INVENTORY. Stock of good items maintained to full fill present & future need of an organization

Transcription

1 INVENTORY SYSTEMS

2 INVENTORY Stock of good items maintained to full fill present & future need of an organization Manufacturer:- Raw material stock, spare parts, semi furnished goods Hospital :- Stock of drugs Book-Store :- Books & Stationary

3 Why inventory is maintained? It provides service to the customer immediately or at a short period Due to absence of stock the company may have to pay high price because of piecewise purchasing Maintaining of inventory may earn price discount because of bulk purchasing Inventory acts as a buffer stock when raw material are received late so many sale orders are likely to be rejected

4 Various costs involved Procurement Cost:- Associated costs are a) Set up cost:- K per cycle b) Purchase or production cost:- Co per unit Holding cost :- C per unit of item per unit of time - Storage Cost - Handling Cost - Record keeping & Administrative cost - Taxes & Insurance costs Shortages or penalty Cost ( Cost of back orders or shortage ) C Cost per unit of back order per unit of time

5 Inventory Decision DECISION How much amount of Items to be ordered? When to order??

6 Terms & Notation a) K = Set up Cost per cycle b) C 0 = Manufacturing or purchase cost per unit c) C = Holding cost per unit per unit of time d) D = Demand rate in units per unit of time e) A = No. of units of item produced (or delivered) per unit of time. f) Q = No. of unit purchase (or produce) per cycle or per order (or per run) g) Ordering Cycle h) t = length of cycle I ) Lead time: The time between the placement of an order and its receipt is called lead time.

7 Deterministic Inventory Model Model :- Single item, Static demand, Production model with no back ordering allowed (Finite and constant delivery rate during the production run) No shortages are allowed (A>D) Lead time is zero S = Maximum level of inventory s = Minimum level of inventory Stock Policy: (s, S) policy.

8 Model-. t Time period during which items are arrived (or produced) until complete order is received. During this time period inventory build up with a rate A-D, from minimum level s to maximum level S

9 Model- t Time period during which arrival stops & demand continues to deplete from S to s t 0 Length of order cycle t 0 =t +t

10 Inventory level S s t t time to

11 Q= At = Dt S=s+(A-D)t =s+(a-d)q/a-----(from) And S=s+(-D/A)Q S-Dt =s

12 Model-. Inventory holding cost = t C I ( t ) d t C I ( t ) d t 0 t 0 t where I(t) is inventory level time t. (3) For [0,t ], I(t) =s + (A-D) t and for [t,t ], I(t)= S-D(t-t ) Substitute value in (3),Inventory holding cost per cycle is =[C s+0.5 C Q(-D/A)] t 0 Thus total cost of an order cycle is function of s and Q i.e., TC(Q, s)= K+C 0 Q+C st C t 0 Q (-D/A)

13 Model-.. Objective:- Minimize TC(Q,s)per unit time = TCU (Q,s) =TC(Q,s) per unit of time = TC(Q,s)/t 0 = k/t 0 + C 0 Q/t 0 + sc + 0.5CQ(-D/A) = C 0 D+KD/Q+0.5 C Q(-D/A)+sC (4) (Using Q=Dt 0 ) From(4) it is obvious that for a given Q, TCU(Q,s) is minimum if s=0,putting s=0 in (4) we get TCU(Q) = C 0 D+KD/Q+O.5 C Q(-D/A) by solving d/dq TCU(Q)= 0,we get value of Q which minimizes TCU(Q)

14 Model-.. d/dq TCU(Q)=0 kd Q*...(5) D C( ) A It can be seen that d /dq TCU(Q)>0 at Q=Q* Q* is called Economic order quantity or Economic lot size (EOQ) t 0 =optimum order cycle=q/d Minimum value of TCU (Q,S)= TCU (Q*,S*) t * o Q * D k D DC ( ) A S*, the optimum value of S is given by S* = Q* (-D/A) (FROM ) D CoD kd( ) C A ( kd( D / A)) / C

15 Optimal Parameters for Model-: (Summary of Model-) *. D k D A S C * o k t D DC A * kd EOQ Q D C A ( *) o D TCU Q C D kd C A * s 0.

16 Model II:- Single item, static Demand, purchase model with no back order, Infinite Delivery rate with no shortage allowed. (when A is infinity in model-) Inventory level Q=S t 0 t 0 t 0 Time(t)

17 Optimal Parameters for Model-. EOQ Q* kd / C t k / DC * o TCU ( Q*) C D C DK o S* Q* kd / C * s 0.

18 Problem The demand of product is 500 units per week and the delivery rate is 000 units per week. If the purchase price is Rs per unit, the ordering cost is Rs 00 per order, the holding cost is Rs.00 per unit per week. Calculate Q*, TCU(Q*).

19 Problem A contractor has to supply 0,000 bearings per day to an automobile manufacturer.he finds that, when he starts a production run, he can produce 5,000 bearings per day.the cost of holding a bearing in stock for one year is 0 paisa,and set up cost of a production run is Rs 80.How frequently should production run be made?

20 Problem 3 You have to supply your customer 00 units of certain product every Monday(and only then).you obtain the product from a local supplier at Rs 60 per unit.the costs of ordering and transportation from the supplier are Rs.50per order.the cost of carrying inventory is estimated at 5 %per year of the cost of the product carried. ()Describe graphically the inventory system. ()Find the lot size which will minimize the cost of the system. (3)Determine the optimal cost.

21 Problem4 An aircraft company uses rivets at an approximate customer rate of,500 kg.per year. Each unit costs Rs 30 per Kg.and the company personnel estimate that it costs Rs.30 to place an order, and that carrying cost of inventory is 0 % per year. How frequently should orders for rivets to be placed? Also determine the optimum size of each order.

22 MODEL-3: Shortage or Back ordering is allowed. Single item, static demand, production model with back ordering Constant delivery rate A Constant usage rate D Maximum shortage( Back order) allowed is B. C is the back ordering cost per unit per unit of time.

23 Inventory Level S s t t t3 t4 to Time (t) B B B

24 Model-3. Shortages are being filled at the rate of (A-D) units per unit of time during time period t Inventory is building at a constant rate of (A-D) unit per unit of time during time period t No arrivals/delivery during time period t 3. Inventory is decreasing at a rate D per unit of time Shortages are building at a rate D units per unit of time during time period t 4

25 Observations: Model-3.. During t B ( A D) t B t () A D During t S ( A D) t () S t. A D During t 3, S Dt (3) 3 S t3. D

26 Observations: Model-3.. During t 4 B Dt4 B t (4) D Also, Q Dt A( t t ) (5) 0 From equation (),(), and (5), Q D S ( A D) B Q B. A A (6) The period of one cycle is t 0 =t +t +t 3 +t 4

27 H olding C ost for one cycle C t t 3 0 I ( t ) dt t t t 3 C ( A D ) td t ( S D ( t t )) d t 0 t t t t 3 S C tdt ( D t3 D ( t t )) dt t 0 t t t t 3 S C tdt D ( t t3 t ) dt t 0 t

28 Holding Cost for one cycle t t t 3 S S C tdt ( t t3 t) dt t 0 t t 3 S S CS ( t t3 ) CS. A D D C D Q B S A C D D D D A A.

29 Production Cost for one cycle K C Q. L e t B ( t ) b e th e le v e l o f b a c k - o r d e r a t tim e t. B a c k o r d e r o r s h o r t a g e c o s t 4 C B ( t ) d t C B ( t ) d t 0 0 t 4 C ( A D ) t d t C D t d t 0 0 t 4 B B C t d t C t d t t 0 t C B ( t t 4 t t t B t f o r o n e ). D C D A 0 c y c le

30 The Total Cost for one cycle TC( Q, B) D Q B A CB K C0Q C D D D D A A. The Total Cost per unit time TCU ( Q, B), is given by TCU ( Q, B) TC( Q, B) / t. 0

31 Now, TCU ( Q, B) D Q B KD A CB K C 0 D C. Q D D Q Q A A Aim: Find Q and B, such that TCU ( Q, B) is minimum or min TCU ( Q, B). ( Q, B)

32 (, ) 0 ( ) / 0.. Since 0 (, ) is m inim um w hen. TCU Q B B D B C C Q C A D Q A B C C C TC U TCU Q B B D Q A B C C C

33 Substituting the value of B in TCU ( Q, B), we get KD D TCU ( Q) C0D CC Q /( C C). Q A Now, TCU ( Q) will be minimum if Q KD( C C). C C D A

34 Optim al Parameters for M odel - 3 The Economic order quantity (OOQ) KD ( C C ) D CC A * Q. T he Economic back order quantity (EBO) B * D D A A ( C C ) C ( C C ) * Q C KDC.

35 * * 0 * * * * * The Optimum order cycle (OOC) ( ) Q. The optimal values of s and S of the (s, S) policy is given by and K C C t D D DC C A KD D s B S Q B A Optimal Parameters for Model ( ) D C A C C C

36 Thus the Optimum(minimum) total cost per unit of time is D KDCC A TCU ( Q *, B * ) C D. 0 C C

37 Problem6 The demand of an item is 5 units per month.fixed cost is Rs 5 each time a production run is made.the production cost is Rs =00 per item & the inventory carrying cost is Rs 0.30 per item per month.if the shortage cost is Rs.50 per item per month, determine how often to make a production run and of what size it should be.

38 PROBLEM 7 The demand for item in a company is 8000 units per year and the company can produce the 3000 per month.the cost of one set up is Rs 500.If the holding cost of one unit per month is 5 paisa, the shortage cost of unit is Rs 0=00 per unit per month. Determine the optimum manufacturing quantity & no of shortages.also determine the manufacturing time & time between setups.

39 Model-4: Single item static demand inventory model with infinite delivery rate (Just put A is infinity in model-3)

40 I n v e n t o r y L e v e l I S S t t0 t Q Time B B B

41 Optimal Parameters for Model - 4 The Economic order quantity (OOQ) KD( C C ) CC * Q. The Economic back order quantity (EBO) B Q C KDC. ( C C ) C ( C C ) * *

42 Optimal Parameters for Model - 4. The Optimum order cycle (OOC) t Q K ( C C ). * * 0 D DCC The optimal values of s and S of the (s, S) policy is given by * * * * * KDC s B and S Q B. C ( C C )

43 Thus the Optimum(minimum) total cost per unit of time is KDC C TCU ( Q *, B * ) C D. 0 C C

44 Remarks : Model-= Model-3 with B 0 and C. Model-= Model-3 with B 0 and C, A. Model-4= Model-3 with A.