A MATHEMATICAL PROGRAMMING MODEL FOR THE INTEGRATION OF AGGREGATE PRODUCTION PLANNING WITH SHORT-TERM FINANCIAL PLANNING Muñoz, M.A.; Ruiz-Usano, R.; Framiñán, J.M.; Crespo, A.; Moreu, P.; León, J.M. University of Seville. Spain. e-mails:
[email protected],
[email protected] ABSTRACT Although the Aggregated Production Planning problem has been extensively studied in literature, few authors have considered the interaction between this problem and the Short-Term Financial Planning problem. In this paper, we develop a fractional programming model that integrates production decisions and short-term financial decisions, in order to maximise the return on investment of the company. Keywords: Aggregated production planning, short-term financial planning, fractional programming, working capital, ROI.
1. INTRODUCTION Great efforts have been traditionally devoted to the decision problems of each company internal organisation. The models assume implicitly that decision making in each of these areas is independent of the decisions to be taken in other areas, but this is not always true. An example is the production-planning problem, which determines production, inventory and employment levels in the factory. These levels affect cash management and Short-Term Financial Planning (STFP), proving that production and financial decisions are not independent. Most classical models try to obtain an aggregate production plan without allowing for the interaction between the Aggregated Production Planning (APP) and the rest of the company subsystems. Specifically, they do not consider economical-financial issues of APP. Dealing with the APP from a financial point of view may have some advantages: 1. A common approach is to solve the APP without allowing for company cash levels within the planning horizon. So, the aggregated production plan may be not feasible at all, although it could be feasible from the point of view of factory capacity and demand meeting. 2. In classical models, first of all an Aggregated Production Plan is established without considering short-term financial and investment decisions. Then, a short-term financial plan that makes the production plan feasible is developed. Instead, one would like to make production and financial decisions simultaneously, optimising both functions jointly. 3. By developing the APP and the STFP simultaneously it is possible to obtain the aggregated production plan, the cash budget, the projected balance sheet and the projected income-statement.
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4. By solving the APP and the STFP simultaneously, the influence of production plans over some financial ratios and variables, i.e. over current ratio or over cash position ratio, may be evaluated. 5. The desired objective in the APP models is frequently cost minimisation. One of this costs is inventory carrying cost. These costs present the following features: a) a high percentage of them aren't out of pocket costs but opportunity costs. b) they don't appear in financial accounting. c) They are difficult to quantify, so some authors (Kirca and Köksalan, 1996) have labelled them as abstracts. Through an integrated point of view, we propose to quantify the influence of the inventories by means of the ROI (Return On Investment), which includes inventories in the denominator as an element of the company's assets. Although APP problem has been widely studied in the literature - Nam and Logendran (1992) give an excellent state of the art on APP -, few authors have considered the economical-financial side of this problem. Damon and Schramm (1972) develop a mo del which incorporates financial and marketing decisions, starting from the classical Holt, Modigliani, Muth and Simon (HMMS) model (Holt et al. –1960-). Baker and Damon (1977) develop a linear programming model for APP and the working capital. Pizzolato (1979) suggests a small variation of the preceding model to make it consistent with accounting principles. All of these models consider only one product. Fisk (1980) describes an interactive game for the financial planning and the production planning based on HMMS model. Kirca and Köksalan (1996) give a linear programming model for the integration of production planning and financial planning in a multiproduct make to order production environment. All previous models consider as objective function the company's profit. In this paper, we develop a fractional programming model, which integrates production decisions and short-term financial decisions trying to maximise firm's ROI. 2. THE FRACTIONAL PROGRAMMING MODEL We next discuss the model in this way: First of all we introduce the elements on the balance sheet. Secondly, we give a complete list of model data and decision variables. Next, we formulate model constraints. Finally, we consider the model objective function. 2.1 Elements on the balance sheet In each of the periods that constituite the planning horizon, the projected balance sheet can be considered. The elements on the balance sheet and the used notation for a generic period are shown in table 1. (Note that we only consider the short-term loan as a financial source, and short term marketable securities are not taken in account). ASSETS CURRENT ASSETS Cash Accounts Receivable Completed Products Raw Materials Value Added Tax FIXED ASSETS
CUAt CAt ARt CPt RMt VAT1 t FAt TOTAL ASSETS At 2
LIABILITIES CURRENT LIABILITIES Accounts Payable Short-Term Loan Accrued Taxes Accrued Dividends Value Added Tax LONG TERM LIABILITIES OWNER’S EQUITY ACCUMULATED UNDISTRIBUTED PROFIT Operating Financial ACCUMULATED DEPRECIATION
CULt APt LOt ATt ADt VAT2 t LTLt OEt AUPt OAUPt FAUPt ADEPt TOTAL LIABILITIES At Table 1: Elements on the balance sheet
2.2 Model parameters T = Time horizon, in periods. N = Total number of products. P = Total number of machines. L = Total number of worker types. M = Total number of raw material types. Dit = Forecast demand for product i in period t (units). b ik = Amount of time of machine k needed to produce one unit of product i (hours/unit). MCkt = Total time available for machine k in period t (hours). k ik = Amount of time of worker k needed to produce one unit of product i (hours/unit). (rm) kt = Total manhours of type k regular labor available in period t (hours). (om) kt = Total manhours of type k overtime labor available in period t (hours). eij = Amount of type j material needed to produce one unit of product i (Kilograms/unit or Kg/unit). t0 = First period of planning horizon in which is projected to make the Value Added Tax (VAT) returns-payment. τ = Number of periods during which VAT is due. rva = VAT rate (per unit). αi = Proportion of the total sales value of product i collected in the current period (rate per unit). βj = Proportion of the total purchases value of material j paid off in the current period (rate per unit). p it = Unit selling price of product i in period t (monetary units/unit or m.u./unit). rkt = Cost per manhour of type k regular labor in period t (m.u./hour). o kt = Cost per manhour of type k overtime labor in period t (m.u./hour). cj = Unit cost of material j (m.u./Kg.) ilo = Short-term financing interest rate (rate per unit per period). OH1 t = General overhead manufacturing charges in period t, except fixed manhour (m.u.). OH2 t = Administrative, commercial and general overhead in period t (m.u.). 3
FEt = Financial expenses of long-term liabilities in period t (m.u.). PBt = Repayment of long-term loans in period t (m.u.). TXPt = Tax payment, corresponding to previous fiscal year, in period t (m.u.). DIVt = Dividend payment corresponding to previous fiscal year, in period t (m.u.). INOt = Cash outcomes, from investments, in period t (m.u.). γit = Unit cost of product i in period t (m.u./unit). dep = rate per unit per period of fixed assets depreciation. SS it = Safety stock imposed to product i in period t (units). lbca t = Lower bound for cash position ratio in period t. ublo t = Upper bound for short-term loan in period t (m.u.). lbcrt = Lower bound for current ratio in period t. In addition to these parameters, we consider the initial balance sheet (subscript 0) as known. 2.3 Decision variables Xit = Amount of product i produced in period t (units). Iit = Inventory level for product i at the end of period t (units). S it = Amount of product i sold in period t (units). Wkt = Manhours of type k regular labor used during period t (hours). Okt = Manhours of type k overtime labor used during period t (hours). Mjt = Amount of type j material procured in period t (Kg.). Yjt = Inventory level for type j material at the end of period t (Kg). CAt = Cash balance at the end of period t (m.u.). ARt = Balance of accounts receivable at the end of period t (m.u.). CPt = Balance of completed products at the end of period t (m.u.). RMt = Balance of raw materials at the end of period t (m.u.). VAT1 t = Balance of Value Added Tax (on Current Assets) at the end of period t (m.u.). APt = Balance of accounts payable at the end of period t (m.u.). LOt = Amount of short-term loan borrowed in period t (m.u.). VAT2 t = Balance of Value Added Tax (on Current Liabilities) at the end of period t (m.u.). OAUPt = Accumulated undistributed trading profit at the end of period t (m.u.). AUPt = Accumulated undistributed profit at the end of period t (m.u.). 2.4 Constraints Production-inventory balance equations. A possible formulation is that one proposed by Hopp and Spearman (1996, p.504): I i,t −1 + X it − I it − Sit = 0 i = 1,..., N ; t = 1,..., T X it , I it , Sit ≥ 0 Constraints on the machines capacity. The machinery capacity is taken into account on these restrictions by means of linear constraints: N
∑ bik X it ≤ MC kt
k = 1,..., P; t = 1,..., T
i =1
Constraints on the manpower availability. A possible formulation is that one suggested by Hax and Candea (1984, p.74): 4
N kik X it − Wkt − Okt = 0 i =1 0 ≤ Wkt ≤ (rm ) kt k = 1,..., L ; t = 1,..., T 0 ≤ Okt ≤ (om) kt Materials balance equations. These equations are quite similar to production-inventory balance equations, but materials (not completed products) are now considered. N Y j ,t −1 + M jt − Y jt − eij X it = 0 j = 1,..., M ; t = 1,..., T i =1 Y jt , M jt ≥ 0 Projected change in the cash balance. In this equations we set that the cash level at the end of period t is equal to the cash level at the end of period t-1 plus what was collected during period t minus what was disbursed during period t. We assume that a certain proportion αi of the sales of product i is collected in the current period and all receivables are collected within the following period. We assume too that a certain proportion βj of the purchases of material j is disbursed in the current period and all payables are paid off within the following period. The problem might be also formulated in another way. Now we must consider the VAT returns-payment problem. Let A be the set of subscripts that reference the periods of the problem: A ={1,...,T}. Let t0∈A be the first period of the planning horizon in which the company must complete the VAT returns-payment. Let τ be the number of periods during which VAT is due. Let B be the subset of A as t − t0 B = { t : t ∈ A, ∈Z } τ where Z is the integers set. So, set A is split into set B which represents the periods during which the VAT returns-payment is performed and the set A-B which represents the remainder periods. According to the previous asserts, cash equations can be formulated as follows:
∑
∑
− CAt + CAt − 1 + -
N
∑ [(1 + rva )α p i
N
it S it
i =1
] + ∑ [(1 + rva )(1 − α i ) pi,t − 1Si,t − 1 ] − i =1
∑ [rkt (rm )kt + oktOkt ] − ∑ [( 1 + rva )β jc j M jt ] − ∑ [( 1 + rva )(1 − β j )c j M j ,t −1 ] + L
M
M
k =1
j =1
j= 1
+ (1 − ilo)LO t − LOt − 1 − OH 1t − OH 2t − FEt − PBt − TXPt − DIV t − INOt = 0 N
− CAt + CAt − 1 +
∑ [(1 + rva)α p S
i it it
i =1
-
∀t ∈ A − B
N
] + ∑ [(1 + rva)(1 − α i ) pi,t − 1Si,t − 1 ] − i =1
∑ [rkt (rm)kt + okt Okt ] − ∑ [( 1 + rva )β jc j M jt ] − ∑ [( 1 + rva )(1 − β j )c j M j,t −1 ] + L
M
M
k =1
j= 1
j =1
+ (1 − ilo)LOt − LO t − 1 − OH 1t − OH 2t − FEt − PBt − TXPt − DIV t − INOt −
− (VAT 2t − VAT 1t ) = 0 ∀t ∈ B Projected change in accounts receivables. The accounts receivable balance at the end of period t is equal to that at the end of period t -1 plus the credit sales in the current period minus the collection of the credit sales of the previous period: 5
− ARt + ARt −1 +
N
N
i =1
i =1
∑ [(1 + rva )(1 − αi ) pitS it ] − ∑ [(1 + rva)(1 − αi ) pi ,t −1Si,t −1 ] = 0
t = 1,..., T
Projected change in completed products inventories. To calculate the economic value of the completed products inventories at the end of period t, we can use the statement suggested by Pizzolato (1979), in which the inventory at the end of the period are computed as the inventory at the beginning plus the cost of goods produced (CGPt) during the period minus the cost of goods sold during the period (CGS t). − CPt + CPt −1 + CGPt − CGS t = 0 t = 1,..., T The cost of goods produced is equal to the used raw materials cost plus direct labour cost plus the fixed overhead charges if we assume a full costing system. M
CGPt =
N
L
∑ c ∑ e X + ∑ [r j
j =1
ij
it
kt
i =1
(rm )kt + okt Okt ] + OH 1t
t = 1,..., T
k =1
To compute the cost of goods sold, we have to know the products unitary cost. The model assume that there is a fixed unitary cost, γit, whichever the adopted production decisions be (Pizzolato, 1979). N
CGS t =
∑γ
t = 1,...,T
it S it
i =1
Projected change in the materials inventories: M
− RM t +
∑ c j Y jt = 0
t = 1,..., T
j =1
Projected change in the VAT (on current assets): M
− VAT 1t + VAT 1t-1 +
∑ rva c
j
M jt = 0 ∀t ∈ A − B
j =1
M
− VAT 1t +
∑ rva c
j
M jt = 0 ∀t ∈ B
j =1
Projected change in accounts payables. The accounts payable balance at the end of period t is equal to that at the beginning plus credit purchases in the current period minus disbursement due to credit purchases of the previous period:
− APt + APt −1 +
∑ [(1+ rva )(1 − β j )c j M jt ] −∑ [(1 + rva)(1 − β j )c j M j,t −1 ] = 0 M
M
j =1
j =1
t = 1,..., T
Projected change in the VAT (on current liabilities). This account change according the VAT that burdens the firm's sales and with VAT returns-payment. N
− VAT 2 t + VAT 2t −1 +
∑ (rva p S
it it
)=0
∀t ∈ A − B
i= 1
N
− VAT 2 t +
∑ (rva p
it S it
)= 0
∀t ∈ B
i =1
Projected income-statement. In each period the income-statement balance is changed by profits of the period. These profits are calculated as the revenues generated by the firm's sales minus the cost of the goods sold minus administrative, commercial and overhead charges, minus depreciation minus financial expenses. 6
N
− OAUPt + OAUPt −1 +
∑
M
pit S it −
i =1
N
∑ ∑ cj
j =1
L
eij X it −
i =1
∑ [rkt (rm) kt + o kt Okt ] − OH1t + k =1
+ CPt − CPt −1 − OH 2t − dep FAt −1 − ilo LO t = 0
t = 1,..., T
AUPt = AUPt −1 + OAUPt − OAUPt −1 − FE t t = 1,..., T Constraints on the current ratio. Considering a value lbcrt as a lower bound to this ratio, it may be imposed the current ratio not to be less than that lower bound. CUA t ≥ lbcrt ⇒ CUL t
⇒ CAt + ARt + CPt + RM t + VAT 1t − lbcrt ( APt + LO t + ATt + AD t + VAT 2 t ) ≥ 0 t = 1,..., T
Constrains on the cash level. We can formu late this constraint, imposing cash position ratio not to be less than a lower bound lbcrt. CAt − lbcat ( APt + LO t + ATt + AD t + VAT 2 t ) ≥ 0 t = 1,...,T Specifications. In addition to the previous constraints, some decision variables must be bounded. We group all this constraints under the title "specifications": X it ≥ 0 i = 1,...,N; t = 1,...,T
I it ≥ SS it
i = 1,...,N; t = 1,...,T
0 ≤ Sit ≤ Dit i = 1,...,N; t = 1,...,T 0 ≤ W kt ≤ (rm ) kt k = 1,...,L; t = 1,...,T 0 ≤ Okt ≤ ( om) kt k = 1,...,L; t = 1,...,T M jt , Y jt ≥ 0
j = 1,...,M; t = 1,...,T
CAt , ARt , CPt , RM t , VAT 1t , APt , VAT 2t ≥ 0
t = 1,...,T
0 ≤ LO t ≤ ublot t = 1,...,T Besides previous constraints there are projected changes that don't depend on taken decisions, that's why they are predictable a priori. These equations are the following: Projected change in fixed assets. The balance of this account will change according with company investments plan, which we assume that is given. Having investments disbursement increase fixed assets (although they could also be used as a way to decrease the debt with immobilized supplier), we have. FAt = FAt −1 + INOt t = 1,..., T Projected change in accrued taxes. Having there's not distribution of profits during the planning horizon, we have: ATt = AT t −1 − TXPt t = 1,..., T Projected change in accrued dividends. According with the previous assumption: ADt = AD t −1 − DIV t t = 1,...,T Projected change in long-term liabilities. Considering the long-term financing pay-back: LTL t = LTL t −1 − PB t t = 1,..., T Projected change in owner's equity. Having that owner's equity don't change during planning horizon, we have: OE t = OE 0 t = 1,..., T Projected change in accumulated depreciation: ADEPt = ADEPt −1 + dep FAt −1 t = 1,..., T
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2.5 Objective function Financial accounting doesn't consider opportunity costs represented by inventory carrying costs, so to penalise inventories we suggest an objective function that quantifies the Return On Investment (ROI), which may be defined as the quotient between income before taxes and interests and average net total assets during the horizon. (Pérez Carballo, 1981, p.222). OAUPT − OAUP0 Max. g = T 1 (CAt + AR t + CPt + RM t + VAT 1t + FAt − ADEPt ) T + 1 t =0
∑
3. TRANSFORMATION OF THE RATIO GOAL The previous objective function
g=
(T + 1)OAUPT − (T + 1)OAUP0 T
∑ (CA + AR t
t
+ CPt + RM t + VAT 1t + FAt − ADEPt )
t =0
originates a fractional programming model, which can be easily transformed in a linear programming model by means of Charnes and Cooper transformation (see for example Goedhart and Spronk, 1995). Given that the objective function denominator is strictly positive through the feasible set, this model can be transformed in a linear programming one by means of a single transformation. Making the following substitutions 1 ω= T
∑ (CA + AR + CP + RM t
t
t
t
+ VAT 1t + FAt − ADEPt )
t =0
X 'it = ω X it ; I it' = ω I it ; S'it = ω Sit ; Wkt' = ω Wkt ; O'kt = ω Okt ; M 'jt = ω M jt ; Y 'jt = ω Y jt ; CA't = ω CAt ; ARt' = ω ARt ; CPt' = ω CPt ; RM 't = ω RM t ; VAT 1't = ω VAT 1t ; APt' = ω APt ; LO 't = ω LO t ;VAT 2't = ω VAT 2t ; OAUPt' = ω OAUPt ; t = 1,..., T we have a linear programming model, in which the decision variables are those defined by de substitutions made. The new linear objective function is g = (T + 1) OAUPT' − (T + 1) OAUP0ω The model constraints are obtained replacing old decision variables as a function of the new ones in each restriction. Also, we have to include the definition constraint on scalar variable ω: T T CA0 + AR 0 + CP0 + RM 0 + VAT 10 + FA0 − ADEP0 + FAt − ADEPt ω t =1 t =1
∑
∑ (CA + AR T
+
' t
t =1
' t
)
+ CPt' + RM 't + VAT t' = 1
8
∑
4. COMMENTS AND FUTURE LINES In this paper a fractional programming model integrating production decisions and short-term financial decisions is developed. This model tries to optimise company's ROI and can be easily transformed in a linear programming one by means of the well known Charnes and Cooper transformation. From this integrating point of view, we try to obtain a set of advantages, summarised in the following points: • The production plan is set taking into account the cash position of the firm during the planning horizon. • Production and short-term financial decisions are taken simultaneously. • The company can get the aggregated production plan, the cash budget, the projected balance-sheet and the projected income-statement jointly. • Financial ratios and variables are included in the model. • Inventory carrying cost don't have to be determined, which is often difficult to estimate. The model can be enriched in some different ways. For example, different short-term financial sources and not only short-term loan might be considered. Marketable securities might also be included on the balance-sheet. Another future research line is the formulation of a fractional program with multiple goals, in which profit, profitability and liquidity might be established as goals. REFERENCES BAKER, K.R. and W.W. DAMON (1977): "A simultaneous planning model for production and working capital". Decision Sciences, 8 (1), 95-108. DAMON, W.W. and R. SCHRAMM (1972): "A simultaneous decision model for production, marketing and finance". Management Science, 19 (2), 161-172. FISK, J. (1980): "An interactive game for production and financial planning". Computers-&Operations-Research, vol.7, no.3, p.157-68. GOEDHART, M.H. and J. SPRONK (1995): "Financial planning with fractional goals". European Journal of Operational Research, 82, 111-124. HAX, A. and D. CANDEA (1984): Production and Inventory Management, Prentice-Hall. HOLT, C.C., F. MODIGLIANI, J.F. MUTH, and A. SIMON (1960): Planning Production, Inventories, and Work Force, Prentice-Hall, Englewood Cliffs, New Jersey. HOPP, W.J. and M.L. SPEARMAN (1996): Factory Physics. Foundations of Manufacturing Management, Richard D. Irwin. KIRCA, O. and M. KÖKSALAN (1996): "An integrated production and financial planning model and an application". IIE Transactions, 28, 677-686. NAM, S.J. and R. LOGENDRAN (1992): “Aggregate production planning – A survey of models and methodologies". European Journal of Operational Research, 61, 255272. PEREZ CARBALLO, A. and J. and E. VELA SASTRE (1981): Gestión financiera de la empresa, Alianza Universidad Textos. Madrid. PIZZOLATO, N.D. (1979): “A note on a simultaneous planning model for production and working capital". Decision Sciences, 10 (2), 334-336.
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