A lot-size model with discrete transportation costs

Computers ind. Engng Vol. 22, No. 4, pp. 397--402,1992 Printed in Great Britain. All fights reserved

A LOT-SIZE

MODEL

0360-8352/92 $5.00+ 0.00 Copyright© 1992PergamonPress Ltd

WITH

DISCRETE COSTS

TRANSPORTATION

OMPRAKASH K. GUPTA Indiana University Northwest, Division of Business and Economics, 3400 Broadway, Gary, IN 46408, U.S.A. (Received for publication 17 April 1992)

Abstract--This paper deals with an inventory model where transportation costs are considered explicitly. In most situations, it is the buyer who must bear the transportation cost of the goods purchased from the supplier. Such costs are either assumed to be fixed and are therefore included in the ordering cost or variable and included in the item cost. In most realistic situations, it is observedthat a fixed cost is incurred for a transport mode such as a truck or wagon irrespective of the quantity loaded. No matter what is the size of the lot, it would always require an integer number of transport modes. Therefore the transportation cost becomes a discrete function. In this paper we develop a lot-size model with discrete transportation costs and present a simple algorithm for the optimal lot-size.

1. INTRODUCTION When a purchasing manager places an order of items, it is important to realize that he/she should take into account m a n y types of costs, including transportation costs of the items purchased. Since inventory and transportation costs influence each other, it would be desirable to arrive at a joint transport-inventory policy rather develop these policies disjoint from each other. Since its inception the Harris-Wilson [1] inventory model has attracted m a n y researchers to the inventory modelling area. One of the main reasons behind this is the fact that the Harris-Wilson model makes so m a n y restrictive assumptions which almost never hold in real life. Most of the research in the inventory modelling area is based on the relaxation of the original assumptions. Several research papers deal with the modifications of the cost structure. The original model considers three types of costs: item cost, inventory holding cost, and the ordering cost. The model assumes that the cost of an item is constant. M a n y papers have appeared relaxing this assumption by accommodating nonconstant unit costs such as with all-unit and incremental quantity discounts. It assumes that holding costs are directly proportional to the value of the items in stock. Several papers consider nonlinear holding costs. Finally the model also assumes that the ordering cost is constant and does not depend on the size of the lot. Researchers have considered various extensions of nonconstant ordering cost such as r a n d o m cost, concave cost, time-dependent cost, etc. Since inventory and transportation decisions influence the overall costs, it would be appropriate to arrive at a joint transport-inventory policy. The incorporation of transportation costs into the lot-size determination analysis has attracted few researchers. The first serious consideration to this issue was perhaps given by Baumol and Vinod [2]. They considered an inventory-theoretic model of freight transport whereby order quantity and transportation alternative can be jointly determined. They considered the problem of minimizing the total of transportation, ordering, and inventory carrying costs. Though their model was developed for the primary purpose of gaining insight into the demand for freight service, it is now being recognized as a pioneering effort in this area. The model however considered a per unit constant transportation cost. Their work was followed by Das [3] who basically worked on the same model with a different set of assumptions. His model provided for the determination of a fixed order quantity, and dealt safety stock sizes from the order size. Buffa and Reynolds [4] extended these works to explicitly include stockout costs and freight discounts based on minimum full truck-loads. Langley [5] considered five different types of freight costs: constant, proportional, exponential, inverse, and discrete. He however did not develop any mathematical model to minimize total costs but showed how explicit enumeration can be used to arrive at the optimal lot size. Constable and W h y b a r k [6] also extended [1] to incorporate 397

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backorder costs. They considered three attributes of different transportation alternatives: the transportation cost, the expected transit time, and the variability in the transit time and determined the transportation alternative and inventory parameters. Even this model considered transportation cost as a part of unit cost. Larson [7] provided a model for economic transport quantity (ETQ) with quantity discounts on freight. He considered different setup costs at origin and destination, but failed to provide an algorithm for determining the ETQ. It was however shown how lot-sizing decision would have impact on company's other policies such as storage capacity and just-in-time production schedules. More recently Russell [8] had considered quantity discounts on freight costs and developed a model where per unit freight cost decreases as lot-size increases. It was shown that in such a situation it may be economical to overdeclare the shipment to take advantage of quantity discount and reduce overall costs. In this paper we consider a more realistic and frequently encountered situation when a fixed cost is incurred for a transport mode such as a truck which has a fixed load capacity. This cost is incurred whether a truck is fully loaded or only partially loaded. Since the number of trucks required would always be a positive integer, the transportation cost would be a discrete function of the lot-size. The total cost would therefore be a nonlinear discontinuous function of the order quantity. A simple algorithm is developed for determining the optimal lot-size which minimizes the sum of the inventory holding, ordering and transportation costs. 2. MODEL DEVELOPMENT We assume the following inventory characteristics: 1. 2. 3. 4. 5. 6. 7. 8. 9.

The demand rate is uniform and constant. No shortages/backorders are allowed. Leadtime is constant. The entire lot is delivered in one batch. There are no quantity discounts. Holding costs are directly proportional to the time-averaged inventory levels. Ordering cost is constant and does not include transportation costs. The inventory planning horizon is infinite. The transportation cost is constant for a truck-load (of a given capacity) even if the quantity shipped is less than a truck-load. 10. The buyer is responsible for the transportation cost. The following notations are used: D = I = Cl = A= K= C2 = Q* = rx-~ =

Annual Demand Inventory holding cost per dollar per year Unit item cost Ordering cost for lot-size Q Capacity of the truck (in units) Cost of transporting upto one truck-load Optimal lot size Lowest integer greater than or equal to x.

For lot-size Q, it would be necessary to hire rQ/KI number of trucks. Therefore the annual average relevant cost for lot-size Q is the sum of the average annual ordering costs, holding costs, and transportation costs, and it can be expressed as:

f(Q) = AD/Q + QIC~ + DC2(rQ/K'I)/Q.

(1)

It is obvious that if m trucks are used for the order quantity Q, then (m - I)K < Q <~mK. If m trucks are used to deliver the lot, we will call it an m-truck policy. The annual average relevant cost for transporting Q units with m trucks is given by:

f,,(Q)=AD/Q +QIC,/2+ DmC2/Q,

( m - 1 ) K
A lot-size model with discrete transportation costs

399

where m is a positive integer. It may be easily seen that { f , , } is an increasing sequence as shown in Fig. 1. The cost function f ( Q ) is discontinuous shown by the solid curve. The functionf,.(Q) attains its unconstrained minimum at Qm ffi sqrt [2(A + C2m)D/IC~]

(2)

and the value of fro(Q) at Qm will be sqrt [2(,4 + C2m)DICI].

(3)

Therefore fro(Q) >t sqrt [2(A + C2M)DICI] for all values of Q. Suppose we consider ordering with one truck (i.e. m = 1). f~(Q)=AD/Q +QIC~/2+DC2/Q,

O < Q ~
The functionf~(Q) has its unconstrained minimum at QI = sqrt [2(A + C~)D/ICI]. If Q~ ~ K, the optimal lot size Q * is obviously Q~. If Q~ > K,f~(Q) will have itsconstrained minimum at Q = K, and the corresponding average annual relevant cost would be f~(Q)= A D / K + KIC~/2 + DC~/K which would be an upper bound, say UB, on the optimal average annual relevant cost. From equation 3 it is clear that ordering with m trucks will not be economical if sqrt [2(./I+ C2m)DICI] > UB. Simplifying, we get: m > [(UB2)/2DICI - A]/C 2 r, say. =

(4)

We can therefore discard all m-truck policies where m > r. Based on these observations the following algorithm is developed. 3. ALGORITHM

Step If Step Step Step

1. Compute the unconstrained minimum Qm by using equation (2). Qm~ r. 4. If all truck policies have been considered, stop. Otherwise, move to the next higher number of truck level policy, and compute the unconstrained optimal lot size Q using equation (2). If Q is also constrained optimal, compute f ( Q ) by (1). If f (Q) < f ( Q * ) , set Q* = Q, and stop. I f f ( Q ) = f ( Q * ) , Q and Q* are both optimal. 1200

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If Q is not constrained optimal, replace Q by the highest feasible value of Q, and compute f ( Q ) . I f f ( Q ) < UB, update Q * = Q, and UB = f ( Q * ) . Go to Step 2. A F O R T R A N listing of the above algorithm is given in the appendix. 4. SENSITIVITY ANALYSIS

In this section we first compute lot sizes and annual relevant costs using the above algorithm for two illustrative examples. We then compute the same without considering transportation costs explicitly. We next compare the results to see how sensitive the values of lot sizes and costs are to the discrete transportation cost structure. In Example 1, we find that the optimal lot-size suggests only full truck loads as no partial truck load is used. In Example 2, the optimal lot-size requires a partially loaded truck.

Example 1. Suppose: A = $20, D = 400 units, I = 0.1, K = 50 units, C1 = $20, and C2 = $50.

Step 1. Compute Q1. QI --- sqrt [2(20 + 50)400/(0.1)(20)] = 167.33 Since Q~ > 50, it is infeasible. Therefore set Q * = 50. Compute f ( 5 0 ) = 20(400)/50+ 50(0.1) (20)/2 + 400(50)/50 = $610.00 Set upper bound UB = 610. Step 2. Compute r. r = [(610)2/2(400) (0.1) (20) - 20]/50 = 4.251

Step 3. Ignore ordering with 5 or more trucks. Step 4. Consider the next higher truck-policy. Compute Q2Q2 = sqrt [2(20 + 100)400/(0.1) (20)] = 219.10 Since Q2 > 2(50), it is also infeasible. Replace it by Q2 = 100, and compute f(100). f(100) = 20(400)/100 + 100(0.1 ) (20)/2 + (400/100) (50) (2) = $580.00 Since f ( 1 0 0 ) < U B , update Q * = 100, and UB = 580. Step 5. Recompute r. r = [(580)2/2(400) (0.1) (20) - 20]/50 = 3.805

Step 6. Ignore ordering with 4 or more trucks. Step 7. Consider the next higher truck-policy. Compute Q3. 03 = sqrt [2(20 + 150)400/(0.1) (20)] = 260.77 Since Q3 > 3(50), it is also infeasible. Replace it by 150, and compute f(150). f(150) = 20(400)/150 + 150(0.1) (20) + (400/150) (50) (3) = $603.33 Since f ( 1 5 0 ) > UB, we do not change Q*. As we have exhausted all truck-policies, stop. The optimal lot-size is 100 units i.e. two full truck-loads. Suppose we do not explicitly consider transportation costs and compute lot size by the conventional Harris-Wilson formula, we get, Lot-size = sqrt [2(20) (400)/(0.1) (20)] = 89.44 units, and the annual relevant costs

sqrt [2(20) (400) (0.1) (20)] + 2(50) (400/89.44) = $626.12. The proposed algorithm therefore provides a bigger lot-size and also a reduction in cost. In this example the cost reduction is about 7.4%.

Example 2. Suppose: A = $10, D = 1,000 units, I = 0.2, K = 65 units, Ct = $4, and C2 = $1.

A lot-size model with discrete transportation costs

Step 1. Compute Q~. Ql = sqrt [2(10 + 1)1000/(0.2) (4)] -- 165.83 Since QI > 65, it is infeasible. Therefore f(65) = 10(1000)/65 + 65(0.2)(4)/2 + 1000/65 ffi $195.23 Set upper bound UB = 195.23 Step 2. Compute r.

401

set

Q* ffi 65.

Compute

r -- [095.23)2/20000)(0.2)(4) -- 10]/l -- 13.82 Step 3. Ignore ordering with 14 or more trucks. Step 4. Consider the next higher truck policy. Compute Q2. Q2 = sqrt [2(10 + 2)1000/(0.2)(4)] ffi 173.21 Since Q2 > 2(65), it is also infeasible. Replace it by 130, and compute f(130). f(130) = 10(1000)/130 + 130(0.2)(4)/2 + (1000/130)(2)(1)

--

$144.31

Since f(130) < UB, update Q* = 130, and UB ffi 144.31. Step 5. Recompute r. r = [044.3)2/20000)(0.2)(4) - 10]/1 = 3.015 Step 6. Ignore ordering with 4 or more trucks. Step 7. Consider the next higher truck-policy. Compute Q3. Q3 -- sqrt [2(10 + 3)1000/(0.2)(4)] -- 180.28 Since Q3 is feasible, computer(180.28). f(180.28) = sqrt [2(10 + 3)0000)(0.2)(4)] ffi 144.22 Since f(180.28) < UB, we stop. The optimal lot-size is 180.28 units, i.e. 2 fully loaded trucks and one partially loaded truck with 50.28 units. Without considering transportation cost, the lot-size can be computed as, Lot size = sqrt [2(10)(1000)/(0.2)(4)]-- 158.11 units, and the annual relevant costs = sqrt [2(10)(1000)(0.2)(4)] + 3(1)(1000/158.11) ffi $145.46. With the proposed algorithm the lot-size was computed as 180.28 units with annual relevant cost of $144.22. In this case also the proposed algorithm provides a bigger lot-size and reduction in cost. The cost reduction is however smaller, about 1%. These illustrations suggest that when the proposed algorithm is applied we may get a different lot size and also a reduction in cost. The actual amount of savings would depend on the values of the specific problem parameters. 5. CONCLUSIONS

In many real-life situations transportation costs of goods are fixed for a finite capacity of a transport mode such as a truck. A fixed cost is incurred when a truck is deployed whether it is fully utilized or not. Most inventory models either consider transportation cost as fixed and include it in the ordering cost or consider it variable and consider as a part of the item cost. In this paper we have developed a transport-inventory model where transportation costs are explicitly considered. The resulting cost function is a discrete function of the lot-size. A simple algorithm is proposed to determine the optimal lot-size. Illustrative examples have been given to demonstrate how the lot-size and operating cost may be sensitive to the transportation costs. These illustrations suggest that when this algorithm is implemented it may reduce operating costs. The magnitude of reduction however would vary and depend on the actual values of specific problem parameters. Acknowledgements--The author is grateful to the anonymous referees for their constructive comments to improve this paper. The author also acknowledges assistance of Mr. P. P. C. Rao in making some of the preliminary computations.

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OMPlU,KASHK. GUPTA REFERENCES

1. F. W. Harris. What quantity to make at once. The Library of Factory Management, Vol. 5, Operations and Costs, pp. 47-52. A. W. Shaw Company, Chicago (1915). 2. W. J. Baumol and H. D. Vinod. An inventory theoretic model of freight transport demand. Mgmt Sci. 16, 413-421 (1970). 3. C. Das. Choice of transport service: an inventory theoretic approach. Logist. Transport. Rev. 10, 181-187 (1974). 4. F. P. Buffa and J. I. Reynolds. The inventory-transport model with sensitivity analysis by indifference curves. Transport. J. 16, 83-90 (1977). 5. C, J. Langley Jr. The inclusion of transportation costs in inventory models: some considerations J. Bus. Logist. 2, 106-125 (1981). 6. G. K. Constable and D. C. Whybark. The interaction of transportation and inventory decisions. Decision Sci. 9, 688-699 (1978). 7. P. D. Larson. The economic transportation quantity. Transport. J. Winter, 43-48 (1988). 8. R. M. Russell. Optimal purchase and transportation cost lot sizing for a single item. DSI Proc. 1989 Annual Meeting, pp. ll09-1111. APPENDIX PROGRAM

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Discrete Transportation Costs

DIMENSION Q(100) REAL,4 I,INVCST WRITE(,.,) 'Give A,D,I,C1,C2,K' READ(, .,)A,D,I,C1,C2,K M=I QN = 2 , ( A + M , C 2 ) , D QD = I,C1 Q(M) = SQRT(QN/QD) J=M,K IF (Q(M).LE.J)GOTO 20 QO=M,K ORDCST= A,(D/QO) I NVCST = Q O , I , C 1 / 2 TRCST = ( D / Q O ) , M , C 2 FQ = ORDCST + INVCST + TRCST IF (M.EQ.1) UB = FQ IF (FQ.LT.UB)UB = FQ IF (FQ.GT.UB)GOTO 45 GOTO 30 QO=Q(M) FQ = SQRT(QN,QD) IF (MEQ.1)GOTO 40 IF (FQ.LT.UB)GOTO 40 UB = FQ GOTO 30 R = ( ( ( U B * * 2 ) / ( 2 , D , I , C 1 ) ) - A)/C2 M=M+I IF(M.GT.R)GOTO 45 GOTO 10 QO = Q(M) WRITE(,.,) 'OPT. ORDER QUANTITY=',QO FQO = FQ WRITE(,.,) ' M I N I M U M COST=',FQO GOTO 50 QO = ( M - 1 ) , K WRITE(,.,) 'OPT. ORDER QUANTITY='.QO FQO = UB WRITE(,.,) ' M I N I M U M COST=',UB STOP END