IP modeling techniques II

1
IP modeling techniques II

• In this handout,
• Modeling techniques
• Either-Or Constraints
• Big M method
• Balance constraints
• Fixed Charges
• Applications
• Multi-period production planning
• Inventory management

2
Modeling technique Either-Or Constraints

• In some situations,
• a choice can be made between two constraints,
• so that only one (either one) must hold whereas
  
• the other one can hold but is not required to do
  so.
• E.g., recall the capacity constraints for the
  furniture manufacturer example
• pine 5xt 1xc 9xd ? 1500 (1)
• oak 2xt 3xc 4xd ? 1000 (2)
• Suppose the furniture can be made from either
  pine or oak but we dont need both.
• How to achieve that in the model?

3
Either-Or Constraints

• Introduce new binary variables. For i1,2
• Only one of (1) and (2) must hold.
• Thus, add a constraint y1y2 1
• We also need y11 implies 5xt1xc9xd ?1500
• y21 implies 2xt3xc4xd ?1000
• How to express these implications
• by linear constraints?

4
Either-Or Constraints

• New idea use the big number method.
• Select a huge positive number M .
• Note that 5xt1xc9xd ?1500M holds for any
  reasonable choices of xt, xc, xd . It is
  equivalent of not putting any restriction on xt,
  xc, xd at all.
• Consider constraint 5xt1xc9xd ?1500M(1-y1)
  (3)
• If y11 then (3) is equivalent to (1)
• If y10 then (3) doesnt impose any restriction
  on xt, xc, xd
• Thus, the set of constraints
• 5xt1xc9xd ?1500M(1-y1)
• 2xt3xc4xd ?1000M(1-y2)
• y1y2 1
• provides that only one of
• 5xt1xc9xd ?1500 and 2xt3xc4xd ?1000 must
  hold.

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k out of p constraints must hold

• Suppose the model includes a set of p constraints
  
• f1(x1, x2, , xn) ? d1
• f2(x1, x2, , xn) ? d2
• ....
• fp(x1, x2, , xn) ? dp
• such that only some k of these constraints must
  hold.
• Generalizing the big M method of the previous
  slide,
• that condition is achieved by the following set
  of constraints
• f1(x1, x2, , xn) ? d1My1
• f2(x1, x2, , xn) ? d2My2
• .
• fp(x1, x2, , xn) ? dpMyp
• y1y2yp p k
• y1, y2,, yp binary

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IP modeling Multi-period production planning

• A manufacturer wishes to schedule production for
  K periods in advance to meet known monthly
  demands for a given product.
• Demand for period i is Di .
• In period i, at most Ci items can be produced
• at cost pi per item.
• The demand of the current period can be satisfied
  by the items produced in earlier production
  periods (aka inventory).
• The cost of holding an item in inventory
• from period i to period i1 is hi .
• No inventory at the beginning of the first
  period. At most Hi items are allowed in inventory
  at the beginning of period i.
• Goal Formulate an IP which will minimize the
  total cost
• while satisfying the demands.

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IP modeling Multi-period production planning

• What variables should we have?
• Define xi number of items produced in period i
  , for i1,...,k .
• wi number of items in inventory at the
  beginning of period i , for i1,...,k1 .
• (xi and wi are nonnegative integers)
• What is the objective function?
• Minimize the total production and inventory
  cost
• Some obvious constraints.
• Production limit xi ? Ci , for i1,...,k
• Inventory limit wi ? Hi , for i1,...,k1
• No inventory before period 1 w1 0

8
Multi-period production planning Balance
Constraints

• Also need constraints satisfying the demands,
• and relating xi and wi .
• Consider the following diagram for period i
• The corresponding constraint for period i
  (i1,,k)
• wi xi Di wi1
• This is known as balance constraint,
• and is often used in multi-period problems.
• Note that the balance constraints provide that
• wi xi Di (since wi1 0), and thus the
  demands are satisfied.

wi
Di
Period i
INPUT
OUTPUT
xi
wi1
9
Multi-period production planning Complete IP
model

• s.t. xi ? Ci , for i1,...,k
• wi ? Hi , for i1,...,k1
• wi xi Di wi1 , for i1,,k
• w1 0
• xi 0 integer for i1,,k
• wi 0 integer for i1,...,k1

10
Multi-period production planning Fixed Setup
Cost

• Note that the demand of the current period can be
  totally satisfied by the inventory carried from
  the previous period.
• Suppose there is a setup cost si for period i
• if we decide to have any production for that
  period
• (and there is no setup cost if there is no
  production).
• How to take the setup costs into account?
• We need a new entry in the objective function
• if xigt0 then si for each period i .
• But this is not a linear function
• (because no conditions are allowed on
  variables).

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Multi-period production planning with setup cost

• To overcome the problem, introduce new variables.
  For i1,...,k,
• Then the setup cost is siyi for period i .
• But we also need new constraints relating xi and
  yi .
• Idea Use the big M method.
• For each i1,...,k , add a constraint
• xi ? Myi .
• Why does it work?
• If yi0 then xi must be 0
• If yi1 then there is no restriction on xi .
• Note that we can take MCi for period i. Thus, we
  dont need new constraints. Simply, replace xi ?
  Ci with xi ? Ciyi .
• Question Can we have yi1 but xi 0 for period i
  ?

12
Multi-period production planning with setup cost

• Summarizing, the modified IP model is
• s.t. xi ? Ciyi , for i1,...,k
• wi ? Hi , for i1,...,k1
• wi xi Di wi1 , for i1,,k
• w1 0
• xi 0 integer for i1,,k
• wi 0 integer for i1,...,k1
• yi binary for i1,,k