Dynamic lot sizing for products with high setup

cost subject to obsolescence

Luca Zeppetella, Elisa Gebennini, Andrea Grassi, Bianca Rimini

Dipartimento di Scienze e Metodi dell’Ingegneria,

Università degli Studi di Modena e Reggio Emilia

18/11/2014, Budapest

Beyond the economic lot

The demand downward trend has determined the success of Lean Practices and in general the necessity of overcoming the concept of economic lot

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1 2 3 4 5 6 7 8 9 10

Setup Holding

Q*

Products’ end of life

- Perishability: products have a fixed life time after whichthey perish and become useless;

- Deterioration: items with a stochastic lifetime;

- Style Goods: have one selling season per year;

- Obsolescence: items whose demand decrease…

- Sudden Obs: unexpected, immediate obsolescence.

In our case we consider a Sudden Obsolescence with several possible products’ lifetimes

The risk of obsolescence

Obsolescence can be related to life-cycle and managers usuallyknow a product’s life-cycle, or anyway they assume to know it, consciusconsly or not, to take decisions

Examples of life-cycle distributions from literature:

- Norton-Bass Model;

- Bass Model;

- Bell-shaped pattern;

- Binomial distribution;

- Beta distribution;

- Triangular distribution;

- Weibull (λ,k) with k > 1;

- Markov Chain and Jump Process;

- …

The probability of obsolescence

• We consider the obsolescence as related to the positioningalong the product life-cycle:

given g(y) the product life-cycle distribution function, the probability of

obsolescence occurrence in t is computed as

• For sake of simplicity we assume a constant risk of obsolescence within a period:

Buyer-supplier relationship

We started from a situation based on blanket orders,

given a long-term forecast:

- supplier has to guarantee an in-stock quantity

- the buyer is forced to purchase the quantityforecasted for the next period, but the buyer can update the successive forecast

Buyer reacts to customers’ demand in very low advance

to avoid inventories and unsold articles

push to the limit the suppliers

Model - scenario

High setup costs make the lot-for-lot policy unsustainable, so the supplier has to produce more than the ordered quantity at its own risk

0

0.5

1

1.5

2

2.5

1 5 9

13

17

21

25

29

33

37

41

45

49

53

57

61

65

69

73

77

81

85

89

93

97

How much risk..? A risk that takes into account the

positioning on the product’s life-cycle

Model – assumptions

- Time-varying demand;

- Uncertain demand;

- Long-term forecast;

- Rolling sales horizon of one period;

- Zero lead time;

- Known product’s life-cycle distribution;

- Lost sales;

- No price mechanism to alter demand.

Math Formulation

Solution

The problem can be solved to the optimum through the branch and cut tecnique

In our tests we used the GLPK solver

Case Study - scenario

Since 1972, Ghepi has been active in the field of plastics. Ghepi is a modernfamily company and it’s involved in Project Development and OrderManagement, starting with consulting on polymers and extending to mouldsand fluid mechanics simulation for components, component and moulddesign, manufacturing and supply according to the customer's logisticstandards.

Case Study - data

• Tests were made on real data of the past 4 years and 100 articles. The articles considered changed their features due to fashion issue and their inventory became useless at once

• We compared the performance of our solution with the real production plans of the company

React to uncertainty

0

0.5

1

1.5

2

2.5

3

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100

0

0.2

0.4

0.6

0.8

1

f(t)

F(t)

Initial life-cycle estimation

t

t

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.5

1

1.5

2

2.5

3

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100

0

0.5

1

1.5

2

2.5

3

React to uncertainty

f(t)

F(t)

Life-cycle revision

Underestimated life lenght

Overestimated life lenght

f(t)

F(t)

Initial life-cycle estimation

0

0.5

1

1.5

2

2.5

3

0

0.2

0.4

0.6

0.8

1

React to uncertainty

F(t)

0

0.2

0.4

0.6

0.8

1

Conditional Probability

Life-cycle revision

f(t)

Cost sensitivity analysis

• We are interested in a managerial insight and robust solutions

• Obsolescence cost and the other costs are difficult to assessprecisely in a small enterprise

• Extreme solutions – costs boundaries:

– No production (all stockout)

– Lot-for-Lot production

– (worst case:1-piece production in every demand period)

Case Study - solutions

• We computed all the combinations of:

– 3 moments for the revision of the life-cycle (50%, 70%, 80%)

– 4 different life length revisions (+/- 10%, +/-20%)

– 4 Setup, Obsolescence, Holding, Stock-out Costs

• Then we consider as winning solution the most robust over all the scenarios

Case Study

…resulting Inventory:

0

1000

2000

3000

4000

5000

6000

7000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Real Production Demand

0

1000

2000

3000

4000

5000

6000

7000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Real Production Optimal Production Demand

Further extensions

- Multi-products case;

- Backorder case;

- Capacity constraints;

- Price mechanism;

- Sthocastic demand;

- Clients probability of staying in line;

- Buyer-supplier optimization;

- Presence of multiple suppliers.

…Thanks for the Attention