analysis of a standing order inventory system with emergency orders

Analysis of a Standing Order Inventory System with Emergency OrdersAuthor(s): Matthew Rosenshine and Duncan ObeeSource: Operations Research, Vol. 24, No. 6 (Nov. - Dec., 1976), pp. 1143-1155Published by: INFORMSStable URL: http://www.jstor.org/stable/169982 .

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OPERATIONS RESEARCH, Vol. 24, No. 6, November-December 1976

Analysis of a Standing Order Inventory System

with Emergency Orders

MATTHEW ROSENSHINE and DUNCAN OBEE

The Pennsylvania State University, University Park, Pennsylvania

(Received original October 29, 1974; final, December 8, 1975)

The advent of relatively sophisticated information systems has mode it possible for many businesses that keep inventories to know their needs very rapidly. However, the producers of these inventories have not all made comparable advances in scheduling production to meet the rapidly changing orders placed by their increasingly sophisticated customers. As a result, leadtimes have been growing rather than de- creasing. Since in most cases the company needing the inventory cannot control the leadtime, it must either accept it or look for ways to mitigate its effect. This paper evaluates one way to cope with the problem of large or growing leadtimes.

AN INVENTORY-CONTROL system is designed to secure an economic balance between the cost of unsatisfied demand and the cost of pre-

venting it. When deinand is stochastic, increases in leadtime (the time between the placing of an order to replenish the inventory and the arrival of that order) result in the increased probability of shortages and higher inventories for a given inventory policy. Under a policy in which inventory is reviewed at fixed intervals, a large leadtime means that several review intervals will pass before an order is delivered. The order size must therefore be sufficiently large to account for the stochastic variations in demand that, during these review intervals, could lead to the occurrence of shortages. In essence greater leadtimes increase the cost of inventory- control systems.

One possible alternative to carrying a large inventory when the leadtime is large is to adopt a standing-order inventory system. Assuming that leadtime is deterministic, orders of fixed size can be placed in advance so that they arrive at the beginning of each period, regardless of the magnitude of the leadtime. Since a fixed-size order will arrive at the beginning of each period, regardless of demand, a standing-order system must allow for emergency order(s) if large shortages occur during any period and sell-offs if small demand results in inventory levels that exceed storage capacity.

A standing-order inventory system has some very attractive attributes.

1143



1144 Matthew Rosenshine and Duncan Obee

The ordering costs associated with placing periodic orders are eliminated and leadtime, as far as the standing order is concerned, does not exist. In addition, it seems more than likely that many suppliers would be willing to provide some form of price break or discount for items delivered under a standing order. On the other hand, there will be some penalties, in particular, a higher cost for items on emergency orders and a loss for selling off surplus inventory.

The objective of this paper is to provide a comparison between a standing- order inventory system with emergency orders and sell-offs and an order- level system with prescribed periodic reviews, stochastic demand, anld order level to be determined. The evaluation is accomplished by modeling the standing-order system, determining its expected costs, and comparing it with expected costs determined for an order-level system with leadtime.

1. THE MODEL AND ANALYSIS OF A STANDING ORDER INVENTORY SYSTEM

Overview To determine the relative effectiveness of the standing-order inventory

system, a measure of effectiveness is expressed for this system in a form that is comparable to a similar measure for the order-level system. The measure of effectiveness is the total expected cost, the sum of the minimum system cost (which includes holding, shortage, emergency penalty, and sell-off costs), and the purchasing cost. The minimum system cost is obtained by formulating the standing-order system as a Markov chain. The Markov chain model is then solved to determine the values of the decision variables that minimize the sum of these costs. The resulting minimum sumi is the minimum system cost, anid the values of the decision variables associated with this cost determine the cost of goods purchased.

The Standing Order Inventory System

Parameters and Variables. A standing order of size q is to be delivered periodically. An emergency order can be placed at most once (primarily for reasons of analytical tractability) in each period for immediate delivery. An emergency order is placed if the inventory level at the end of a period will be less than or equal to a predetermined level, IMIN. The size of the demand is known at the beginning of each scheduling period and is assumed to occur uniformly during the scheduling period. Alternatively, if the size of the demand is unknown at the beginning of the period, some form of monitoring, with its associated costs, is required to determine when the inventory level drops to IMIN-q.

The scheduling period, tp, is fixed and therefore is a parameter not subject to control. The emergency order point IMIN for an emergency order is also a parameter that can be either positive or negative. The order level IOL for an emergency order is a decision variable that defines the size of an



Inventory System with Standing and Emergency Orders 1145

emergency order (IOL-IMIN). The remaining decision variable of the system is the size of a standing order, q.

Constraints. The inventory level of the system is constrained not to exceed the system's storage capacity I,MAX. The inventory may be positive or negative, negative inventory occurring as the consequence of backlogging.

Assumptions. The demand sizes x during each of the scheduling periods are independent, identically distributed, nonnegative random variables.

As a result of the ordering policy and the constraints, the system incurs a number of costs that are assumed time invariant and are expressed as a fraction of the purchasing price RC of a unit of inventory. The ordering cost of a standing order is zero. However, the ordering cost of an emergency order includes a penalty cost CE per unit ordered so that the cost of an emergency order is proportional to its size (IOL-IMIN). When the inventory level exceeds the storage capacity, IMAX, the excess inventory is sold off at a cost of CO per unit sold. Thus the sell-off cost is proportional to the average excess inventory. The inventory holding cost is proportional to the number of units held in inventory and the length of time the units are held. CH is the holding cost per unit, per period. Similarly, the shortage cost is proportional to the size of the shortage and to the elapsed time between the occurrence of a shortage and the delivery of the back order caused by the shortage. CS is the shortage cost per unit, per period.

A shortage of one unit is assumed less desirable than carrying the unit or placing an emergency order for it. In addition, a unit of inventory in excess of the inventory storage capacity IMAX is less desirable than carrying the unit. These assumptions are included in the following inequalities:

CH<CE<CS, (1)

CHKCO<CS, (2)

where RC: unit purchasing price of inventory, $/unit, CH: unit cost of holding inventory, fraction of RC/period, CE: unit penalty cost on the purchasing price of an emergency order, fraction of RC, CS: unit cost of inventory shortage, fraction of RC/period, and CO: unit sell-off cost, fraction of RC. Of course, the cost comparisons in (1) and (2) are meaningful only if they are applied over the same length of time, a period.

The size of a standing order q is assumed to be equal to or greater than the average demand during the scheduling period. This assumption is intuitively reasonable, particularly if CE and CS are appreciably larger than CH. The order level, IOL, for an emergency order is constrained in the Markov model by a lower bound to ensure that the number of emergency orders does not exceed one during each scheduling period. The size of an emergency order is, therefore, greater than some calculable lower bound.




The Markov Chain Model

The states of the model are defined as the discrete inventory levels of the system. The demands during each scheduling period have been assumed to be independent, identically distributed random variables. If the inventory level at the beginning of the current scheduling period is some specified value, then the inventory levels at the beginning of subsequent scheduling periods will form a sequence of random variables. The probabilities associated with making a transition from the current inventory level to an inventory level at the beginning of the next scheduling period depend only on the current level and the demand size. The sequence of inventory levels at the beginning of successive periods, therefore, forms a Markov chain.

The inventory level at the beginning of the next period equals the level at the end of the current period plus q. Therefore, by defining state i as the inventory level at the beginning of the scheduling period and state j as the inventory level at the end of the scheduling period plus q, we can cal- culate puj, the transition probability from state i to state j.

The system states range from the emergency order point plus one, IMIN+ 1, to the inventory storage capacity, IMAX. The emergency order point IM\IN is not a system state because an inventory jump to the emergency order level IOL is made as soon as the system's inventory level reaches the emergency order level IMIN. The emergency order is made only if without it the state j would have been less than or equal to IM\IN. If an emergency order is not necessary to ensure a state j greater than IMIN, then the system may pass through IMIN but is raised back above IMIN by the arrival of the standing order at the end of the period. In either case the system does not occupy IMIN at the beginning or end of a period and therefore IMIN is not a system state.

We restrict the number of emergency orders to a maximum of one for each scheduling period by (3).

L+IMIN+ 1?_IOL <JIMAX, (3)

where L: largest demand size x, IMIN: emergency order point, IOL: emergency order level, and IMAX: inventory storage capacity.

Transition Matrix. The elements pij of the Markov chain transition matrix are dependent on the demand size probability distribution p(x). However, the demand size that leads to a particular transition is not always readily apparent. To facilitate the calculation of the transition matrix, we make the following classifications:

Category 1 state-a state i is a category 1 state if a standing order alone is not sufficient to assure a state j that is greater than IMIN.

Category 2 state-a state i is a category 2 state if a standing order is sufficient to assure a state j that is greater than IMIN.




Category A state-a state j is a category A state if j is less than IMAX and the transition was made without an emergency order.

Category B state-a state j is a category B state if a transition is impossible from a state i.

Category C state-a state j is a category C state if j is equal to IMAX and the transition was made without an emergency order.

Category D state-a state j is a category D state if j is less than IMAX and the transition was made with an emergency order.

Category E state-a state j is a category E state if j is equal to IMAX and the transition was made with an emergency order.

The various categories into which the states of the system can fall can be specified in terms of the system's parameters and variables: IMIN, IMAX, IOL, L, and q. Each ij combination of states corresponds to precisely one combination of categories for i and j, and the general form for pij is com- pletely determined by this combination of categories. Thus, knowing i specifies the category of state i and knowing the demand in addition to i determines the category of state j (see Fig. 1) . The appropriate equation to use to determine pij is given by the number at the right of Fig. 1 and the equations corresponding to these numbers are presented in Table I.

Cost Matrix. The cost matrix is composed of elements cij, each represent- ing the cost of the one-step transition from state i to j. Each transition cost cij is obtained from one of the following general equations:

cti = CH( ,) ( tiltv) + CS (1I2) ( t2/tp) I ( 9)

ci= CH(11) (tlltv) +CS(12) (t2/tp) + CEE(IOL-IMIN), (10)

c=EK==1MXA X {[H(I)(t/p) + CS ( 2) ( t2/tp)II

+CO(K-IMAX)][p(i-K+q) ]},

where ci.: cost of transition between i and j, fraction of RC/period, I,: amount held in inventory, units, til/t: proportion of scheduling period, tp,

in which inventory is held, 12: amount of shortage, units, t2/tp: proportion of tp in which a shortage occurred, and p( ): probability of demand of size ( ) *

As indicated in Fig. 2 the entries in the cost matrix are obtained according to the combination of categories of states i and j and the conditions on these states. The equations corresponding to the numnbers given at the right of Fig. 2 are presented in Table II. Equations (12), (13), and (14) are derived from (9) and cover the situations where there is neither an emergency order nor a sell-off. Equations ( 15), (16), and ( 17) are derived from ( 11) and deal with the situations in which sell-offs occur but emergency orders do not. Equations (18), (19), (20), and (21) are derived from (10) and provide the costs when emergency orders are required but sell-offs are not. In no situation will an emergency order and a sell-off be required during a single period since the emergency order brings the inventory up to IOL,




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TABLE I TRANSITION PROBABILITY EQUATIONS

Equation number General transition probability equation

4 Pi = p(i-j+q) 5 P= = o 6 Ptj - EK=IMAX P(i K + q) 7 Pij = p(i-IMIN + IOL-j + q) 8 Pi} = p(i - IMIN + IOL - IMAX + q)

which cannot exceed IMAX. Since tranisitions to Category B states are not possible, no costs need be obtained for these transitions.

Analysis

Cost Function and Emergency Order Probabilities. The cost of operating the inventory system is assumed to be the sum of the period costs. As the number of periods goes to infinity, so does the cost. On the other hand, the finiteness of the average or expected cost per period depends on the cost and system parameters and the values chosen for the decision variables.

Condition of State i Transition State Categories and State j Cost

Involved in Transition i

> 0 Positive Iniventory Equation < q Negative Inventory cij

i > ? j q q1

Category (I or 2) State to ii < j q 13

Category A State

i>0 1 q 4

> i j ; > q 15

Category (1 or 2) State to i < O j < q

Category C State

i > 0 j < q 17

:i< O j > q 18

Category 1 State i <0 i .:q 19 to

Category (D or E) State q 20

> 0 ; > 20

Fig. 2. The relation of state categories and condition of states to transition cost equations.




If the transition matrix is regular and irreducible, then it has an associated stationary probability vector. An element 4i of this vector may be in- terpreted as the expected proportion of the time the systemn spends in state i. The expected cost ecij is equal to the product of the transition probability pij and the transition cost ci unless j is a Category C state. The transition cost between a Category 1 or 2 state and a Category C state is already the expected cost, as can be seen from (15), (16), and (17). Therefore, the

TABLE II TRANSITION COST EQUATIONS

Equation Transition cost equation number

12 ci; = CH(i +j - q)/2 13 cij= CS(-i-j+q)/2 14 ci* = [CH(i)2 + CS(q - j)2]/2(i - j + q) 15 Cij = EK+qIMAX {[CH(i + K - q)/2

+ CO(K - IMAX)][p(i -K + q)]} 16 Cij = ~EKIMAX {[CS(-i - K + q)/2

+ CO(K - IMAX)][p(i - K + q)]} 17 ci j ZEK=IMAX {[ (CH(i)2 + CS(q - K)2)]/ 2(i - K + q)

+ CO(K - IMAX)[p(i - K + q)]} 18 cij = CH(IOL -j + q)(IOL + j + q)/2(i - IMIN + IOL - j + q)

+ (-CS)(i - IMIN)(i + IMIN)/2(i - IMIN + IOL - j + q) + CE(IOL - IMIN)

19 cij = -CS(i + IMIN)(i - IMIN)/2(i - IMIN + IOL - j + q) + CS(q-j)(q-j)/2(i-IMIN + TOL-j + q) + CH(IOL)(IOL)/2(i - IMIN + IOL - j + q) + CE(IOL - IMIN)

20 cij = {CH[i2 + (IOL + j - q)2] + CS(IMIN)2}/2(i - IMIN + IOL - j + q) + CE(IOL - IMIN)

21 cij = {CH(i2 + IOL2) + CS[IMIN2 + (q - j)2]}/2(i - IMIN + IOL - j + q) + CE(IOL - IMIN)

cost equation-e.g., (16)-is the expected cost of the transition, ecij, when j is Category C. The summation of the expected cost ecij for a given state i over all states j is the expected cost of being in state i. The product of 40 and this summation is the expected cost contribution of state i to the cost of the system. The expected cost of the system is the summation of these products over all states i. Equation (22) is thus the expected cost of the standing order inventory system with decision variables q and IOL. The minimum system cost is determined by minimizing (22) over all possible pairs of decision variables as indicated in (23). The pair(s) of decision variables that minimize (22) are the optimal values of the decision variables.




C(Si,q,jOL) = Zi=IMINA1 o ,j=IMIN+1 eci jRC, (22)

C(S1) = minq,IoL C(S1,q,I0L), (23)

subj ect tofaverage demand < q, JIMIN+L+I < ?IOL < IMAX,

where SI: standing order inventory system, C(Si,q,IOL): expected cost of a Si system with size of a standing order q and an emergency order level IOL, $/period, oi: ith element of the stationary probability vector, ecij: expected cost of a transition between i and j, fraction of RC/period, and C(Si) minimum system cost of an Si system, $/period.

The minimum system cost was calculated using a Fortran program that computed the expected costs for all possible combinations of the decision variables q and IOL. Results obtained from this program suggested that C(S5,q,IOL) is a convex function when viewed as a continuous function of q and IOL. This convexity has been demonstrated so that the calculations can be reduced to only those required to find a relative minimum.

An emergency order is possible only from a Category 1 state. The probability of an emergency order is dependent on the probability of a transition between a Category 1 state and either a Category D state or a Category E state. The probabilities of an emergency order for a state i and for the the system are expressed in (24) and (25), respectively.

P= -Zj=-IMIN+IoL-L+q pi, (24) Ps x

:L2r1MIN-q , 1 25' iS =<=IMINtl iPi, V J

where Pi: probability of an emergency order if system is in state i, and Ps1: probability that an SI system will experience an emergency order during a scheduling period.

COMPARISON OF THE STANDING ORDER INVENTORY SYSTEM WITH THE

ORDER LEVEL INVENTORY SYSTEM

Measure of Effectiveness for the Standing Order Inventory System

The measure of effectiveness of the standing order inventory system is total expected cost as given by (26). The first term in the expression is the minimum system cost. The second term is the expected purchasing cost of an emergency order, excluding the penalty cost, for a scheduling period. The last term is the purchasing cost of a standing order, which includes a price break a.

TC(S) = C(S1) +[(IOL-IMIN)Ps?+(q) (a) ]RC, (26)

where TC(Si): total expected cost of SI system, $/period, and a: price break (fraction of regular cost charged) for items in a standing order. The




TABLE III HYPOTHETICAL VALUES OF THE PARAMETERS OF EACH SYSTEM

Parameter Standing order system Order level system

Unit holding cost 0.2 RC/period 0.2 RC/period Unit shortage cost 2.0 RC/period 2.0 RC/period Unit ordering cost for 0.0 RC 0.1 RC

nonemergency orders Unit penalty cost for Varies between 0.4 RC and 0

emergency orders 1.8 RC Unit sell-off cost for ex- 0.5 RC 0

cess inventory Leadtime 0 Varies between 0 period and

7 periods, inclusive

IMIN and IMAX were set at -1 and 50, respectively, for the standing order system. The probability distribution of demand was taken as:

p(3) = 0.014 p(6) = 0.306 p(4) = 0.111 p(7) = 0.222 p(5) = 0.222 p(8) = 0.111

p(9) = 0.014

components of this measure-C(S1), IOL, q, and Ps,-are determined for the range of CE values indicated in Table III. The optimal ordering policy and associated minimum system cost for the entire range of unit penalty costs are presented in Table IV. In addition, Table IV provides the probability Ps, that the system will experience an emergency order during a scheduling period. These components are used to determine the price break that equates the total expected costs of the two systems for different leadtimes and unit penalty costs.

TABLE IV OPTIMAL RESULTS FOR THE STANDING ORDER SYSTEM

Emergency Otml Optimal order Probability of Otmlepce order unit Otml level for an Otmlepce standing order anemrnc penalty cost size emergency order omeren Ps cost C(SI) CE sieI OLorePs

0.4 RC 4 9 0.0392 2.5518 RC 0.6 RC 4 9 0.0392 2.6618 RC 0.8 RC 4 9 0.0392 2.7718 RC 1.0 RC 4 9 0.0392 2.8819 RC 1.2 RC 4 9 0.0392 2.9920 RC 1.4 RC 4 9 0.0392 3.1019 RC 1.6 RC 4 8 0.0420 3.2120 RC 1.8 RC 4 8 0.0420 3.3212 RC




Measure of Effectiveness for the Probabilistic Order Level System

The measure of effectiveness that is used to determine the optimal order level Zo is the expected cost for a scheduling period. The minimum expected cost for an order level system with leadtime 1 is given in reference 1. This cost plus the ordering and the purchasing cost comprise the total expected cost of this system, which is comparable to the total expected cost of the standing order system. The expression for the total expected cost of the probabilistic order level system is

TC(S21) =C(S21) + (C3Q+ Q) RC, (27)

where, S21: probabilistic order level system with leadtime 1, TC(S21): total

TABLE V PROBABILISTIC ORDER LEVEL SYSTEM, S2, SIMULATION RESULTS

Average Optimal order Optimal expected Measure of Lead-time I regular order level Z cost (C(S effectiveness

size Q lee ot((2) TC (S21)

0 4.556 6 0.9348 RC 5.9464 RC 1 4.556 12 1.3417 RC 6.3533 RC 2 4.556 18 1.7089 RC 6.7205 RC 3 4.556 24 2.0316 RC 7.0432 RC 4 4.556 29 2.3172 RC 7.3288 RC 5 4.556 34 2.5819 RC 7.5935 RC 6 4.556 40 2.7737 RC 7.7853 RC 7 4.556 44 2.9487 RC 7.9603 RC

expected cost of S21 system, $/period, C(S21): minimum expected cost of a S21 system, $/period, C3: unit ordering cost, fraction of RC, Q: average order size per period, units/period. RC: unit purchasing price of inventory, $/period. Equation (27) is evaluated for values of the parameters indicated in Table III, and the results are presented in Table V. These results were obtained by simulation because the analytic methods given in reference 1 become increasingly cumbersome when 1 increases. As expected, the increases in leadtime result in inereased inventory levels and system costs.

Price Break

Under the standing order inventory system, a price break ae is negotiated between the purchaser and the supplier so that each item delivered on a standing order costs a RC rather than RC. The supplier may be willing to offer a substantial price break because the standing order demand is deterministic.

The price break a on the purchasing price of a standing order that equates the measure of effectiveness for the two systems is presented in Table VI as




TABLE VI PRICE BREAK a THAT EQUATES THE MEASURE OF EFFECTIVENESS OF THE STANDING

ORDER SYSTEM S1 AND THE PROBABILISTIC ORDER LEVEL SYSTEM S2

Leadtime Emergency order unit penalty cost CE (fraction of RC)

1 Periods 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

0 0.711 0.684 0.656 0.629 0.601 0.574 0.547 0.519

1 0.813 0.786 0.758 0.730 0.703 0.675 0.648 0.621

2 0.904 0.877 0.850 0.822 0.794 0.767 0.741 0.713

3 0.986 0.958 0.930 0.903 0.876 0.848 0.821 0.794

4 1.057 1.029 1.002 0.974 0.947 0.919 0.892 0.865

5 1.123 1.095 1.068 1.041 1.013 0.985 0.959 0.931

6 1.171 1.114 1.116 1.088 1.061 1.033 1.006 0.979

7 1.214 1.187 1.159 1.133 1.105 1.078 1.051 1.023

a function of the order level system leadtime and the standing order system emergency penalty cost. For example, a price break of 0.758 is required to equate the total expected cost of a standing order system with an emergency order unit penalty cost of 0.8 RC to the total expected cost of an order level system with one period leadtime. In other words, if RC is equal to $10.00 and leadtime is one period for an order level system and the unit penalty

TABLE VII LEADTIME 1 THAT EQUATES THE EXPECTED COST OF THE Two SYSTEMS WHEN THE

PRICE BREAK a IS 1.0

Emergency order unit penalty cost CE L eadtime (periods) (fraction of RC) I

0.4 4 0.6 4 0.8 4 1.0 5 1.2 5 1.4 6 1.6 6 1.8 7




cost is $8.00 for a standing order system, then a unit price of $7.58 for a standing order is required for the two systems' costs to be equal. If the price break is such that the unit purchasing price is less than $7.58, then the standing order is preferred; if the unit purchasing price exceeds $7.58, then the order level system is preferred.

For a leadtime of four periods or longer, Table VI indicates that the standing order system is more economical than the order level system if the unit penalty cost is equal to 0.8 RC. Thus, Table VI can be used to discern the relative effectiveness of the standing order inventory system.

From another point of view, if the price break is set at 1.0 and the leadtime at which the two systemns have equal costs is determined as a function of the unit penalty cost, then these leadtimes are an indication of the relative effectiveness of the standing order inventory system. These leadtimes are presented in Table VII. If the unit penalty cost, CE, is equal to the value indicated, then for any leadtime equal to or more than the one listed, the standing order system is more economical than the order level system regardless of the price break.

CONCLUSIONS

In a standing order inventory system, the effects of leadtime are mitigated by an ordering policy that eliminates leadtime. A comparative evaluation of the standing order inventory system has been accomplished by modeling the standing order system, determining its expected costs and, for reasonable hypothetical values of system parameters, comparing it with the expected costs of an order level system with leadtime. In general, the comparisons show that the standing order system becomes more economical as the order level system leadtime becomes appreciable. Also, as the unit penalty cost on the purchasing price of an emergency order decreases, so does the leadtime at which the standing order system becomes more economical. The order level system is more economical in cases where the leadtime is insignificant and the unit penalty cost on the purchasing price of an emergency order is high.

ACKNOWLEDGMENT

The authors would like to thank the referees for their detailed reading and thoughtful suggestions. Their comments, which are gratef-ully ac- knowledged, led to several improvements.

REFERENCE

1. E. NADDOR, Inventory Systems, Wiley, New York, 1966.



analysis of a standing order inventory system with emergency orders

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