Inventory Control with TimeVarying Demand

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Inventory Control with Time-Varying Demand
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Lecture Topics

• Week 1 Introduction to Production Planning and
  Inventory Control
• Week 2 Inventory Control Deterministic Demand
• Week 3 Inventory Control Stochastic Demand
• Week 4 Inventory Control Stochastic Demand
• Week 5 Inventory Control Stochastic Demand
• Week 6 Inventory Control Time Varying Demand
• Week 7 Inventory Control Multiple Echelons

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Lecture Topics (Continued)

• Week 8 Production Planning and Scheduling
• Week 9 Production Planning and Scheduling
• Week 10 Managing Manufacturing Operations
• Week 11 Managing Manufacturing Operations
• Week 12 Managing Manufacturing Operations
• Week 13 Demand Forecasting
• Week 14 Demand Forecasting
• Week 15 Project Presentations

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Characteristics

• Demand varies from period to period
• The demand for each period is exactly known
• Costs may vary from period to period
• Capacity may vary from period to period

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Big versus Small Buckets

• Big time-bucket models
• Items produced/ordered in a period can be used to
  satisfy the demand for that period
• Small time-bucket models
• Production/supply leadtimes can take multiple
  periods

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The Lot Sizing Model
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Assumptions for the Basic Model

• Demand varies from period to period but is
  exactly known for each period.
• Demand in each period must be satisfied during
  the same period (backordering is not allowed).
• There are no limits on how much can be produced
  or ordered.
• Items produced/ordered in a period are available
  to satisfy demand during the same period (big
  bucket model).
• Setup/ordering, production/purchasing, and
  inventory holding costs can vary period to
  period.

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Objective

• Determine the optimal order quantity (lot size)
  in each period so that the demand in each period
  is met while the sum of ordering, purchasing, and
  inventory holding costs are minimized.

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Notation

• t a period (e.g., day, week, month) t 1,
  ,T, where T represents the planning horizon
• Dt demand in period t (number of units)
• ct unit purchasing/production cost
• At ordering/setup cost associated with placing
  an order (or initiating production) in period t
• ht cost of holding one unit of inventory from
  period t to period t 1
• Qt the size of the order (or lot size) in period
  t a decision variable

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Example
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The Lot for Lot Solution
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The Fixed Order Quantity Solution
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The Fixed Order Period Solution
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A Mixed Integer Linear Program (MILP) Formulation
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Solution Approach

• Solve as a standard MILP (using for example a
  branch and bound algorithm) several commercial
  MILP solver software tools are available
• Develop a customized solution that takes
  advantage of structural properties specific to
  the problem (e.g., the Wagner-Whitin algorithm)

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Property 1

• Under an optimal lot-sizing policy either the
  inventory carried to period t1 from the previous
  period will be zero or the production quantity in
  period t1 will be zero.

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The Basic Idea of the Wagner-Whitin Algorithm

• Using property 1, either Qt0 or QtDtDk for
  some k.
• If jk t last period of production (or
  ordering) in a k period problem, then we will
  produce (or order) exactly Dt Dt1 Dk in period
  jk.
• We can then consider periods 1, , jk-1 as if
  they are an independent jk-1 period problem.

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The Basic Idea of the Wagner-Whitin Algorithm
(Continued)

• Construct an algorithm where the decision is
  whether or not to order in a given period. If we
  order, then the order quantity should be just
  enough to cover demand until the next period in
  which we order.
• Solve a series of smaller sub-problems (a one
  period problem, a two period, ., N period
  problem), where the solution to each sub-problem
  is used in solving the next subproblem.

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Example
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Example

• Step 1 Obviously, just satisfy D1 (note we are
  neglecting production cost, since it is fixed).
• Step 2 Two choices, either j2 1 or j2 2.

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Example (Continued)

• Step3 Three choices, j3 1, 2, 3.

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Example (Continued)

• Step 4 Four choices, j4 1, 2, 3, 4.

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Property 2

• If jkt, then the last period in which
  ordering/production occurs in an optimal k1
  period policy must be in the set t, t1,k1.

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Property 2 (Continued)

• In the Example
• We order in period 4 for period 4 of a 4 period
  problem.
• We would never order in period 3 for period 5 in
  a 5 period problem.

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Example (Continued)

• Step 5 Only two choices, j5 4, 5.
• Step 6 Three choices, j6 4, 5, 6.
• And so on.

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Example Solution
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Example Solution (Continued)

• Optimal Policy
• Order in period 8 for 8, 9, 10 (40 20 30
  90 units)
• Order in period 4 for 4, 5, 6, 7 (50 50 10
  20 130 units)
• Order in period 1 for 1, 2, 3 (20 50 10 80
  units)
• Note we order in 7 for an 8 period problem, but
  this never comes into play in optimal solution.

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A Network Representation
The lot sizing problem can be represented as a
network, where each node t represents a period
and an arc from node t to node t represents the
fact that we order (or produce) in both periods
t and t but not in periods in between.
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The Network
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1
2
5
3
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Node 6 is a pseudo node representing the end of
the problem.
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Example Path
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Interpretation Order (or produce) in periods 1,
3, and 4 so that Q1 D1 D2 Q2 0 Q3 D3
Q4 D4 D5 and Q5 0.
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Arc Costs
The cost ct,t of reaching node t from t is the
cost of ordering in t but not in t1, t2,
, t-1
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Key Insight
Finding the minimum cost solution is equivalent
to finding the least costly path (shortest path)
in the network to go from node 1 to node T1,
where T is number of periods.
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A Dynamic Programming Algorithm to Find the Least
Costly Path
Step 1 t 1, zt 0 Step 2 t t1. If t gt
T1, stop. Otherwise go to step 3. Step 3 For
all t 1, 2, , t - 1, Step 4 Compute
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Step 5 Compute (that is, choose the period t
that minimizes ) Step 6 Go to
to step 2. The optimal cost is given by
The optimal set of periods in which
ordering/production takes place can be obtained
by backtracking from

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Example
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Example (Continued)
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Example (Continued)
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Example (Continued)
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Example (Continued)
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Example (Continued)
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Example (Continued)
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Example (Continued)
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Example (Continued)
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Example
t Dt At ct ht 1 10 40 2 1 2 2 40 2 1 3 12 40 2 1
4 4 40 2 1 5 14 40 2 1
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Example
z1 0 c1,2 40 20 60 z2 z1 c1,2
06060 p2 1 c1,3 40 242 66 c2,3 40
4 44 z3 min (z1 c1,3, z2 c2,3) min
(66, 104) 66 p3 1
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Example
c1,4 114, c2,4 80, c3,4 64 z4 min (z1
c1,4, z2 c2,4, z3 c3,4) min (114, 140, 130)
114 p4 1 c1,5 134, c2,5 96, c3,4 76,
c4,5 48, z5 min (134, 156, 142, 162)
134 p5 1 c1,6 218, c2,6 166, c3,6 132,
c4,6 86, c5,6 68 z6 min (218, 226, 198,
200, 202) 198 p6 3
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Heuristics

• Instead of solving the problem optimally, we
  could use a heuristic (a rule) that leads to
  reasonably good solutions but not necessarily
  optimal.
• The advantage of heuristics is ease of
  implementation and lower computational effort to
  reach a solution.

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Example Heuristics

• Choose a fixed order quantity and order in
  multiples of this order quantity. Order again
  when demand in a period cannot be met from
  available inventory.
• Choose a fixed order period P. Then, every P
  periods order all the demand for the next P
  periods.
• Use a greedy heuristic such as the Silver-Meal
  heuristic.

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The Silver-Meal Heuristic
Starting with a period t, order for the next k
periods if the resulting average cost per period
zt,tk is smaller than the average cost per
period if we ordered only for the next (k-1)
periods.
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The Silver-Meal Algorithm
Step 1 Set t 1 Step 2 Step 3 tt1 Step
4 If t gt T, go to step 7. Otherwise go to
step 5 Step 5
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Step 6 If zt,t ? zt,t-1 , set t t1 and go
to step 4. Otherwise go to step 7. Step 7 Set
Qt Dt Dt1 Dt-1 Step 8 Set
tt. Step 9 If t gt T, stop. Otherwise, go to
step 2.
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Example
t Dt At ct ht 1 10 40 2 1 2 2 40 2 1 3 12 40 2 1
4 4 40 2 1 5 14 40 2 1
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Example
z1,1 40 20 60 z1,2 (60 4 2)/2
66/233 lt 60 z1,3 114/3 38gt33 So the
heuristic sets Q1 12. Next, z3,3 64 z3,4
76/2 36 lt 64 z3,5 132/3 44gt36 So, Q3 16.
Next,
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z5,5 68. Since we have reached the end of the
planning horizon, the heuristic sets Q5 14.