Alternative lot sizing schemes Anadolu University Industrial Engineering Department Source: Nahmias, S., Production And Operations Analysis, McGraw-Hill /Irwin, Fifth Edition, 2005, ISBN: 0073018651

The three major control phases of the productive system Information requirements for each end item over the planning horizon Master Production Schedule Phase 2 Lot sizing rules and capacity planning Materials Requirements Planning Phase 3 Requirements for raw material Detailed shop floor schedule

The explosion calculus Explosion calculus is as term that refers to the set of rules by which gross requirements at one level of product structure are translated into a production schedule at that level and requirements at lower level. Child level (level2) Parent level (level 1) End item level End item A(2) 1 week C(1) 2 weeks D(2) 1 week B(1) 2 weeks C(2) 2 weeks E(3) 2 weeks

Example 7.1 The Harmon Music Company produces a variety of wind instruments at its plant in Joliet, Illinois. Because the company relatively small, it would like to minimize the amount of money tied to inventory. For that reason production levels are set to match predicted demand as closely as possible. In order to achieve this goal, the company has adopted an MRP system to determine production quantities. One of the instruments produced is the model 85C trumpet. The trumpet retails for $800 and has been a reasonably profitable item for the company. Valve: ses dugmesi Trumpet (end item) Bell Assembly (1) Lead Time: 2 weeks Valve casing (1) Lead times 4 weeks Slide assemblies (3) Lead Time : 2 weeks Valves (3) Lead times : 3 weeks Figure gives the product structure diagram for the construction of the trumpet.

Explosion of trumpet BOM Week 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Demand 77 42 38 21 26 112 45 76 Trumpet Week 8 9 10 11 Scheduled receipts 12 6 23 trumpet at the end of the week 7 week 8 9 10 11 12 13 14 15 16 17 net predicted demand 42 32 26 112 45 76 38 Bell Assembly Week 6 7 8 9 10 11 12 13 14 15 16 17 Gross requirements 42 32 26 112 45 76 38 Net requirements Time-phased net requirements planned order relase (lot-for-lot) Time-phased net requirements 42 32 12 26 112 45 14 76 38 planned order relase (lot-for-lot) Valve Casing Assembly Week 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Gross requirements 126 96 36 78 336 135 42 228 114 Scheduled receipts On-han inventory 186 60 30 Net requirements 66 Time-phased net requirements planned order relase (lot-for-lot) valves

Alternative Lot-Sizing Schemes EOQ lot sizing The silver-meal heuristic Least Unit Cost Part Period Balancing

EOQ Lot Sizing To apply the EOQ formula, we need three inputs: The average demand rate,  The holding cost rate, h Setup cost, K

EOQ Lot Sizing- Valve casing assembly in Ex.7.1 Suppose that the setup operation for the machinery used in this assembly operation takes two workers about three hours. The worker average cost: $22 per hour K= (22)(2)(3) = $132 The company uses a holding cost on a 22 percent annual interest rate. Each valve casing assembly costs $141.82 in materials and value added for labor, Holding cost, h = (141.82)(0.22) / 52 = 0.60 Lot-for-lot policy requires Total holding cost for Total setup cost = 0 = (132)(10) = $1320

EOQ Lot Sizing- Valve casing assembly in Ex.7.1 Week 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Gross requirements 42 32 26 112 45 76 38 Net requirements Time-phased net requirements Planned order relase (EOQ) 139 Planned deliveries Ending inventory 97 55 23 124 106 92 117 Lot-for-lot policy requires Cumulative ending inventory Total holding cost for = (0.6)(653) = $391.8 Total setup cost = (132)(4) )= $528 The total cost of EOQ policy )= $528 + $391.8 = $ 919.8 97 +55 +23 +11 +124 +12 +106 +92 +16 +117 = 653

The silver meal heuristic named for Harlan Meal and Edward Silver is a forward method that requires determining the average cost per period as a function of the number of periods the current order is to span, and stopping the computation when this function first increases. C(T) as the average holding and setup cost per period if the current orders spans the next T periods.

The silver meal heuristic Let (r1,…,rn) be the requirements over n-period horizon. Consider period 1. If we produce just enough in period 1 to meet the demand in period 1, then just incur the order cost K. C(1) = K If we order enough in period 1 to satisfy the demand both periods 1 and 2 then we must hold r2 for one period. C(2)= (K + hr2)/2 Similarly C(3)= (K + hr2 + 2hr3)/3 and in general, C(j)= (K + hr2 + 2hr3 + … + (j-1)hr3) /j Once C(j)>C(j-1), we stop and set y1 = r1 + r2 + … + rn , and begin the process again starting at period j.

The silver meal heuristic – Example 7.2 A machine shop uses the Silver-Meal heuristic to schedule production lot sizes for computer casing. Over the next five weeks the demands for the casing are r = (18, 30, 42, 5, 20). The holding cost is $2 per case per week, and the production setup cost is $80. Find the recommended lot sizing. Starting in period 1 C(1) = 80 C(2) =[80 + (2)(30)] /2 = 70 C(3) = [80 + (2)(30)+ (2)(2)(42)] /3 = 102.67 stop because C(3)>C(2). Set y1 = r1 + r2 = 18 + 30 = 48 Starting in period 3 C(2) =[80 + (2)(5)] /2 = 45 C(3) = [80 + (2)(5)+ (2)(2)(20)] /3 = 56.67 stop Set y3 = r3 + r4 = 42 + 5 = 47 Because period 5 is final period in the horizon, we do not need to start the process again. Set y5 = r5 = 20 y = (48, 0, 47, 0, 20)

Least unit cost The least unit cost (LUC) heurist Similar to Silver – Meal heuristic except instead of dividing the cost over j periods by the number of periods, j, we divide it by the total number of units demanded through period j, r1 + r2 + … + rj . Define C(T) as the average holding and setup cost per unit for a T-period order horizon. Then, C(1) = K/ r1 C(2)= (K + hr2)/(r1 + r2) … C(j)= [K + hr2 + 2hr3 + … + (j-1)hr3] /(r1 + r2 + … + rj ) As with the Silver-Meal heuristic, this computation is stopped when C(j)>C(j-1), and production level is set to r1 + r2 + … + rj-1 . The process is then repeated, starting at period j and continuing until the end of the planning horizon is reached.

Least Unit Cost – Example 7.4 Assume the same requirements schedule and costs as given in Ex.7.2 Starting in period 1 C(1) = 80/18 = 4.44 C(2) =[80 + (2)(30)] /(18+30) = 2.92 C(3) = [80 + (2)(30)+ (2)(2)(42)] /(18+30+42) = 3.42 stop because C(3)>C(2). Set y1 = r1 + r2 = 18 + 30 = 48 Starting in period 3 C(1) = 80 /42 = 1.90 C(2) =[80 + (2)(5)] /(42+5) = 1.92 stop Set y3 = r3 = 42 Starting in period 4 C(1) = 80/5 = 16 C(2) =[80 + (2)(20)] /(5+20) = 4.8 As we reached the en of the horizon, we set y4 = r4 + r5 = 5 + 20 =25. The solution obtain by LUC heuristic y = (48, 0, 42, 25, 0)

Part Period Balancing Although Silver-Meal heuristic seems to give better results in a greater number of cases, part period balancing seems to be more popular in practice. The method is set the order horizon equal to the number of periods that most closely matches the total holding cost with the setup cost over that period. The order horizon that exactly equates holding and setup costs will rarely be an integer number of periods.

Part Period Balancing- Example 7.5 Consider Example 7.2. Starting in period 1 Because 228 exceeds the setup cost of 80, we stop. As 80 is closer to 60 than to 228, the first order horizon is two periods. y1 = r1 + r2 = 18 + 30 = 48 We start again in period 3 We have exceeded the setup cost of 80, so we stop. Because 90 is closer to 80 than 10, the order horizon is three periods. y3 = r3 + r4 + r5 = 42+ 5 + 20= 67 y = (48,0,67,0,0) Order horizon Total holding cost 1 2 60 (30)(2) 3 228 (30)(2)+(2)(2)(42) Order horizon Total holding cost 1 2 10 (5)(2) 3 90 (5)(2)+(2)(2)(20)

Comparison of results Silver-Meal LUC Part Period Balancing Demads Solution y = (48, 0, 47, 0, 20) y = (48, 0, 42, 25, 0) y = (48,0,67,0,0) Holding inventory 30 + 5 = 35 30+20 = 50 30 + 5+(2)(20)=75 Holding cost (35)(2) = 70 (50)(2) =100 (75)(2)=150 Setup cost (3) (80) = 240 (3)(80) = 240 (2)(80) = 160 Total cost 310 340

Incorporating lot-sizing algorithms into the explosion calculus Consider valve casing assembly in the Ex.7.1 Time phased net requirements for valve casing are The setup cost the valve casing is K=$132, and holding cost h=$0.60 per assembly per week. Silver Meal heuristic. Starting in week 4 C(1) = 132 C(2) =[132 + (0.6)(42)] /2 = 78.6 C(3) = [132 + (0.6)[42+ (2)(32)]] /3 = 65.2 C(4) = [132 + (0.6)[42+ (2)(32)+(3)(12)]] /4 = 54.3 C(5) = [132 + (0.6)[42+ (2)(32) +(3)(12) +(4)(26)]] /5 = 55.92 , stop. Set y4 = 42+ 42+ 32 + 12= 128 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Time-phased net requirements 42 32 26 112 45 76 38 planned order relase (lot-for-lot) Valve Casing Assembly

Incorporating lot-sizing algorithms into the explosion calculus Starting in week 8 C(1) = 132 C(2) =[132 + (0.6)(112)] /2 = 99.6 C(3) = [132 + (0.6)[112+ (2)(45)]] /3 = 84.4 C(4) = [132 + (0.6)[112+ (2)(45)+(3)(14)]] /4 = 69.6 C(5) = [132 + (0.6)[112+ (2)(45)+(3)(14)+(4)(76)]] /5 = 92.12 , stop. Set y8 = 26+ 112+ 45 + 14= 197 y12 = 76 + 38 Week 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Gross requirements 42 32 26 112 45 76 38 Net requirements Time-phased net requirements Planned order relase (S-M) 128 197 114 Planned deliveries Ending inventory 86 44 171 59

S-M policy requires Lot for lot : TCL4L=$1320 EOQ : TCEOQ=$919.8 Total holding cost for = (0.6)(424) = $254.4 Total setup cost = (132)(3)= $396 The total cost of EOQ policy )= $396 + $254.4 = $ 650.50 Lot for lot : TCL4L=$1320 EOQ : TCEOQ=$919.8

Lot sizing with capacity constraints Capacity constraint clearly makes the problem far more realistic. However it also makes the problem more complex. Even finding a feasible solution may not be obvious. Consider the Example: r=(52, 87, 23, 56) and capacity for each period c=(60, 60, 60, 60). First we must determine if the problem is feasible On the surface the problem looks solvable, 4*60 =240>218 But problem is infeasible. ! 60+60=120<139 Feasibility condition Even feasibility condition is satisfied, it is not obvious how to find feasible solution.

Lot sizing with capacity constraints-Example 7.7 Checking for feasibility: r1 = 20 c1 = 60 r1 + r2 = 60, c1 + c2 = 120, r1 + r2 + r3 = 160, c1 + c2 + c3 = 180, r1 + r2 + r3 + r4 = 195, c1 + c2 + c3 + c4 = 240, r1 + r2 + r3 + r4 + r5 = 275, c1 + c2 + c3 + c4 + c5 = 300, r1 + r2 + r3 + r4 + r5 + r6 = 350, c1 + c2 + c3 + c4 + c5 + c6 = 360, r1 + r2 + r3 + r4 + r5 + r6 + r7 = 375, c1 + c2 + c3 + c4 + c5 + c6 + c7 = 420, Setting y=r’ gives a feasible solution. 50 60 r‘ = ( 20, 40, 100, 35, 80, 75, 25 ) c =( 60, 60 40 60 60 55 60 60

Improvement step Modified requirements schedule r’ is feasible. Is there another feasible policy that has lower cost. Variety of reasonable improvement rules can be used For each lot that is scheduled Start from the last and work backward to the beginning Determine whether it is cheaper to produce the units composing that lot by shifting production to prior periods of excess capacity By eliminating a lot, one reduces setup const in that period to zero, but shifting production to prior periods increases the holding cost.

Example 7.8 Assume that K=$450 and h=$2 r=(100, 79 , 230, 105, 3, 10, 99, 126, 40) c=(120, 200, 200, 400, 300, 50, 120, 50 , 30) Feasibility check feasible. r'=(100, 109, 200, 105, 28, 50, 120, 50, 30) cum. c 100 179 409 514 517 527 626 752 792 cum. r 120 320 520 920 1220 1270 1390 1440 1470 1 2 3 4 5 6 7 8 9 r' 100 109 200 105 28 50 120 30 c 400 300 y Excess capacity 20 91 295 272

Example 7.8 y=(100, 109, 200, 263, 0, 0 , 120, 0 , 0) r=(100, 79 , 230, 105, 3, 10, 99, 126, 40) Setup cost = 5 x 450 = 2250 Holding cost = 2 x (0+30+0+158+155+145+166+40+0) = 2 x 694 = $ 1388 Total cost of this policy $3638, ($4482 for initial feasible policy). 1 2 3 4 5 6 7 8 9 r' 100 109 200 105 28 50 120 30 c 400 300 y 263 158 108 58 Excess capacity 20 91 137 295 142 192 242 272 1 2 3 4 (4 period) ($2 per unit per period) (30 unit) =$240 <K=$450

Optimal Lot-Sizing for the time Varying Demand Heuristic techniques (EOQ lot sizing, The silver-meal heuristic, Least Unit Cost, Part Period Balancing) are easy to use and give lot sizes with costs that are near the true optimum. Lot size problem can be expressed as a shortest path problem. The dynamic programming can be used to find shortest path. Assume that| Forecasted demands over the next n periods are known and given by vector r=(r1,r2,…,rn). Costs are charged against holding at $h per unit per period and $K per setup.

Example 7A.1 The forecast demand for an electronic assembly produced at Hi-tech, a local semiconductor fabrication shop, over the next four weeks is 52, 87, 23, 56. There is only one setup each week for production of these assemblies, and there is no back-ordering of excess demand. Assume that the shop has the capacity to produce any number of assemblies in a week. Let y1, y2, y3, y4 be the order quantities in each week. Clearly y152 , if we assume that ending inventory in week 4 is zero, then y1 218 (52+87+23+56) y1 can take any one of 167 possible values It is thus clear that for even moderately sized problems the number of feasible solution is enormous.

Wagner-Whitin Algorithm The Wagner – Whitin algorithm is based on the following observation: Result. An optimality policy has the property that each value of y is exactly the sum of a set of future demands(exact requirement policy). y1 =r1, or y1=r1+r2+…, or y1=r1+r2+…+rn y2 =0, or y2=r2 or y2=r2+r3+…, or y2=r2+r3+…+rn … yn =0, or yn =rn. The number of exact requirement policy is much smaller than the total number of feasible policies.s

Wagner-Whitin Algorithm Because y1 must satisfy exact requirements, it can take only values : 52, 139, 162, 218 Ignoring value of y1, y2 can take 0, 87, 110, 166. Every exact policy is the form (i1,i2,…,in) , where ij are either 0 or 1. For example the policy (1,0,1,0) means that production occurs in period 1 and 3 only, that is y= (139, 0, 79, 0). For this example, there are exactly 23 = 8 distinct exact requirement policies.

Wagner-Whitin Algorithm A convenient way to look at the problem is as a one-way network with the number of nodes equal to exactly one more than the number of periods. Every path through to network corresponds to a specific exact requirements policy. For any pair (i,j) with i<j, if the arc (i,j) is one path, it means that ordering takes place in period i and the order size is equal to the sum of requirements in periods i, i+1, i+2 ,…, j-1. Period j is the next period of ordering. All paths end at period n+1, The value of each arc, cij, is defined as setup and holding cost of ordering in period i to meet requirements through j-1. 1 2 3 4 5 c12 c23 c34 c45 c13 c14 c15 c24 c25 c35

Example 7A.2 r = (52, 87, 23, 56), h=$1, K=$75 The first step compute cij for 1i4 and 2j5 c12 =75 (setup cost) c13 =75 + 87 = 162 c14 =75 + (23 x 2) + 87 = 208 c14 =75 + (56 x 3) + (23 x 2) + 87 = 376 c23 =75 c24 =75 + 23 = 98 c25 =75 + (56 x 2) + 23 = 210 c34 =75 c35 =75 + 56= 131 c23 =75 (setup cost) c45 =75 Path Cost 1-2-3-4-5 300 1-2-4-5 248 1-2-5 285 1-2-3-5 281 1-3-4-5 312 1-3-5 293 1-4-5 283 1-5 376 i \ j 1 2 3 4 5 75 162 208 376 98 210 131

Solution by Dynamic Programming The total number of exact policies for a problem of n periods is 2n-1. As n gets large, total enumeration is not efficient. Dynamic programming is a recursive solution technique that can significantly reduce the number of computations required. Dynamic programming is based on the principle of optimality. Define fk as the minimum cost starting at node k. Initial condition is fn+1 =0

Solution by Dynamic Programming