Production Planning and Profit Design in P 3 System

Management 2014, 4(3): 64-70 DOI: 10.5923/j.mm.20140403.02 Production Planning and Profit Design in P 3 System Pradeep J. Jha 1, Viranchi Shah 2,* 1 LJ College of Engineering & Technology, Ahmedabad, India 2 Saga Laboratories, Ahmedabad, India Abstract This study paper characterizes dominant contribution of *P 3 system in (1) Production Planning and (2) study of impacts of different factors on sale price. As a result it helps study of variations in profit under fluctuations in factors like cost price, fixed price, and variable cost index, etc. in a well defined * P 3 system. (*P 3 = Production, Planning and Procurement). P 3 system is designed keeping the notion of CPFR (Collaborative Planning, Forecasting and Replenishment) in the centre. It assures perspectives based on synergistic notion in all the three (vendor, distributor and buyers) fundamental components of complete production cycle. This system, beyond contradiction, nullifies the probable shortages and it closely parallels to VMI system-a well proven and broadly accepted system. We have constructed a linear model for the profit function to study break-even analysis and the effects of various parameters of the cost function and dispatch routines. Keywords Safety Stock, Emergency Stock, Lightening Stock, P 3 System, Profit Function, Break-Even Abbreviations (1) VMI - Vendor Managed Inventory (2) CPFR - Collaborative Planning, Forecasting and Replenishment (3) ECR - Efficient Customers Response (4) JIT - Just In Time Notations Some of the mostly used notations during the discussion and mathematical model are given below. C 1 : Cost Price per unit amount of raw material R 1 : Raw material required for one unit production S: Sale Price/Unit (S = S 1 + α S 1 ; clarified in point (5)) F 1 : Fixed cost of Production F 2 : Fixed cost of dispatch h 1 : Holding cost per unit per unit time Q: Sum Total of Normalized Demand of all buyers D: Harmonic Average of mean demands of all buyers r 1 : Fraction of % discount on Safety stock K 1 : Safety Index K 2 : Emergency Index K 3 : Lightening Index Assumptions * Corresponding author: viranchi@sagalabs.com (Viranchi Shah) Published online at http://journal.sapub.org/mm Copyright 2014 Scientific & Academic Publishing. All Rights Reserved (a) Vendor, distributors and buyers share the inventory status on on Line basis. They communicate the current market trends and exchange information. (b) Maintenance and replenishment of inventory is, in principle, an ethical responsibility of the buyer. In short, the shortages are not allowed. (c) A pre-fixed discount and relaxation in payment terms on the same is a part of terms and conditions agreed upon by all, the vendor and the buyer. (d) New buyers dire to be a part of the system can join but only at the beginning of the new cycle and same applies to any buyer considering leaving the system. (e) The P 3 system has interwoven and inherits the major of ideals of VMI, CPFR, Consignment model system, JIT (for buyers and task force engaged in production line) and all targeting to achieve and maintain Efficient Customers Response. (f) The vendor, to remain dominant over the competitors, and pass maximum benefit to the buyers, enjoys privileges to determine sale price. The mathematical formula derived helps him to determine his profit margin. 1. Introduction It is known that the production rate depends on continuous supply of raw material required by the production capacity of machinery units employed for the purpose. The number of units to be produced in a given cycle is a managerial decision. This depends on the survey carried out during the consumption period of supply of first lot along with safety stock. Total production should be at least equal to the sum total

Management 2014, 4(3): 64-70 65 of different types of supply made by the vendor from time to time. (1) First supply of normalized Demand along with the safety stock (2) Second prompt supply of Emergency Stock by the distributor (3) Last supply of Lightening stock by the vendor This consumption period is approximated by the average (harmonic) demand calculated on the basis of average usage (sale) performance done by each buyer in a unit time period. Every time on new production cycle begins, the vendor incurs a fixed set-up cost and variable cost per unit production remains proportional to the number of units produced. Total production units should closely approximate the total demand and it may require more than one production cycles to meet with total demand. In this case partial production and partial supply will increase fixed costs associated with production and supply. This is the situation in some cases; the vendor might have agreed to some other buyers to send the goods on or before some pre-fixed due date failing which the buyer might have to face with penalty clause. In the case, if production amount exceeds the total demand of all the buyers then the vendor has to face problems of holding cost of the excess amount till the surplus stock has been merged with in the next lot production. As the number of cycles increases the set up costs increases and if the production units exceed the demand vendor will have to confront the problems of holding cost and hence many other critical aspects like obsolescence, buyers unwillingness to purchase and sell the additional or surplus stock of the first production cycle etc. In such cases the aftermath is reflected in the total profit which eventually declines. In order to further analyze this problem we need to study and analyze the mathematical model of the profit function and safely plan the system in such a way that production crosses the number of units necessary to meet with break-even criterion. The number of units produced in one lot is a fact and the target production is the expected figure derived from the study of past records. Any variation in these two figures could be due to many factors like, short supply of raw material, failures or break down of some machines, non-availability of trained operators etc. 2. Objective of Study The inspirational force driving to work in leading area of P 3 system is the responsibility of the designer of the system to meet the cross questions regarding any component of the system and different types of enquires that may come across from the current buyers or the new buyers who may like to join the system at a given point of time. The system, without much contradiction and controversies should be acceptable. This is only possible only if the system operations at all stages are transparent, the system is steady in operations and progressive in nature, all that concerns the end users through the channels of buyers must be consistently reliable i.e. consumers reliability in the product remain steady over a long period of time, and the product must have unique features in comparison to those of the competitors. All these features relate to the majors components like purchase of the best quality raw material at a competitive rate on long term contract, steady production rate, and time-efficient operations of packing and forwarding. This poses the next question: What should be the average production rate and the sale price per unit? 3. Basic Features and Dominant Tenets of P 3 System The system begins its operations with a study and analysis of past records. The terms derived exploring the past records are as follows. in the sequence of operations, they are (a) Normalized Order Size (b) Safety Stock (c) Emergency Stock (d) Lightening Stock In order to have a look at the above cited points, we analyze the records and use important derivations in our proceedings. There are N buyers and say, we have their sales record for a period of reasonable months. We find average sales record (d i ) of each buyer and find their grand average as harmonic mean (D). At about an end, each buyer places an order (q ib ) for the coming period; but vendor, taking all past performances in consideration, the vendor normalizes demand (q i ). (a) Normalized demand Normalized demand = q i = q ib.(d / d j ) = Actual supply to each buyer (1) (b) Safety Stock: This is the additional amount of stock that the vendor dispatches to each buyer. This stock stands as safe guard against shortages. There are some additional advantages also, as mentioned in assumptions (A 2, c), like the buyer gets prefixed % discount on the purchase price of this amount and a good margin of time to make the payment against amount raised on this safety stock. Normally, it is expected that the payment should be made on or before the time the buyer places a request for the emergency stock. Safety stock amount and liberal terms of payment concept parallels the concepts of consignment inventory model there in the vendor taking higher risks, submits the lot to the buyer with a remote control on goods. Safety Stock for the i th buyer = K 1. q i (2) where, K 1 = ii=1 qq ii qq iiii / ii=1 qq iiii (3) K 1 is interpreted as absolute deviation per unit demand. 0 < K 1 < 1

66 Pradeep J. Jha et al.: Production Planning and Profit Design in P 3 System (c) Emergency Stock: Emergency stock is the amount of additional units that the registered buyer can receive directly from the distributer. The buyer can either directly contact the distributor or can request the vendor to direct the emergency stock lying at the distributor s end. Carrying charge for the stock is, as agreed upon, on the perspective buyer picking up his pre-determined amount of Emergency stock. The vendor, depending upon availability dispatches the emergency stock for each one of his registered buyers. Emergency stock is in proportion to the normalized demand of each buyer. The time since the first buyer lifts his emergency stock; the vendor remains alert and forwards the lightening stock; as mentioned in assumptions (A 2, a and b). On the other end, the vendor starts purchase of necessary amount of raw material and new set-up process for the production of the next batch begins. Some minor changes in the shape, size, color combinations of the production units, if at all are there, are attended at this stage. Emergency index is constant emergency stock is proportional to normalized demand. defined Emergency index K 2 ; K 2 = NN ii=1 dd ii DD NN ii=1 dd ii (4) While the emergency stock for the i th buyer = K 2 q i (5) (i = 1 to N- buyers) (d) Lightening Stock: On the time, just about the completion of the emergency stock, the vendor being online and well aware, dispatches the lightning stock to each buyer. Lightening stock is in the proportion of the normalized demand. The time slot between the buyer lifting the emergency stock and before the lightening stock is exhausted is very important and minor suggestions pertaining to minor changes in production are being attended during this period. Some suggestions in specifications, like small variation in size, shape, color combinations, addition in printed instruction, and changes in packing formats, if at all are there, are also taken into consideration. Lightening Stock Index = KK 1 + KK 2 = KK 2 3 (6) Lightening Stock = K 3. q i, where q i is the normalized demand of the i th buyer (7) It is known that, D being the average demand of all the buyers and calculated over a reasonably long period of time, the lighten stock will last approximately over NN ii=1 KK 3 qq ii Period of time (8) In this case the vendor knows the range of the time Lightening stock, on the basis of the average demand, will last for a period, for the i th buyer = KK 3qq ii DD for i = 1 to N This time period is very important and crucial for the vendor, (as mentioned in assumption (A 2, a, b, and e) as his total attention will be on different areas like (1) Probable purchase of raw material (stock taking end) (2) Preparing the set up for the next lot (3) Analysis of the market summary (4) Determining the production lot size (5) Calculation of different types of stock required to be forwarded to different buyer From time to time and many more issue of financial aspects. 4. Construction of Linear Model 4.1. Facts We construct an algebraic model in this situation. First, we identify the given set-up and then work mathematics on it. Buyers: B 1 B 2 B 3 B 4 B N Average Sales: d 1 d 2 d 3 d 4.d N Normalized Supply Quantity : q 1 q 2 q 3 q 4.q N Safety Stock: K 1 q 1 K 1 q 2 K 1 q 3 K 1 q 4..K 1 q N Emergency Stock: K 2 q 1 K 2 q 2 K 2 q 3 K 2 q 4..K 2 q N Lightening Stock: K 3 q 1 K 3 q 2 K 3 q 3 K 3 q 4..K 3 q N Total Supply in one Cycle: K 4 q 1 K 4 q 2 K 4 q 3 K 4 q 4..K 4 q N [Where K 4 = 1+ K 1 + K 2 + K 3 K 1 = ii=1 qq ii qq iiii / ii=1 qq iiii = Safety Index K 2 = ii=1 dd ii DD ii=1 dd ii = Emergency Index K 3 = (K 1 + K 2 ) / 2 = Lightening Index

Management 2014, 4(3): 64-70 67 and D = NN ii=1 ( 1/dd ii ) = Harmonic Mean [Normalized Supply Quantity = q ib (D/ d j ) Where q ib is the intended order size of the i th buyer.] As a special case, each unit produced goes to the packing department. in this department, as planned, the packer along with the produced unit, puts three more items ( like small cutter, cleansing pad, an injection unit etc.) which makes one complete unit. These additional items are necessarily completely safe and packed under possibly safe environment and conditions as the specifications are given or necessarily required. As it should be, the unit sale price includes profit on the production unit and a reasonably marginal profit on the additional items. If we consider S as the sale price of one complete unit then we division this in two parts; S = S 1 + S 2 Let us, assume that, S 2 = sale price of additional items included equals some fraction of the main unit. As it can be derived from the above table, the total supply to all the N buyers is, 4.2. Identification of Costs Thus, S = S 1 + ααss 1 where 0 < αα < 1 (9) K 4. ii=1 qq ii = K 4.Q where K 4 = 1+ K 1 + K 2 + K 3 As a result, total revenue generated on sale to all N buyers = T = S K 4.Q (10) Now, we identify different cost components. Let us assume that one unit needs R 1 units of raw material and it costs C 1 Rs./unit. For the total units produced the raw material costs =R 1 C 1 K 4.Q (11) Production Cost = set-up cost + variable cost = F 1 + αα 1. K 4.Q (12) Forwarding Cost being assumed variable and dependent on number of items dispatched, we have total forwarding cost = F 2 + αα 2 (K 4.Q) (13) As a basic format, each buyer gets a pre-fixed discount on the cost price of number of units in safety stock. This discount is say.rr 2 % This comes to an amount discounted = r 2 S.K 1. Q (14) The vendor bears holding charge of the items, Lightening Stock, produced and stored. This can be some, pre-known percentage of the sale price of the units in the lightening stock. We, note that this is applicable on the period beginning from the first dispatch till the time the lightening stock is forwarded. The carrying cost per unit per unit time is a fraction of the sale price S. Total carrying cost = h 1.S (K 3 Q) [(K 4 K 3 )Q / (N.D) ] (15) [The terms in the last bracket shows the time till all stock except the lightening stock is consumed. This helps calculate the holding cost of lightening stock.] 4.3. Profit Function We denote the profit function as P( S);and from the above data we write P(S) = S K 4.Q { R 1 C 1 K 4.Q + F 1 + αα 1. K 4.Q + 3F 2 + αα 2 (K 4.Q) + r 2 S.K 1.Q + h 1.S (K 3 Q) [(K 4 K 3 )Q / (N.D)} (16) P(S) = [ (K 4 r 2.K 1 )Q -- h 1 KK 3 QQ 2 (KK 4 KK 3 ) ] (S) [ (α 1 + α 2 + R 1 C 1 ) K 4.Q + F 1 + 3F 2 ] (17) This is a linear function in S. [When S = 0, P(S) = [ (α 1 + α 2 + R 1 C 1 ) K 4.Q + F 1 + 3F 2 ] When S = [ (α 1 + α 2 + R 1 C 1 ) K 4.Q + F 1 + 3F 2 ] / [ (K 4 r 2.K 1 )Q -- h 1 KK 3 QQ 2 (KK 4 KK 3 ) ] it gives break even value of S. Any value of S greater than the one given by above relation will give profit. i.e. For P(S) > 0; S > [(α 1 + α 2 + R 1 C 1 ) K4.Q + F 1 +F 2 ] / [ (K 4 r 2.K 1 )Q -- h 1 KK 3 QQ 2 (KK 4 KK 3 ) ] (18) This is a guiding result and helps in deriving the sale price per unit in different cases. Terms like F 1, F 2, R 1, and C 1 are constants on a given time but one or more of them are likely to change in depending upon different types of market situation from time to time. In all cases, in this system, the result obtained above result preserves consistency.

68 Pradeep J. Jha et al.: Production Planning and Profit Design in P 3 System 5. Conclusions The above discussion and derivations in the view of P 3 system closely inherit the fundamental notion of the widely accepted VMI system. The designed P 3 system also incorporates the salient features of other known inventory management and production system in accordance with CPFR. The derived formula helps determine the critical value of sale price with respect to the various values of purchase price of the raw material and other values treated constant on a given time. It is then a matter of discretion on vendor s end to fix-up the sale price and make further deals. Linear model of profit reflects many parameters but we have focused on the unit price of the raw material and derived break even sales price. As the cases demand, one can change other factors like, values of constants α 1, α 2, set-up costs F 1, and F 2 and derive sales price accordingly. This model is open-ended and is capable of including changes and is still sustainable in different cases as we have already put it on sharp tests. 6. Illustration We have five buyers on the list and a sales record of last six months. All these buyers have come out with their expected demand. On the basis of this, we generate the entire operation procedure in P 3 system. Sales Record of Last Six Months Buyer 1 Buyer 2 Buyer 3 Buyer 4 Buyer 5 Total H.M = D Month 1 210 180 270 200 180 Month 2 220 190 260 200 170 Month 3 190 200 270 190 200 Month 4 250 200 280 200 200 Month 5 200 220 300 210 210 Month 6 230 190 250 210 200 Sum Dem 1300 1180 1630 1210 1160 6480 Ind. Ave. 217 197 272 202 193 1080 213 Deviation 4.01 15.99 59.01 10.99 19.32 109.32 Buyer s demand 210 210 250 200 180 1050 Normalized. Demand 206 227 196 211 198 1038 Abs Diff 3.89 17.07 54.30 10.90 17.99 104.15 K 1 = safety Index 0.0992 K 2 = Emergency Index 0.1012 K 3 = Lightening Index 0.1002 K 1 q i ( Safety Stock) 20.44 22.52 19.41 20.92 19.64 103 K 2 q i (Emergency Stock) 20.86 22.99 19.81 21.35 20.04 105 K 3.q i (Lightening Stock) 20.65 22.75 19.61 21.13 19.84 104 Total Production in a given month = 1350 Supplied 268 295 255 274 258 1350 Let the set-up (Production) Cost = 300 and variable cost α 1 = 0.1 Let the fix forwarding Cost be Rs. 100 and variable Cost α 2 = 0.15 Let R 1 = 1.2 units of raw material to make one unit final unit. Let C 1 = 1 cost per unit of raw material Let h 1 = 0.05 carrying cost per unit per unit time Let r 2 = 0.1 discount on sale price given on safety stock K 4 = 1 + K 1 + K 2 +K 3 = 1.3006 Now, we find the minimum value of S which gives break-even for the total amount Q = ii=1 qq ii

Management 2014, 4(3): 64-70 69 S > [(α 1 + α 2 + R 1 C 1 ) K 4.Q + F 1 + 3F 2 ] / [ (K 4 r 2.K 1 )Q -- h 1 KK 3 QQ 2 (KK 4 KK 3 ) ] Substituting the values we derive the lower break even limit on S. S > 3078.8242 / 1753.8788 = 1.755 Any value of S greater than 1.755 per unit will generate profit. This value of S is logically convincing as the raw material cost to produce one unit is 1.2 X 1 =R 1 X C 1 = 1.2 along with other fixed and variables costs as mentioned above. We may consider different cases and determine the values of S. We consider variation in cost and evaluate the lower limit on S. [This is logical as the price of raw material keeps on changing as it is affected by many external factors; what we are concerned with is the purchase of quality material and that too with competitive rate.] 7. Graph and Behavioural Pattern of S against C We extract data from the work-sheet; Different values of C are probable assumptions and dependent values of S are the mathematical output and draw a graph of S against variations in the value of C. COST B- Even 'S' 0.5 1.15 0.6 1.27 0.7 1.39 0.8 1.52 0.9 1.64 1 1.76 1.1 1.88 1.2 2.00 1.3 2.12 1.4 2.25 1.5 2.37 Unit Cost VS.Break-Even 2.50 2.00 Break-Even 'S' 1.50 1.00 0.50 B-Even'S' 0.00 0 0.5 1 1.5 2 Unit Cost- Raw Material

70 Pradeep J. Jha et al.: Production Planning and Profit Design in P 3 System 8. Conclusions It is clearly seen that the graph is a linear one. Break-even trend of S, as seen from the graph, Is sympathetic to the trend in the value of unit price of raw material. In the same we, we can change fixed costs of production, fixed cost of dispatch, and units of raw material required to produce one unit etc. In each case we can calculate the value of S. An important feature of this model is that we can combine more than one variables described above and can determine behaviour of S. This study helps one take rational decision. REFERENCES [1] Goyal, S,K. (1995), A one-vendor multi-buyers integrated inventory model-. a comment. European journal of operation Research 82, 209-210. [2] Jha, Pradeep, J (june-2013) Buyer- Vendor integrated system- the technique of EOQ dependent shipment size to achieve steady level and cost minimization ISOR journal of mathematics (iosr-jm) 46-57. [3] Rajmanohar, T,P, (2008) Inventory Optimization Introduction The ICFAI University Press. [4] Jha, Pradeep, J: Shah, Viranchi, A (nov-dec 2013) Production planning and stocking of life saving medicines at Vendors end and Buyers end ISOR journal of business management ( iosr-jbm) 54-63. [5] Jha, Pradeep, J: Shah, Viranchi, A (May- june 2014) Maintaining a seamless supply chain of Essential Medicines [A combination of various concepts converging in novel P 3 system, European Journal of Research and Reflection in Management Science.