An Inventory-Location Model:Formulation, Solution Algorithm and

Computational Results

Mark S. Daskin, Collete R. Coullard and Zuo-Jun Max Shen

presented by Zümbül Bulut

Content

• Problem Definition

• Assumptions

• Model

• Model Properties

• Solution Approach

• Results

• Further Research Areas

Example• Local blood bank supplied 30 hospitals,

• Platelets; the most expensive and most perishable of all blood products,

• Highly variable demand for platelets,

• Hospitals supplied by the blood bank collectively

- own the blood bank

- set the prices

• Little incentive to manage inventories sufficiently,

• Large hospitals order more than required

- need to destroy

• Other hospitals order on an emergency basis

- expensive transportation costs

Solution Approach• Selected hospitals would maintain an inventory of platelets

for use in neighbouring hospitals

take advantage of the risk-pooling effect

• Hospitals at which inventory would be maintained are determined by P-median model

• The solution approach did not account directly for:

- working inventory costs,

- safety stock costs,

- fixed cost of establishing the facilities,

- transportation costs.

More Accurate Model

Other Hospitals

Transportation cost

• fixed cost

•working inventory cost

•safety stock inventory cost

Hospitals serving as DCs

Blood Bank

Transportation cost

Problem Definition

• The determination of

- the optimal number of distribution centers,

- their locations,

- the retailers assigned to each distribution center,

- optimal ordering policy at the distribution center

while

- minimizing the total cost,

- satisfying certain service level

Assumptions• The locations of the suppliers and retailers are known,

• The suppliers have infinite capacity,

• The DCs receive the product from the plant with the smallest total shipping cost to DC, which depends only on

- distance btw. DC and the supplier plant

• The plant to DC lead time is the same for all plant/DC combinations

• The variance-to-mean ratio of demand at each retailer is identical for all retailers

• The inventory problem is modelled using (Q,r) inventory policy with type 1 service level

• (Q,r) policy is approximated by assuming that DC orders inventory from plant using EOQ model

• The reorder point is determined to ensure that the probability of a stockout at the DC is less than or equal to some specified value

Assumptions

• The customers to be assigned to a DC are known in advance (temporary assumption)

Model

Minimize

Jj Iiiji

Iiiji

Jjj

Jj Iiijijj

Jj Jj Jjijiijjj

YLhz

YaYgFh

YdXf

2

2

Subject to

.,,1,0

,,1,0

,,,

,,1

JjIiY

JjX

JjIiXY

IiY

ij

j

jij

Jjij

Model

Objective Function

Min fixed cost of locating DCs + the local delivery cost +

total working inventory cost + safety stock inventory cost

Constraints

subject to

1. Each retailer must be assigned to a DC,

2. Retailers can be assigned to an open DC,

3. All of the demand at a retailer must be assigned to the same DC

4. Standard integrality constraints

Model Properties• Instead of the nearest DC, it may be optimal to assign retailers to

a more remote distribution center

increase in the transportation cost < decrease in the inventory and supplier-to-DC transport cost

Inventory related costs are large relative to other costs

( is large relative to )

Model Properties• The assumption of identical variance-to-mean ratios for

demands of retailers

- reduces the non-linear terms in the objective function

- it is never optimal to open a DC at a node and then to serve the demands from that node from other DC

• Each DC has a region of service

If retailer “a” is optimally assigned to DC1, then it is optimal

to assign retailers “b and c” to DC1.

a b c DC1

Solution Approach• Lagrangian relaxation embedded in branch and bound

1. Finding a lower bound:

Solve the Lagrangian problem by relaxing the constraint that each retailer must be assigned to a DC and use fixed Lagrange multipliers in order to find a lower bound for objective function.

2. Finding an upper bound:

Assign the unassigned retailers to the open DC which increases the total cost the least based on assignment made so far. After the assignment of all retailers an upper bound for objective function is obtained.

Solution Approach3. Retailer Reassignments:

- Try to improve the upper bound further by considering all possible single retail moves from the DC to which the retailer is currently assigned to another DC,

- Do not remove an open DC from consideration until no improving reassignment can be found,

- continue looping until one entire loop is completed without finding an improved reassignment,

- Remove the open DCs with no assigned demand,

- Update the upper bound of the objective function.

Solution Approach

4. DC exchange algorithm improvements:

- For each DC in the current solution find the best substitute DC that is not in the current solution,

- Assign retailers to the DC which increases the cost the least,

- If a DC exchange is found to improve the solution make the exchange and try single retailer reassignments to the best DC configuration,

- Restart the search for improving exchanges,

- Stop as soon as no other improving exchange is found.

Solution Approach5. Variable Fixing:

- Performing branch and bound algorithm on all of the DC locations,

- Decide on which DCs to exclude or include.

6. Branch and Bound:

- If lower bound = upper bound, the solution corresponding to the upper bound is optimal.

- If lower bound< upper bound and some candidate DC locations are not forced in or out of the solution branch and bound method is employed

- branch first of all on the DC that is in the best-known solution

- if all DC locations corresponding to the best solution are forced into the solution, force other DCs first out and then into the solution

Results

• As the transport costs increase, the number of DCs goes up

• In order to decrease the transportation cost more DCs are required

0

5

10

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10

Number of DCs

Tra

ns

po

rtat

ion

Co

st

Results • As inventory costs increase, the number of DCs goes down

• In order to decrease inventory costs, less DCs are required

0

20

40

60

80

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Number of DCs

Inv

en

tory

Co

sts

Results

• As the relative importance inventory costs goes up, the number of DCs located goes down and as the importance of transportation costs goes up, the number of DCs increase.

• If the fixed cost of placing an order decreases significantly, as might be the case with e-commerce technologies

- the total cost decreases,

- the number of DCs goes up.

• Response time decrease as number of DCs goes up.

Further Research Areas

• Solve the same problem when

- the variance to mean ratio is not identical,

- multiple items,

- maximum allowable inventory at DCs,

- maximum demand that can be served by a supplier,

- different future demand scenarios,

- different travel cost scenarios.

QUESTIONS?