Applying LP to Plan Production at Jos A Bank

Transcription

1 Applying LP to Plan Production at Jos A Bank

2 Learning objectives today Understand Jos A Bank production Issues in production planning Formulating an LP to plan production Issues in formulation Solving the LP using Xpress

3 Goal(s) Maximize long-term Economic Profit by manufacturing and selling suits produce the right product and offer it at the right price (service goal) minimize the cost to deliver (cost goal) This way of thinking (separating the revenue goal from the cost goal) is quite common in business and industry

4 General Business Process Customer orders Marker-making Inventory data How are orders batched together for marker-making? Marker Spreading What is the lead-time for inventory, and how are the fabrics and colors selected? Fabric from inventory

5 General Business Process, cont d Marker Spreading Fabric from inventory Cutting Fitting Sub-assembly Final Ass y Press/Inspect Technological and labor requirements for each process step

6 Marker driven batch operations Cut Match & Tag Bundle cut All these are the same part for the same style/size Pull out and tag all the parts for the same suit (style, size, and color) Bundle together identical parts, so they can all be sewn at one time How many bundles are created from each marker?

7 Estimating spreading/cutting labor Depends on the number of layers Depends on the type(s) of fabric Depends on the length of the table Issue: how to size capacity

8 Marker-Making Extremely difficult problem Good marker minimizes waste by getting the maximum number of parts onto the marker A suit comes from a single layer Want to spread as much fabric as possible (layers) because the maximizes cutting productivity

9 Order driven batch operations bundles subassembly subassembly Final assembly One or more orders

10 Estimating assembly labor content Depends on the technology Depends on the style, fabric, pattern Depends on the experience of the operator Depends on the bundle size Traditional IE role is to set standards

11 An operator Works on many bundles during a shift Has a buffer of bundles waiting, to insure no idle time

12 To finish a suit The pieces for the suit are cut at the same time, then fitted and tagged But then they are bundled with pieces for other suits (of the same style, size, fabric) The five different bundles, traveling five different routes, all come together at final The bundles go to different operators; variability in queue time and process time means they don t all arrive together at final assembly

13 Design Questions How many operators are needed in assembly? What is the manufacturing cycle time? (or how much WIP on average?) What is the minimum production quantity for a size/style/color?

14 Number of Operators Std hrs for each operation Number of suits of each type Total std hrs Hrs avail per operator (take out meals, breaks, and allowances average over the year/season

15 Examining Impacts labor Inventory, delay Operating Cost per suit Average operator utilization Queuing phenomenon!

16 Design Issue Each operator is a server How many servers do we need? Trade-off cost of operators (labor) against the cost of waiting (inventory and customer service) Capacity is not a free variable

17 Why Study Queues? Because understanding Little s Law, and the behavior of queues helps us to understand the behavior of Jos A Bank s sewing room, and understand the interplay between the number of sewing operators, the manufacturing cycle time, the work-in-process inventory, and the total cost of garments produced.

18 How can Jos A Bank plan production over the season?

19 Hypothetical Sales Forecast Units Required Week Coats Trousers

20 Fundamental Decisions How much work to release each week, for both trousers and coats X(t,1), X(t,2) How much inventory to carry in anticipation of peak demand I(t,1), I(t,2)

21 Objective Minimize the total inventory carried over the duration of the season Sum of the week-ending inventory values T [ I( t,1) + = t 1 I( t,2)]

22 Types of Constraints Technical Financial Work practices Conceptual

23 Let s simplify cutting sewing yds/garment min/garment coats trousers capacity 5000 yds/week 6 operators 3.2X ( t,1) + 2.2X ( t,2) 5000 t = 1,... T 7.34X ( t,1) X ( t,2) 6*420 t = 1,... T

24 Lead-time Suppose we have done the queuing or simulation analysis, and determined that the lead-time is five weeks (note: we know, in reality, that leadtime will be a function of loading and operator utilization, so it actually will vary as we change the release rate)

25 Consider only trousers, and leadtime = zero weeks prod p(2) p(3) p(4) inv i(2) i(3) i(4) d(1) d(2) d(3) d(4) i(t) = i(t-1) + p(t) - d(t) Boundary conditions for first and last periods Inventory balance (flow conservation) equation

26 Suppose Lead-Time = 1 Week prod p(2) p(3) p(4) 1 inv i(2) i(3) i(4) d(1) d(2) d(3) d(4)

27 Lead-Time = 2 weeks prod p(2) p(3) p(4) inv i(2) i(3) i(4) d(1) d(2) d(3) d(4)

28 Visual Xpress Model for Jos A Bank model JABank! Start a new model uses "mmxprs"! Load the optimizer library declarations nprod = 2!number of products--coats, trousers iprod = 1..nprod nper = 26!number of weeks in the plan iper = 1..nper LT = 5!production lead time in weeks ilt = 1..LT D: array(iper, iprod) of real!demand in each period for coats and trousers y: array(iprod) of real!cutting standards for coats and trousers s: array(iprod) of real!sewing standards for coats and trousers X: array(iper, iprod) of mpva!quantity of coats and trousers to be started in each period I: array(iper, iprod) of mpvar!inventory of coats and trousers at the beginning of each period end-declarations

29 Visual Xpress Model for Jos A Bank y := [3.2, 2.2] s := [7.34, 5.25] YPW := 6000!cutting capacity in yards per week SOP := 5!sewing operators available initializations from 'jabank.dat' D end-initializations forall(i in iprod) s(i) := s(i)/60!convert sewing standards into hours from minutes

30 Visual Xpress Model for Jos A Bank!objective function; total inventory carried during the planning horizon obj_fcn:= sum(j in iper) I(j,1)+sum(j in iper) I(j,2)!capacity constraints forall (j in iper) cutting (j):= y(1)*x(j,1) + y(2)*x(j,2) <= YPW!can't cut more than 5000 yds per week forall (j in iper) sewing (j):= s(1)*x(j,1) + s(2)*x(j,2) <= SOP*40!can sew more than SOP*40 hours per week!balance constraints forall (j in LT+1..nper) bal_coats (j):= I(j-1,1) + X(j-LT,1) -D(j,1)-I(j,1) = 0 forall (j in LT+1..nper) bal_trou (j):= I(j-1,2) + X(j-LT,2) -D(j,2)-I(j,2) = 0!boundary conditions forall (j in ilt) init (j):= I(j,1)+I(j,2)=0 minimize(obj_fcn)

31 Visual Xpress Model for Jos A Bank! Print out the solution writeln("solution ") writeln(" Objective: ", getobjval) writeln writeln(" Coat Trouser") writeln("period Production Inventory Production Inventory") writeln forall (j in iper) writeln(" ", strfmt(j, 2), " ", strfmt(getsol(x(j,1)),6,0)," ", strfmt(getsol(i(j,1)), 6,0), " ", strfmt(getsol(x(j,2)), 6, 0), " ", strfmt(getsol(i(j,2)), 6, 0)) end-model

32 Data File for Jos A Bank D: [ ]

33 Xpress-IVE

34 Solution--Trousers Demand Production Inventory

35 Solution--Coats Demand Production Inventory

36 This Model: Enforces the cutting capacity constraint Enforces the limit on sewing operators Enforces inventory balance Satisfies all demand Minimizes the total unit-weeks of inventory

37 What s tricky about this? Lead-time!

38 NOTE: I can use this model to determine the minimum possible number of sewing operators for a given lead-time I can vary the lead-time (What s the relationship between lead-time and number of operators?)

39 Summary Jos A Bank production process Formulating the production planning LP The inventory balance constraint Dealing with lead-time Practices in using Xpress