JUST-IN-TIME (JIT) INVENTORY SYSTEM

Transcription

1 CHAPTER 8 SOME ADVANCED MODELS IN JUST-IN-TIME (JIT) INVENTORY SYSTEM 8.1 INTRODUCTION: 1 lie models pertaining to the concept of JIT have been derived lor deterministic situations. In all the models, the demand is uniform and known. But, tins is not always true. There is many situations where demand is not exactly known, instead of that probabilistic distribution of demand is known i.c. we may have a probabilistic situation for a demand pattern. This chapter deals with some such models which are stochastic or probabilistic m nature. The control variable in such cases is assumed to be either the scheduling period or the order level or both depending upon the type of the model. The optimum order-levels are thus derived by minimising the total expected cost. We consider specifically the following probabilistic models in this chapter. MODEL I : JIT model with uniform demand and continuous units for a single period problem. MODEL II : Comparison of JIT with W1P. Each of the above two models developed here is also illustrated by suitable numerical illustrations. means of 8.2 MODEL I : A JUST-IN-TIME (JIT) MODEL WITH UNIFORM DEMAND AND CONTINUOUS UNITS FOR A SINGLE PERIOD PROBLEM INTRODUCTION: Let us consider an inventory pioblem attached to a warehouse or departmental store. The system adopted is a Just-in- 1 ime approach and the demand i is probabilistic in nature with some uniform pattern of variation over a single period. The units to be procured continuously. A decision is to be taken about the optimum ordot quantity so that the expected total cost becomes minimum. are M2 t 1 (155)

2 8.2.2 NOTATIONS: rhe following notations are used in this model: D : Demand (units) Q : Order quantity (units) A : Cost of placing an order (Rs/order) : Aggregate cost per shipment N : Number of shipments H : Cost of holding ASSUMPTIONS: Model is derived under the following assumptions: Since I. Demand is probabilistic in nature and the probability distribution of demand is uniform distribution over a period. 2. Reorder time is fixed and known. 3. The rate of replenishment is instantaneous. Lead time is negligible. 5. Shortages are not allowed to occur. FORMULATION OF THE MODEL: demand D is probabilistic in nature, if f(d) represents the, of demand which is a continuous distribution, we ; have f /( D)d!) - distribution 1 and by virtue u1 of f(d) it is obvious that j(d) > 0 for all D. The order quantity Q will be such that > D Let S denote Q. the order-level then we write Q' = S + 6 where h is a const ant. Hence, the expected cost is given by,/ fj Q Q' I) m + 2 Q DO UQ*7 + (A +.VP)(D ~Q ) 1 m. 2D (ID (i.i) (156)

3 From above, we find that the expected total cost for one period is given by the following expression TC(Q) = H 2N L ~ Q' D 2/O0 H Q 2 Q - 1 f{d)dd 2 + 2N JQI 21) CO J(D)DD +{A + (D-Qf 2D We want to terminate the optimal order quantity Q so that is minimum, for this we should set the necessary condition = 0 subject to sufficient conthe result for dition -ffi?) > 0. To obtain [TC(Q)\ we use differentiating under the integral sign and simplification gives the result w[mid+jl ym»-(a+np)j 00 / > - Q ~ pom H f(d)dl) dq l) [f/c + 00 Q', D (A + = NP0 which gives where F(Q ) 0 gives F(Q ) f(d)dd+jÿ /1-f NP [II + (A + NP)} (A + NP) r*f{d) Th sufficient tlf(d)dd. condition [H + (A + N))L_ II )j-dd > 0 J Q=Q" Hence optimum order quantity and expected minimum total cost can be obtained. > Hence, / d2 TC(Q) dq 2 {A T NP) r f(d) (ii + (A + NP))jQI I) dd > 0 gives (Q0) the condition for finding the optimum value of and HYPOTHETICAL ILLUSTRATION: A baking company sells cakes by weight. The cost of holding is Its.0.20 per cake and ordering cost is Rs and setup cost is Rs The demand is assumed lo be rectangular between 1000 and T he optimum daily amount to be baked is given by the following procedure: f(x) W ; a< x < ; otherwise (157)

4 r Fills implies that f(d)dd + oo b f{x)dx Q D a I Q' f(d)dd = -Ax a A + AfP [H + (A + AT)] Ihe solution to this equation i value between 150 and t Q i 2.00 dd + Q -dd = D /(8+l.OlQ) is obtained by trial and error method which gives the REMARKS: Here it is observed that the order quantity on a daily basis is for 150 to 200 cakes. This incurs smaller cost because this quantity gets sold off easily, thus preventing the holding of extra inventory. In contrast to this, under a traditional approach, the quantity ordered and produced would be more thus incurring many overhead costs. 8.3 MODEL II : COMPARISON OF JUST-IN-TIME (JIT) WITH WORK-IN-PROCESS (WIP) INTRODUCTION: Manufacturing Resource Planning (MRP), Just in-time (JIT), Optimisation Produc tion Technology (OPT) and Flexible Manufacturing Systems (FMS) are al! different management philosophies that are quite in vogue today. The major aim of using these philosophies in different companies is to improve production eficiency by cutting the work-in-process (WIP) stocks. The systems derived from Master Production Scheme (MPS) such as MRP and OPT require a precise demand forecast for each item in order to achieve zero stockouts. In JIT systems, the stocks between the successive production process is minimised by smoothing techniques and stable forecasts. Today, all companies, irrespective of the technique adopted, are concentrating on eliminating WIP or reduce it. to minimum amounts. The target of zero inventories is achieved if the necessary ite;ms are pronecessary quantities at necessary times (Nordan, 1985). duced and delivered in Many companies winch follow WIP system are changing to pseudo-jit sysl 1987; Krajewski, 1987; Murray, 1987) e ms (Kell her, 1 (158)

5 Thus, here system whether a JIT an attempt is made to discuss the decision rules for a production or a WIP system should be utilised. This work is based on the study conducted by Sipper and Shapiira (1989) NOTATIONS: 1 he following notations are used in this model: l : inventory level of work-in-process at any point in time in this case between work-station 1 and work-station 2. I can lx: taken as a constant. a : c : rate of production in units per unit time, equal for both workstations. material cost of an item in WIP. k : penally per unit per unit time for late delivery. i : capital cost per unit time. t : time interval between successive machine failures, T : planning horizon. AT : repair duration. K : total cost over planning horizon T. rn : number of times a failure occurs during the planning horizon T ASSUMPTIONS: Model is derived under the following assumptions: 1. mat 7, and therefore the length of time where WIP is below I, is negligible (which means that the total time for repair duration for failure situation is much less than the planning horizon rf), 2. The level of work in process is replenished to its level I. each time a breakdown repair terminates. Therefore, i can be taken as a constant (159)

6 8.3.4 FORMULATION OF THE MODEL: The model pertaining to the comparison between WIP and JI I as below which gives T AT A'(at*!)di, where 0 < 8 < I. [< = ic. 1 dt + m l/a, at"+l 8 +1 AT 5+1 has been formulateds (fr p a(at) - 1ST- a 6+1 a K = ic.it+ A - It = icit + mk 8+ 1 i/a = ic/t + m/( I ' /AY = icit + mk tn af AT) js+ 1 as(8 +1) V+1(AT)g+1 - Is+i - IATas(8 + I)-I2(8 + 1)' aÿiÿs -f- 1 ) Here the value of 8 will be of primary interest : I2 a 1. If 5 = 0 K = icit + rnk aat / (/ + AT +1) But, AT the repairing time is very small. Hence it is negligible. icit + rnk oa7 ~/2 / which implies a JIT policy. a 1 bus A 2. If 8 = 1 a(a'f)2 p pi K = icit + - IAT 2 2a2 a Neglecting (AT) and higher terms we get K icit rnk which implies a pure WIP policy., I2 al2+ 2a REMARKS: In this model, a comparative analysis between JIT and WIP is carried out The repairing time in any of the advanced philosophies is as low as possible. Thus. AT is taken almost as a very small quantity, so the higher powers arc almost negligible. The different values of 5 specify the state of the system, whether JIT or WIP. it i seen that when WIP is minimised, it almost tends to JIT because the ultimate atm of JIT is reduction of waste which is also in the form of WIP. 147 s / 1 (160)

7 8.4 CONCLUSIONS: In this chapter two models are dealt under a probabilistic approach. I he optimum order quantity is determined in Model-I which is on a JIT basis. The quantity ordered is small so that it could be catered wholly, in order to see that the costs associated with its stocking are minimised. The second model gives a comparative analysis between JIT and WIP. The ul timate aim is to reduce the amount of WIP in order to achieve a more efficient production system. It is shown through equations that lesser the amount of WIP, more efficient is the system overall. We have discussed uptil now in all these chapters regarding different types of Jl 1 models as per their applicability in related areas. We shall end up this chapter by taking their overall view by means of concluding remarks. 8.5 CONCLUDING REMARKS 1 his thesis attempts to formulate certain mathematical models in Just-In-Time In ventory System, keeping in view both the requirements and their practicability. As research is a never-ending process no suggestions that are offered can be regarded as perfect and ultimate. There is always a scope for improvement in the kind of work undertaken in this thesis. Hence we feel that this thesis will be incomplete without a suggestion as to the direction in which future improvements could be carried out. It. is our humble attempt to indicate some areas for research in future study is based upon the various aspects of the Just-in-Time (JIT) schedule are classified into various categories as discussed earlier. The purchasing models of JIT are studied by incorporating the model proposed by Ramasesh (1990). The first model is derived for the case of deteriorating items. This is extended for the case of stock-dependent demand in the second model. Further analysis can be done for ' both these models by utilizing the concept of quantity discounts. The incentive of providing a discount helps the supplier reduce his inventory level. For such an offered discount, the price that may be charged from the buyer could also help the bu\ cr from a long-term commitment with the supplier. In Model III for the purchasing aspect, the objective was to investigate how pric ing decision affects the overall economies of the firm. This problem can further be visualised for those issues where the other aspects like cost as well as profit for both (161)

8 the parties can be studied. These issues could be studied for the cases where there is a mutual understanding between the vendor and the buyer and for those where there is no co-operation between them. These results could be compared internally and interpreted further. Model IV deals with the determination of economic ordering policy for a single supplier - single buyer. In this model the ordering cost parameter was varying. This can he further extended by taking one more parameter i.e. the transportation cost. By incorporating the transportation cost, it can also be found that the optimal cost in reaching the buyer s firm finally reduces the total inventory cost drastically. In the model pertaining to cyclic scheduling, the equations are derived for two cases viz.;- One with the fixed set-up cost, and Liu: other with the varying set-up cost. I his could be extended further to those situations where there are many sub assemblies for each production line. The total elapsed time for each sub-assembly is to be minimised which in turn, would also minimise the total production time. This in turn can be helpful in reducing the cost. The last model of purchasing aspect deals with Lead time is also treated as a decision variable. giving some more mathematical treatment. an important factor like lead time. This can be extended further by Fite demand can be assumed to follow some known form of probability distribution to deal with the stochastic version of the model. The manufacturing models are all dealt with by using the basic model proposed by Crout and Seastrand (1987). in all the models the lead time is assumed to be zero. This can further be developed for the realistic situation of non-lead time, head time is very much reduced under a JIT schedule as combÿygÿt fco that inventory system. The mathematical models can be analysed for a production svstem with variable lead times for procuring the raw materials. The results obtained under zero lead time can be compared with those for variable lead Limes. In the model where a limit has been imposed on the amount tied in a. traditional up as stocks a limit can also be imposed on the floor-space and other related constraints. Then the whole contract quantity can be split up into small lots over some months of time. The optimal order quantity then would depend upon these constraints. They can also be analysed for those cases where there is a known frequency of replenishment orders. The model pertaining to vendor-buyer relationships can be studied further by varying the set-up costs, ordering costs etc. The sensitivityy analysis can also be done 1-19 C-. t 1 (162)

9 further b}' treating order quantity to increase by a small abount as well as lor the cases of changing parameter values. The total cost functions for both the cases could be compared and evaluated. ' * The Kanban models are treated using the basic ideology of Kanban and the basic model developed by Goodman and Souder I 1988)* The number of Kanbans to be withdrawii can also be determined by finding the reorder point. This wouid also give an idea about the average usage per day and the lead time. The number of Kanbans can also be calculated under constant and variable cycle time, 1 he models under financial management can he analysed for various factors per taining to manufacturing aspects. The models have been proposed by incorporating t he purchasing aspect. Apart from ordering, set-up and holding costs, additional costs like transportation, materials handling costs can also be included. Further analysis can be done by taking demand to follow some probability distribution, for this pur pose, Poisson exponential, Gamma and Beta distributions may be applicable as per suitability of the model considered. I he pricing models could also be extended to situations where apart from price discounts, quantity discounts can also be offered. The models can also be studied for various demand functions mentioned in economic theory. The various relationships between demand and supply could be incorporated to deal with the relevant The next form of models will situations. All models arc hey could also be extended to various cases pertain to those treated under tfie competitive developed under fixed and varying market conditions. by adding some cases. more parameters like labour cost, material cost, set-up time and transportation cost. From the above discussion it may be visualised that there is tremendous scope for further theoretical development of the work presented in this thesis. Besides that it has also ample scope for different applications in industries. In the next chapter we shall highlight some studies made in this context with special reference to Indian Industries. 150 t 1 (163)