periods b i = requirement 

39. 39 Strongly NP-hard problem When shift-columns contain a contiguous set of 1’s, the LP-relaxation solution is integer (note: LP optimization in polynomial time) Other special cases of shift scheduling also solvable: cyclic staffing problem Shift scheduling ILP model properties

40. 40 Cyclic staffing problem Special case of shift planning problem m-period cyclic schedule: period m is followed by period 1 Personnel requirement is b i per period Each person works k consecutive periods, and is free for the remaining m-k periods Example: (3,5)-cyclic staffing problem: Columns of A do not always have a contiguous set of ones  LP solution close to ILP solution

41. 41 Exact cyclic staffing algorithm STEP 1: solve LP-relaxation  solution vector x* if x* integer  done, else: go to step 2 STEP 2: form two linear programs LP’ and LP’’, so that these constraints are added respectively: LP’’ has an optimal solution that is integer if LP’ is not feasible  LP’’ solution optimal otherwise: LP’ has an optimal solution that is integer, and the optimal solution is the best of the solutions of LP’ and LP’’

42. 42 Extensions of cyclic staffing Days-off scheduling with a fixed cycle –each employee 2 days off a week –every other weekend off (e.g. 1st, 3rd, 5th, etc.)  solved by column generation Cyclic staffing with overtime –3 work shifts: 8:00-16:00, 16:00-0:00, 0:00-8:00 –matrix A:  solved by algorithm 9.4.1 (cyclic staffing algorithm)

43. 43 Extensions of cyclic staffing (cont.) Cyclic staffing with linear penalties for under- and overstaffing –varying demand b i –penalty c i ’ and c i ’’ for under- and overstaffing –understaffing = x’  overstaffing = b i - (a i1 x 1 + … + a i1 x 1 ) - x i ’ model: if c-c’’A  0 and c’-c’’  0  solvable by alg. 9.4.1

44. 44 Crew scheduling Applications in airline/transportation industry Each crew performs a number of jobs (flight legs) in a so-called ‘round trip’ There are m jobs (flight legs), and n feasible round trips A round trip has costs c j objective: choose a set of round trips, so that each flight leg is covered exactly once, by one and only one round trip  set partitioning problem solution method: column generation heuristic on ILP model

45. 45 ILP model for crew scheduling N round trips m jobs