Models for Integrated Production Planning and Order Selection

7/28/2019 Models for Integrated Production Planning and Order Selection

1/6

Models for Integrated Production Planning and Order Selection

Joseph Geunes, H. Edwin Romeijn, and Kevin TaaffeDepartment of Industrial and Systems Engineering

University of Florida

Gainesville, FL 32611

Abstract

The vast body of literature on dynamic requirements planning problems in make-to-order production systemsassumes that demands in all periods are predefined. In many practical contexts, however, a producer has discretionto accept or deny production orders during the production planning process. That is, given a set of orders for

production at a single stage in a set of time periods, along with fixed plus linear production costs and variableholding costs in every period, the producer must determine the most profitable set of orders to fulfill. We providemodels that address these integrated production planning and order selection decisions.

KeywordsRequirements Planning, Inventory Control, Production Planning.

1. IntroductionProduction planning managers regularly face the challenge of determining how best to schedule production in order

to meet demands. In many production environments, production plans are driven by customer requirements asexogenously determined by the sales and marketing division of the firm. Sales and marketing personnel often relyon limited input from production and manufacturing personnel when committing to requests for the firms

production outputs. When the manufacturer has limited production capacity relative to the total demand for itsoutputs, it becomes necessary to select only a subset of customer orders and to deny certain orders that cannot befilled. In contexts where capacity limits are non-binding, it is not always clear that committing to a particular

customer order is in the best interest of the firm, even if the unit price the customer will pay exceeds the variableproduction cost. This latter phenomenon can arise in environments with significant fixed production costs; in suchcases justifying fixed production costs might require an ability to aggregate the order in question with additional

orders to offset fixed production costs. Decisions on the collective set of orders the firm should accept can be critical

in determining the overall profitability of the firm. Past modeling literature has not addressed customer orderacceptance and denial decisions in a manner that accounts for the impacts these decisions have on operations costs

and, therefore, profitability. This paper partially fills this gap by presenting models and solution approaches forintegrated production planning and order selection decisions in single -stage, make-to-order production systems.

Developing new models for dynamic requirements planning problems has been one of the most consistently activeresearch areas in operations management over the past four decades, start ing with the classical paper by Wagner andWhitin [19]. Since their seminal paper addressed the basic uncapacitated dynamic requirements problem, the

literature has seen many extensions and generalizations of this problem class, including extensions to handlebacklogging [20], serial systems [15], multistage assembly systems [1], and general multistage structures [2]. Arelated stream of literature focuses on the same basic problem under finite production capacity limits (see [5], [9],

and [10]). The capacitated lot sizing problem in particular has received increased attention more recently, due to thedevelopment of advanced solution techniques that include the use of strong valid inequalities (e.g., [6], [13], and

[16]). In addition to the literature cited above, a variety of heuristic approaches (too numerous to mention) have beensuccessfully employed in solving the complex capacitated and multistage/multi-item versions of the basic dynamicrequirements planning problem. [4], [12], and [17] collectively provide an excellent overall analysis of the variousgeneralizations and solution approaches for dynamic requirements planning problems.

This vast body of research on dynamic requirements planning almost universally employs two critical assumptions:A1. The objective seeks to minimize total production and inventory -related (holding and shortage) costs; and

A2. All demands are pre -specified by time period and all demands must be completely filled at the time theyoccur (or after they occur in models that permit backlogging).


2/6

In contrast, the models we discuss do not employ assumptions A1 and A2; rather, these models address contexts inwhich production planning and marketing/sales managers can select from a set of outstanding orders only those

orders the facility will satisfy in order to maximize net profit from production. Each order has associated with it (i)the quantity of the good desired, (ii) the time at which the customer desires demand satisfaction, and (iii) the per unitprice the customer is willing to pay for the good. At one extreme, the managers can choose not to accept any orders,

thereby incurring no costs and receiving zero revenue. At the other extreme, if production capacity limits exceedtotal demand, the firm can accept all orders, regardless of the impact on total profit. This latter extreme might applywhen the firms objective is to maximize market share. We assume that the firm wishes to maximize net

contribution to profit and must, therefore, judiciously select the orders it will accept. This paper exploresuncapacitated and capacitated versions of the single-stage production planning problem with order selectionflexibility. The main contributions of this work include defining a new class of production planning problems that

integrate elements of order selection flexibility, and the development of optimization-based modeling and solutionmethods for this class of problems. We provide a polynomial time algorithm for solving the uncapacitated version ofthe problem and briefly discuss a Lagrangian relaxation scheme for solving capacitated problem instances.

A recent paper [14] considered a related problem that also violates A1 and A2, i.e., deterministic dynamicrequirements planning problems where the producer maximizes net profit from sales of a single item and does not

require satisfying all demands. Their model, however, assumes that only one outstanding order exists in everyperiod, while we allow the firm to consider any number of requests in every period, each with a unique associatedper unit revenue, i.e., we allow for customers with different reservation prices, and/or different delivery costs. In this

respect, their model represents a single-order special case of one of the models we propose. They do, however,establish a lower limit on inventory in every periodthis lower inventory represents a minimum safety stockrequired to buffer against demand uncertaintywhile our model places no lower limit on inventory. More recently,

[11] considered contexts that violate A2, where demands can be met either earlier (through early production anddelivery) or later (through backlogging) than specified without penalty, provided that demand is satisfied withincertain demand time windows for the uncapacitated, single-stage lot sizing problem. Their model assumes, however,

that all demand must be filled during the planning horizon, while our approach does not consider time windows.

This paper is organized as follows. Section 2 presents the basic uncapacitated version of the production planning

problem with order selection flexibility. We provide a solution approach for this problem that generalizes theWagner-Whitin [19] shortest path solution method for single-stage dynamic requirements planning problems.Section 3 briefly explores the implications of finite production capacity limits on the order selection problem. We

discuss a Lagrangian relaxation scheme that is amenable to solution via the methods developed for the uncapacitated

version of the problem discussed in Section 2. Section 4 provides conclusions and directions for future research.

2. The Uncapacitated Order Selection ProblemConsider a producer that manufactures a good to meet a set of outstanding orders over a finite number of timeperiods, T. Producing the good in period trequires a production setup at a cost ofStand each unit costs an additional

pt to manufacture. The total number of order requests in period tequals m(t). We initially assume that the producerfaces no production capacity limits in any period and that no shortages are permitted (i.e., no backlogging; Section 3discusses finite production capacity limitations). Items can be held in inventory at a cost of ht per unit remaining at

the end of period t. Let dmtdenote the quantity of the good requested in order m in period t, for which the customerwill pay rmt per unit, and suppose the producer can choose any quantity between zero and dmt in satisfying orderm inperiod t(i.e., the customer will take as much of the good as the supplier offers, up to dmt). The producer thus has the

flexibility to decide which orders it will choose to meet in each period and the quantity of demand it will satisfy foreach order. If the producer finds it unprofitable to satisfy a certain order in a period, the producer can choose to

reject the order at the beginning of the planning horizon. The producer, therefore, wishes to maximize net profit overa T-period horizon, defined as the total revenue from orders satisfied minus total production (setup + variable) andholding costs incurred over the horizon. To formulate this problem we define the following decision variables:

xt = Number of units produced in period t,

yt =1, if we setup in period ,

0, otherwise,

t

It = Producers inventory remaining at the end of period t,vmt = Number of units of the product used to satisfy orderm in period t.

We formulate the uncapacitated version of the Order Selection Problem (UOS) as follows:


3/6

[UOS] maximize ( )( )

1 1

T m t

mt mt t t t t t t t mr v S y p x h I

= = (1)

subject to:Initial Inventory: I0 = 0, (2)

Inventory Balance: It-1 +xt=( )

1

m t

mtmv

= +It, t= 1, , T, (3)Setup Forcing: 0 xtMtyt t= 1, , T, (4)Demand Bounds: 0 vmtdmt t= 1, ..., T, m = 1, , m(t), (5)Nonnegativity: It 0 t= 1, , T, (6)Integrality: yt {0, 1} t= 1, ..., T. (7)

The objective function maximizes net profit and constraint (2) requires that no initial inventory exists. Constraint set

(3) represents inventory balance constraints, while (4) ensures that no production occurs in period t if we do notperform a production setup in the period. If a setup occurs in period t, the production quantity is unconstrained by

settingMt equal to a large number. Alternatively, we can set Mtequal to( )

1

T m

mt md

= = without loss of generality,since this is the maximum amount of demand that could be satisfied by period t production (in the absence ofbacklogging). Constraint set (5) encodes our assumption regarding the producers ability to satisfy any proportion of

orderm up to the amount dmt, while (6) and (7) provide nonnegativity and integrality restrictions on variables .

2.1 Solution Properties and Shortest Path Representation

Problem [UOS] can be represented as a fixed-charge network flow problem as illustrated by the example in Figure1, where T= 4and m(t) = 2 for t= 1, , T. The network contains three types of arcs: production arcs, inventoryarcs, and order selection arcs. The dummy source node implies that it is not necessary to satisfy all demandthe

dummy source can supply the entire demand over the horizon if necessary. We also add a dummy sink node that canreceive flow from both the production source and the dummy source. Flow on a production arc implies that a setupoccurs in that period, while flow on an order selection arc implies that we satisfy at least some of that order in the

corresponding period. Since the flow cost on each arc is concave or linear (and hence also concave), the objectivefunction (1) is convex and [UOS] maximizes a convex function over a polyhedron. This implies that an optimalextreme point solution exists for [UOS]. Since the problem is a network flow problem, this implies that an optimal

spanning tree solution exists (see [3]), in which the subgraph induced by the arcs with positive flow in a solutionforms a spanning tree. We exploit this spanning tree property to derive certain properties of optimal solutions to[UOS]. Note that [UOS] generalizes the Wagner-Whitin single-stage requirements planning problem under dynamic

demand and we will later extend their dynamic programming solution method to solve our order selection problem.

Figure 1: Fixed charge network flow representation of [UOS]

The existence of an optimal spanning tree solution for [UOS] implies the following property:

All-or-Nothing Order Satisfac tion Property: Given the choice to satisfy any quantity of demand less than or equalto dmtfor orderm in period t, an optimal solution exists with eithervmt equal to 0 or dmt for all m and t, i.e., for eachorder-period combination (m, t)the producer either provides dmt units or none at all.

We next consider how to extend the Wagner-Whitin dynamic programming solution method for solving UOS. Theirdynamic programming solution method can be equivalently posed as a shortest path problem on a graph containing

Demand, dmt

Inventory arcs, flow

cost = ht

Order selection arcs,

revenue = rmtvmt

Production Source,Supply =

( )

1 1

T m t

mtt md

= =

Production flow arcs, flow cost =St+ptxtifxt> 0, and 0 otherwise.

Dummy Source,

Supply =( )

1 1

T m t

mtt md

= =

Dummy Demand,

Demand =( )

1 1

T m t

mtt md

= =


4/6

T+ 1 nodes (see Figure 2). Note that this method relies on the existence of an optimal Zero-Inventory Ordering(ZIO) policy in which a setup only occurs in period tif we hold no inventory at the end of period t 1 (the validity

of this property can also be shown to hold for [UOS] as a result of the spanning tree property of an equivalent fixedcharge network representation of [UOS]). Since the Wagner-Whitin approach minimizes total cost, each arc (t, t') inthe graph is assigned a cost, c(t, t'), where c(t, t') equals the setup cost in period tplus the variable production and

holding costs incurred for satisfying all demand in periods t, t+ 1, , t' 1 using only the setup in period t. Thisapproach ensures that a path exists in the shortest path network for every feasible combination of setups and that thecost of a path corresponds to the minimum cost incurred in using the setups to satisfy all demand.

Figure 2: Shortest path network structure for UOS

Shortest Path Solution Approach for UOS

Since the ZIO property also holds for the uncapacitated order selection problem, we can solve the UOS problemusing a shortest path graph containing the same structure as that used for solving the Wagner-Whitin problem. Thearc length calculation for UOS, however, requires a new approach. The order selection problem seeks to maximize

net profit and so we interpret arc lengths in terms of net contribution to profit and seek the longest path in the graph.The method used for arc length calculation proceeds as fo llows. We interpret the length of arc (t, t') as the maximumprofit possible from satisfying orders in periods t, , t' 1 assuming that the only setup available to satisfy demand

in these periods must occur in period t, if at all. Suppose we choose to perform the setup in period tand incur itscorresponding cost, St. To offset the cost of this setup we will satisfy the demand for order m in period tif and only

ifrmtpt, i.e., if the unit revenue from order m is at least as great as the unit production cost in period t. Similarly,for all periods such that t< < t', we will satisfy orderm in period if and only if

rmpt+1

kk th

= , (8)i.e., if the unit revenue from orderm in period exceeds the unit production cost in period tplus the per unit holding

cost incurred from period tto period . Let ( )t

O t denote the set of orders in period tsuch that rmtpt, and let ( )tO

denote the set of orders in period such that (8) holds for> t. Then the maximum profit possible if we do a setupin period tand use that setup to satisfy demands in periods t, , t' 1, which we denote by MPS(t, t'), is given by:

MPS(t, t') = ( ) ( )' 1 ' 2 ' 1

( ) 1 ( )t t

t t t

mt t mt mk t t m O t k m Or p d h d S

= = = + . (9)

If MPS(t, t') is greater than 0 we set the length of arc (t, t') equal to MPS(t, t'); otherwise we set the length of arc (t, t')

equal to zero and assume no setup occurs in period tif the optimal solution (the longest path in the graph) traversesarc (t, t'). After finding the longest path in the graph we can determine which orders to satisfy in each period by

checking the elements of the sets ( )t

O t and ( )t

O for all arcs (t, t') contained in the longest path.

Note that we can extend formulation [UOS] to accommodate any fixed and/or variable costs incurred when choosingto satisfy orderm in period t. For example, suppose we incur a fixed cost Fmt and a variable cost of cmtper unit oforderm satisfied in period t. The fixed cost, Fmt, might include, for example, a shipping cost plus any sales and

administration expenses required for satisfying orderm in period t.We can account for a variable transportation costby subtracting cmtfrom rmt without any change in formulation [UOS]. Accounting for the fixed cost, Fmt, would

require introducing a new set of variables, mt, where mt equals 1 if we satisfy order m in period t, and zero

otherwise. We subtract the additional term( )

1 1

T m t

mt mt t mF

= = from the objective function (1) and replace each vmt onthe right hand side of constraint (5) with dmtmt. With the addition of such cost terms the objective function is still

convex, so the optimality of the all-or-nothing and ZIO properties still holds. We must, however, alter our rules for

constructing the sets ( )t

O and for calculating MPS(t, t') to account for these additional costs. The index set ( )tO t

now includes all orders in period tsuch that (rmt cmtpt)dmtFmt, and ( )tO includes orders in period such that

(rm cm pt1

kk th

= )dmFm . (10)

1 2 3 4 5


5/6

Letting ( ){ }1,...,

maxt T

m m t=

= , the total computational effort of arc cost calculations is bounded by ( )2O mT and the

shortest path calculation is no worse than O(T2), so the worst case complexity of this algorithm is bounded

by ( )2O mT . More recent work on the uncapacitated lot sizing problem (e.g., [7] and [18]) has reduced the

complexity of the problem from the O(T2) bound to O(T log T) (or even O(T) in certain special cases). Theseapproaches, however, rely on an important property that holds for the uncapacitated lot sizing problem (ULSP). This

critical property requires that the cumulative demand satisfied as we increase the number of periods in a probleminstance be non-decreasing. That is, for the ULSP, the total demand satisfied in periods 1 and 2 is at least as great asthat satisfied in period 1. Interestingly, we can show that this property does not hold for the UOS problem. Considerthe following simple three-period example, where m(t) = 1 fort= 1, 2, 3. We define the following parameter values:

S1 = S2 = 50, S3 = 1,000;p1= 1.50,p2 = 1.25, c3 = 1.20; h1 = h2 = 0

Table 1: Example problem parameters

Order Period Demand Revenue

1 1 20 $36

1 2 20 $80

1 3 10 $100

It is straightforward to verify that (i) the optimal solution for the 1-period problem alone satisfies no demand, (ii) theoptimal solution for the 2-period problem satisfies a total of 40 units of demand in periods 1 and 2 using the setup in

period 1 only, and (iii) the optimal solution for the 3-period problem satisfies a total of 30 units of demand inperiods 2 and 3, using the setup in period 2 only. So the total demand satisfied in the 3-period problem is less thanthat satisfied in the 2-period problem, i.e., cumulative demand satisfied is not non-decreasing. (Note that we can

provide a similar example under which cumulative demand satisfied from period t to T is not necessarily non-decreasing as we move backwards in time, i.e., as t decreases). This example illustrates why we cannot applymethods previously developed to reduce the complexity of ULSP to O(Tlog T) in an effort to reduce the complexity

of our algorithm to, say, ( )logO mT T , since cumulative demand satisfied is not necessarily non-decreasing for theUOS problem.

3. The Capacitated Order Selection ProblemWe have thus far assumed that production was uncapacitated. This led to a dynamic programming solution to the

unrestricted order selection problem. For cases involving production capacity limits, we must decide on whichorders to select and, among the selected orders, how much of the order we will satisfy while obeying our capacity

limits. We provide a brief overview of our approach for solving the capacitated version of the order selection

problem. Let Ct denote the production capacity in period t. To formulate the capacitated order selection (COS)problem, we simply add the following constraints to our original [UOS] formulation (1 7):

Production Capacity Limits: xtCt, t= 1, , T. (11)

We can show that the resulting COS problem is NP-Hard, since it generalizes the capacitated lot sizing problem.Given this, we would like to find an efficient method for obtaining good solutions to the problem. We use aLagrangian relaxation approach, which, as our computational results indicate, is quite effective for solving COS.

Letting denote a T-vector of nonnegative Lagrangian multipliers for capacity constraints (11), we dualize these

constraints by adding ( )1

T

t t ttC x

= to the objective function of [COS]. For any fixed value of , the resulting

Lagrangian relaxation formulation is equivalent to an instance of [UOS], for which we have the polynomial timealgorithm discussed in the previous section. We use subgradient optimization (see [8]) in an attempt to minimize the

Lagrangian dual function over 0. Our Lagrangian scheme also incorporates a heuristic algorithm that adjusts theLagrangian relaxation solution at each iteration to obtain a feasible solution for [COS]. Results of our Lagrangianrelaxation algorithm implementation applied to a broad set of 3600 randomly generated test problems indicate that,on average, the algorithm produced solutions guaranteed to be within 0.2% of the optimal solution value.

4. Summary and Future Research DirectionsDeterministic production planning models typically assume that a set of fixed demands exists that the production

facility must meet. In many practical contexts, the manufacturer has some leeway in deciding whether it will acceptall production orders. The models presented in this paper allow the manufacturer to select a subset of orders thatprovides the greatest contribution to profit. These models exploit a new source of order selection flexibility


6/6

previously ignored in the modeling literature. We have discussed both uncapacitated and capacitated versions of theorder selection problem and provided solution approaches for both classes of problems. Promising directions for

further research in this area include modeling problems where a single customer places a set of orders in multipleperiods and the manufacturer must choose to either satisfy the customers entire set of orders or not to satis fy any ofthe customers orders. In the uncapacitated version of the problem discussed in this paper, the all-or-nothing order

satisfaction property guarantees that we will either satisfy an entire order or reject the order. The capacitated versionof the problem we have discussed implicitly allows the manufacturer to satisfy only a portion of an order, i.e., topartially fill an order. In future work, we will also consider the capacitated version of the problem under the

condition that the manufacturer must fill each accepted order in its entirety.

References1. Afentakis, P., B. Gavish, and U. Karmarkar, 1984, Computationally Efficient Optimal Solutions to the Lot-

Sizing Problem in Multistage Assembly Systems,Management Sci.,30, 222-239.2. Afentakis, P. and B. Gavish, 1986, Optimal Lot-Sizing Algorithms for Complex Product Structures, Oper.

Res., 34, 237-249.3. Ahuja, R., T. Magnanti, and J. Orlin, 1993, Network Flows: Theory, Algorithms, and Applications, Prentice

Hall, Englewood Cliffs, NJ.

4. Baker, K.R., 1993, Requirements Planning, Ch. 11 in Handbooks in Operations Research and ManagementScience v4, Logistics of Production and Inventory (S.C. Graves, A.H.G. Rinnooy Kan, and P.H. Zipkin, eds.),North-Holland, Amsterdam.

5. Baker, K.R., P. Dixon, M. Magazine, and E. Silver, 1978, An Algorithm for the Dynamic Lot Size Problemwith Time-Varying Production Capacity Constraints,Management Sci., 24, 1710-1720.

6. Barany, I., T.J. Van Roy, and L.A. Wolsey, 1984, Strong Formulations for Multi-item Capacitated Lot Sizing,

Management Sci ., 30(10), 1255-1261.7. Federgruen, A., and M. Tzur, 1991, A Simple Forward Algorithm to Solve General Dynamic Lot-Sizing

Models with n Periods in O(n Log n) orO(n) Time,Management Sci., 37, 909-925.

8. Fisher, M.L., 1981, Lagrangian Relaxation Method for Solving Integer Programming Problems, ManagementSci . 27(1), 1-18.

9. Florian, M., and M. Klein, 1971, Deterministic Production Planning with Concave Costs and Capacity

Constraints,Management Sci., 18, 12-20.10. Florian, M., J. Lenstra, and A.H.G. Rinnooy Kan, 1980, Deterministic Production Planning: Algorithms and

Complexity,Management Sci ., 26, 669-679.

11. Lee, C.Y., S. etinkaya, and A.P.M. Wagelmans, 2001, A Dynamic Lot Sizing Model with Demand Time

Windows,Management Sci., 47(10), 1384-1395.12. Lee, H.L., and S. Nahmias, 1993, Single-Product, Single-Location Models, Ch. 1 inHandbooks in Operations

Research and Management Science v4, Logistics of Production and Inventory (S.C. Graves, A.H.G. RinnooyKan, and P.H. Zipkin, eds.), North-Holland, Amsterdam.

13. Leung, J., T.L. Magnanti, and R. Vachani, 1989, Facets and Algorithms for Capacitated Lot Sizing, Math.

Programming, 45, 331-359.14. Loparic, M., Y. Pochet, and L.A. Wolsey, 2001, The Uncapacitated Lot Sizing Problem with Sales and Safety

Stocks,Math. Programming, 89, 487-504.

15. Love, S., 1972, A Facilities in Series Inventory Model with Nested Schedules,Management Sci .,18, 327-338.16. Pochet, Y., 1988, Valid inequalities and separation for capacitated economic lot sizing, Oper. Res. Letters, 7,

109-116.

17. Shapiro, J.F., 1993, Mathematical Programming Models and Methods for Production Planning andScheduling, Ch. 8 in Handbooks in Operations Research and Management Science v4, Logistics of Production

and Inventory (S.C. Graves, A.H.G. Rinnooy Kan, and P.H. Zipkin, eds.), North-Holland, Amsterdam.18. Wagelmans, A.P.M., S. Van Hoesel, and A. Kolen, 1992, Economic Lot Sizing: An O(n log n)-algorithm that

Runs in Linear Time in the Wagner-Whitin Case, Oper. Res ., 40, S145-S156.19. Wagner, H. and T. Whitin, 1958, Dynamic Version of the Economic Lot Size Model, Management Sci.,5,

89-96.20. Zangwill, W., 1969, A Backlogging Model and a Multi-Echelon Model Of a Dynamic Economic Lot Size

Production System,Management Sci ., 15, 506-527.

Models for Integrated Production Planning and Order Selection

Documents

Transcript of Models for Integrated Production Planning and Order Selection