An Order Quantity Decision System for the Case of Approximately Level Demand

Transcription

1 Capter 5 An Order Quantity Decision System for te Case of Approximately Level Demand Demand Properties Inventory problems exist only because tere are demands; oterwise, we ave no inventory problems. Inventory systems in wic te demand size is known will be referred to as deterministic systems. Demand rate is te demand size per unit time. 5. Assumptions Leading to te Basic EOQ () Te demand rate is constant and deterministic. () Te order quantity need not be an integral number of units. (3) Te unit variable cost is independent of te replenisment quantity. (4) Te cost factors do not cange appreciably wit time (i.e. no inflation). (5) Te item is treated entirely independently of oter items. (6) Te replenisment lead time is of zero duration. (7) No sortages are allowed. (8) Te entire order quantity is delivered at te same time.

2 5. Derivation of te EOQ Data D = demand rate in units per year m = production rate in units per year (For te models tat ave finite production rates) A = fixed cost of a replenisment order v = unit variable cost of production (or purcase) = inventory carrying cost per unit per year, usually expressed as = rv, were r is te annual inventory carrying cost rate π = sortage cost per unit sort per year (For te models tat allow backorders) Decision Variable Q = replenisment order quantity b = maximum backorder level permitted (For te models tat allow backorders) T = cycle lengt, te lengt of time between placement of replenisment orders T RC(Q) = total relevant costs per unit time Te average inventory carrying cost per cycle is te area under te inventory triangle. QT = Q D Te average cost per cycle is te sum of procurement and inventory carrying cost. A + Q D To obtain te average annual cost, T RC(Q), we multiply te cost per cycle by te number of cycles per year, D/Q. Doing tis and writing = vr, we get T RC(Q) = AD Q + rvq Te optimum value of Q can be found by solving T RC(Q) Q = AD Q + vr = 0

3 since T RC(Q) is convex function in Q. (note) A differentiable function f(x) is convex in x, if te second derivative is nonnegative. For te above model, Consequently, T RC Q EOQ = and te minimum average annual cost will be = Q 3 0 vr T RC(EOQ) = Advr 5.3 Sensitivity Analysis Penalty for using a wrong EOQ Let Q = ( + p)eoq. Tat is, 00p is te percentage deviation of Q from te EOQ. (proof) P CP = T RC(Q ) T RC(EOQ) T RC(EOQ) P CP = 50 ( p + p ) 00 (cf.) penalty for wrong estimation of cost parameters 5.5 Quantity Discounts We assume all units discount wic is te most common type of discount structure. { v0 if 0 Q < Q v = b v 0 ( d) if Q b Q T RC(Q) = Qv 0r + AD Q + Dv 0, 0 Q < Q b 3

4 T RC(Q) = Qv 0( d)r + AD Q + Dv 0( d), Q Q b tradeoff between extra carrying cost vs. a reduction in te acquisition costs Figure 5.6 TRC under All Units Discount Algoritm (Step ) Compute EOQ(discount) = v 0 ( d)r (Step ) If EOQ(d) Q b, ten EOQ(d) is optimal (case (c)). If EOQ(d) < Q b, go to Step 3. (Step 3) Compute TRC(EOQ) and TRC(Q b ). If TRC(EOQ) TRC(Q b ), EOQ is optimal (case (b)). If TRC(EOQ) > TRC(Q b ), Q b is optimal (case (a)). 5.6 Accounting for Inflation r = continuous discount rate i = inflation rate P V (Q) = (A + Qv) + (A + Qv)e iq D e rq iq D + (A + Qv)e D e rq D ( ) + = (A + Qv) + e (r i)q D + e (r i)q D + Te optimal Q satisfies = (A + Qv) e (r i)q D e (r i)q D = + ( ) ( ) A r i v + Q D (proof) 4

5 Approximating e x by + x + x gives Q = v(r i) = EOQ i r 5.7 Limits on Order Sizes 5.7. Maximum Time Supply or Capacity Restriction () self life of te commodity If T EOQ = EOQ D = A Dvr > SL, Q SL = D(SL). () Even witout a self life limitation, an EOQ tat represents a very long time supply may be unrealistic for oter reasons. (3) Tere may be a storage capacity limitation on te replenisment Minimum Order Quantity Discrete Units Te best integer value of Q as to be one of te two integers surrounding te real value of Q. 5.8 Finite Replenisment Rate Here tere is a finite production rate m rater tan infinite replenisment rate. Note tat in te above figure Q T m + T D = D, I max = (m D)T m, Imax = DT D. Consequently, I max = ( D m )Q 5

6 Te average inventory carrying cost per cycle is te area under te inventory triangle. T I max = Q D ( D m )Q = Q D ( D m ) Te average cost per cycle is te sum of procurement and inventory carrying cost. A + Q D ( D m ) To obtain te average annual cost, T RC(Q), we multiply te cost per cycle by te number of cycles per year, D/Q. Doing tis and writing = vr, we get T RC(Q) = AD Q + vrq ( D m ) Te optimum value of Q can be found by solving T RC Q = AD Q + vr ( D m ) = 0 since T RC(Q) is convex function in Q. Consequently, F REOQ = vr( D ) m and te minimum average annual cost will be T RC(F REOQ) = vr( D m ) Note tat if m, F REOQ =. (Reduces to EOQ) vr 5.9 Incorporation of Oter Factors 5.9. Nonzero Constant Lead Time tat is known wit Certainty Wen te inventory level its DL, an order is placed and it arrives exactly L time units later just as te inventory its zero. 6

7 5.9. Different Type of Carrying Carge Suppose tat tere is an additional carge (in additin to te usual inventory carrying carge r) of w dollars per unit time per cubic foot of space allocated to an item Derivation! Multiple Setup Costs: Freigt Discounts A + Qv if 0 < Q Q 0 Setup Cost = A + Qv if Q 0 < Q Q 0 3A + Qv if Q 0 < Q 3Q 0 Aucamp (98, EJOR) as sown tat te best solution is eiter te standard EOQ or one of te two surrounding integer multiples of Q Joint Replenisment Problem Suppose a company carry more tan one item. If te purcasing manager purcase tose items from a same vendor, it may be a good idea to order tose items togeter so tat e/se can save ordering cost. For te simplicity, we study two items case. Data D = demand rate for item in units per year D = demand rate for item in units per year A = fixed cost of a replenisment order v = unit cost of purcasing item v = unit cost of purcasing item r = annual inventory carrying cost rate Decision Variables Q = order quantity for item Q = order quantity for item T = common cycle lengt, te lengt of time between placement of replenisment orders 7

8 Since T is same for bot items, we get T = Q D = Q D. Te average inventory carrying cost for item per cycle is te area under te inventory triangle for item. rv Q T = rv Q Te average inventory carrying cost for item per cycle is te area under te inventory triangle for item. rv Q T = rv Q Te average cost per cycle is te sum of procurement and inventory carrying cost. D D A + rv Q + D rv To obtain te average annual cost, T RC(Q, Q ), we multiply te cost per cycle by te number of cycles per year, D /Q ( D /Q ). Doing tis, we get Q D T RC(Q, Q ) = AD + rv Q + rv Q Q By substituting Q = D D Q, we get T RC(Q ) = AD + rv Q + rv D Q Q D Te optimum value of Q can be found by solving since T RC(Q ) is convex function in Q. T RC = AD + r(v D + c D ) = 0 Q Q D For te above model, T RC Q = Q 3 0 Consequently, Q = r(v D + v D ), Q = r(v D + v D ) 8

9 And te optimal common replenisment interval is T A = r(v D + v D ) Different Order Arrivals In te classical EOQ (Economic Order Quantity) Model, assume tat wen we order Q units, we receive our order in two parts. Te first part arrives immediately and contains αq(0 < α ) and te second part arrives T units of time after te order and contains te rest of our order, i.e., ( α)q. We assume no sortages are allowed. See te following figure for understanding. Note tat ere T is not a variable but a parameter (given data). Since we assume tat tere are no sortages, αq DT ( ) Since αq x T = D, we get x = αq DT Since ( α)q+x T f Consequently, = D, we get T f = Q DT D T c = T + T f = Q/D 9

10 Te average inventory carrying cost per cycle is te area under te inventory triangle plus te area of trapezoid. T f[x + ( α)q] + T [αq + x] = T (αq DT ) + DT ) (Q D Te average cost per cycle is te sum of procurement and inventory carrying cost. A + T (αq DT ) + DT ) (Q D To obtain te average annual cost, T RC(Q), we multiply te cost per cycle by te number of cycles per year, D/Q. Doing tis, we get T RC(Q) = AD Q + αdt D T Q + (Q DT ) Q Te optimum value of Q can be found by solving T RC Q = AD Q + = 0 since T RC(Q) is convex function in Q. Consequently, Q = Tere are possible cases. (i) If Q = satisfies ( ), i.e., DT α, ten Q = (ii) If Q = doesn t satisfy ( ), i.e., Q = DT α < DT α, ten A Special Opportunity to Procure 0