Manufacturing lot-sizing under MRP II environment: An improved analytical model & heuristic procedure

Pergamon

Computers ind. Engng Vol. 35, Nos 3--4, pp. 423-426, 1998 © 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S0360-8352(98)00124-7 o360-8352/98 $19.oo + 0.00

M A N U F A C T U R I N G L O T - S I Z I N G U N D E R M R P II E N V I R O N M E N T : AN I M P R O V E D A N A L Y T I C A L M O D E L & A H E U R I S T I C P R O C E D U R E Hyun-Joon Kim, Yasser A. Hosni Departmentof Indusa'ial Engineeringand ManagementSystems,Universityof Central Florida Orlando, Florida 32816, U.S.A. ABSTRACT Under single-level lot-sizing problem, well known Wagner-Whitin algorithm based on Dynamic Programming (DP) provides optimal order schedule. Order schedule does not work properly under Manufacturing Resource Planning (MRP II), where it is considered multi-level capaeitated problem and requirements of subcomponents are depend on the parent product. In this paper, we formulate a multilevel capacitated optimization model and develop a relatively efficient heuristic working under MRP II environment which considers work center capacities and interrelationship between levels in lot-sizing computation. © 1998 Published by Elsevier Science Ltd. All rights reserved. Keywords : Lot-sizing, Capacity Planning, MRP II, Linear Programming (LP) INTRODUCTION Previous researches on lot-sizing have focused primarily on the single-level problem. Wagner-Whitin Algorithm based on Dynamic Programming (DP) provides optimal order schedule. However, DP algorithm is essentially a shortest-path procedure on a specially structured "network" and requires lengthy calculations [1]. Therefore, other heuristic algorithms have been developed which considers computational efficiency and achieve sub-optimal schedule. However, order schedule based on the algorithms does not work properly for multi level products under MRP II environment, where capacity limitation is a major constraints. Under MRP environment and considering multi-level multi-product case, the lot size selected for any final product highly effects the gross requirements of related subcomponents. Zangwill [2] formulated the problem as an aeyclie network. A dynamic programming algorithm was presented to solve the problems. Crowston et al.[3] presented DP algorithm for which solution time increases exponentially with the number of periods, but only linearly with the number of levels. They also presented branch-and-bound (B&B) algorithm based on an enumeration technique. Optimal approaches, such as DP or B&B, are limited to small problems. Steinberg and Napier [4] presented a generalized network approach to find an optimal solution to the multi-level problem. However a mixed integer Linear Programming is used because the network formulation does not permit the use of an efficient network algorithm. To build the network model and transfer to the LP code are very complicated, and difficult to implement by practitioners. ARentakis et a l [5] introduced a new formulation of the lot-sizing problem in multistage assembly systems which simplifies its decomposition by a Lagrangean relaxation method. Computational results showed that medium-size problems (up to 15 levels and 18 schedule periods) could be solved reasonably well. However, the basic assumption of [4] and [5]'s model is infinite capacity. In addition, the lead time from one level to the next level is not considered at all. Work-center capacities and its loading profiles can greatly influences the lot size and the schedule. Research in multi-level capaeitated lot-sizing problem is extremely limited due to the nature of its complicate and difficult approaches. Singh and Rajamani [6] presented LP model with zero-one integer variable under capacity consideration. However, the model assumed single resource capacity and setuptime is not considered in the model. Chin [7] extended the model of [4] with additional consideration of 423

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capacity constraints. The capacity is unrealistically defmed as the maximum producible quantity for each item. However, under MRP II environment, resource loading at a certain work center at a specific time period is cumulated based on the production mix of different components. Therefore, the capacity should be defined as the maximum level of resource (constant or time-varying) for each work center. This paper presents a zero-one LP model under consideration of multi-resource capacitated MRP II problem with multi-product, multi-level product structures. The model provides optimal lot size plan for small problems. A heuristic algorithm, as an efficient solution procedure, is developed for large problems or complex product structures. MODEL FORMULATION The model directly applies LP and quick to provide first cut solution, even if it is high cost. The model considers capacity of work centers and interrelationships between parent items and their child items to compute lot sizes. Lead time information is considered to compute the gross requirements of child items. The model does not allow shortage. The decision variables of this model is the quantities to be ordered at each schedule periods. The objective function is to minimize the sum of ordering cost and inventory cost through the planning period. The model assumed that the constant demand for each time period and use average inventory levels to account inventory holding cost. The model can be mathematically expressed as the following: T

N

Minimize ~

~

(Csp*Xpi +CIp*AIpi)

O)

Subject to T

T

Qp, > ~ D~-Blpo i=l

(2)

i=l

Dp, = Q v*,* * C ~

, subcomponents only

(3)

N

L,j = ~

( Spj*Xp, + Opj*Qpi ) < Cij

Blpi = Elpi.i + Qpi, EIpi = BIpi - Dpi, AIpi = (Blpi + EIpi)/2 Qpi < M*Xpi

(4)

(5), (6), and (7) (S)

Xp,: Integer, Xpi > 0, Xpi < 1

(9)

Qpi = Ypi * Lp, Ypi : Integer

(10) (ll)

All variables > 0

Where, T : Planning Period N : Number of items C,p : Ordering Cost or Setup Cost of item P Xpi : 1, order placed at i, 0 otherwise Clp : Inventory carrying cost of item P Qp, : Order Quantity of item p at i Dp, : Demand of item p at i P+ • Parent item of subcomponent P i+ = i + Lp,, Lp+ : Lead time of parent product Lij • Resource Loading on Work centerj at i Spj : Setup time of item p at Work Center j Opj : Operation time for unit item p at WC j C~j : Capacity ofjth WC at i M : Large Positive Value Yp, : Number of lots produced of item p at i. I~ : Lot size ofitem p Alpi, BIpi, Elpi :Average, Beginning, and Ending Inventory level of item p at i. Cp~ : The number of subcomponcnt P required to produce one unit of parent item P÷

Equation (3) calculates the gross requirements of lower level components with lead time offsetting. Equation (4) shows that resource loading for each work center at a certain period is cumulated by several resource segments associated with a production mix of final products and subcomponents. Equation (I0) is optional for fixed lot-sizing rule. Qpi arc dependent variables of Ypi. Alternative lot sizes, Lp, are very limited in practice. For example, even though mathematical optimal lot size is 9 or 13, practical lot sizes

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can be 10 or 12 from the consideration of shop-floor information. The next section presents a heuristic procedure to provide sub-optimal lot-sizing plan with iterative improvement from an initial plan. HEURISTIC ALGORITHM The algorithm consists of two phases. First, it starts with an initial plan, usually proposed master schedule, without consideration of work center capacity constraints. A backward approach is used to find otit overloaded work centers and periods. The translated quantity of final product will be shiR to the previous scheduling period until a feasible initial plan is obtained. Then, iteratively improve the plan with merging two consecutive order quantities with consideration of finite resource capacity. The algorithm follows the steps: Step 1. Check to see if the initial plan is feasible with consideration of finite resource capacity IfL~j < C~j, for all i and j, the initial plan is a feasible plan. Goto step 3. If not, goto step 2. Step 2. Find an initial feasible plan under finite resource capacity. Goto subroutine, Heuristic Capacity Requirement Planning (HCRP).

Step 3. lteratively improve the plan with merging two consecutive orders into the first one. The iteration will stop either if the new merging plan does not save cost throughout all levels or if the plan violate resource capacity constraint. Goto HCRP for each evaluation. Heuristic Capacity Requirement Planning (HCRP) Step 1. Find out when and which work centers are overloaded. The algorithm starts backward search to find the period when a work center is overloaded. The pseudo code for the step rule is: For i = (from) T to 1 For j = 1 to J If L,j > Cij then current index is ( i, j ) end end if next if i = 1 then STOP else next If j is within j+, goto step 2.1, j+ : work centers used for the final product If j is within j , goto step 2.2, j : work centers used only for suhcomponents Step 2.1. Translate overloaded resource segments into the corresponding quantities of the final product to be producible. Overloaded resource segment can be translated into the quantity with the following equation under lot-for-lot lot sizing rule. L,j - Cij = OIj*Qli, forallj

(12)

The above equation can be translated into: Qli = ( Lij - Cij) / Ojj, for all j

(13)

M = Maxj Qli

(14)

And

The quantity to be shiRed, Q,~0, to the previous period is: Q~,) = M, ifM is less than Dli = D~i, ifM is greater than or equal to D~, Goto step 3.

(15)

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Step 2.2. Translate overloaded resource segments into the corresponding quantities of the final product to be producible. This is difficult to answer because in a certain time bucket resource profile is cumulated with resource segments for several subcomponents which must be available in order for the final product schedule to be achieved after a certain rnanufacturing lead time. N Lii - Cii =~'. ( Ogni * Qli.l * Cp] ), for lead time of 1 N = E ( OIg2)J* Qli+2 * Cpl ), for lead time of 2 N = E ( OKL)J* Qli*L * Cpl ), for lead time of L

(16)

p~

where, P(/) is subcomponent item P whose lead time to the final product is I. M = Mint Maxi ( Q,+l, Qli+2 ..... Q,+L )j

(17)

The quantity to be shifted, Q,tm), to the previous period is: Qg~+o = M, ifM is less than corresponding D~i+/ = D~,+~, ifM is greater than or equal to the corresponding D~i+t

(18)

Goto step 3. Step 3. Update lot-size schedule for the final product and subcomponents. Qli = Q,i "Q,(i), Qli.i = Qli-J + Q,
(19) and (20)

Qli+l = Qli+l " Qgi+o, Qli+t.i = Qli+/.1 + Q
(21) and (22)

Dpi = Q v,i+ * Cpp+, equation (3) Go to step 1. CONCLUSION In this paper we presented a heuristics for determining the production lot size for a multi-level multiproduct under MRP II system. However, the model and the associated procedure is "another" heuristics. It could have a value to be used by practitioners. The fact that the procedure is simple and all data pertinent to the product and its levels are available in database; developing a spreadsheet based on the procedure makes it of practical value. REFERENCES [1 ]. Gupta, Yash P and Keung, Ying, 1990, A Review of Muiti-stage Lot-sizing Models, International Journal of Operations and Production Management, Vol 10, No9, pp 57-73. [2]. Zangwill, W., 1969, A Backlogging Model and a Multi-echelon Model e r a Dynamic Economic Lot Size Production System: A Network Approach, Management Science, Vol. 15, No. 9, pp 506-27. [3J. Crowston, W., Wager, M., mad Williams, J., 1973, Economic Lot Size Determination in Multi-stage Assembly Systems, Management Science, Vol. 19, No. 5, pp 517-27. [4]. Steinberg, E., and Napier, H., 1980, Optimal Multi-level Lot Sizing for Requirements Planning Systems, Management Science, Vol. 26, No. 12, pp 1258-71. [5]. Afentakis, P., G-avish, B. and Karmarkar, U., 1984, Computational Efficient Optimal Solutions to the Lot-sizing Problem in Multi-stage Assembly Systems, Management Sci., Vol. 30, No. 2, pp 222-39. [6]. Singh, N. mad Rajamani, Divakar, 1991, An Incremental Cost and Resource Smoothing Heuristic for the Capacitatod Lot Sizing Problem, Computers and Industrial Engr., Vol. 20, No. 4, pp 469-474. [7]. Chiu, Huan Neng, 1993, A Cost Saving Technique for Solving Capacitated Multi-Stage Lot-sizing Problems, Computers and Industrial Engineering, Vol. 24, No. 3, pp 367-377.