“Make to order or Make to Stock Model: and

Application” S.Rajagopalan

By: ÖNCÜ HAZIR

Content of PresentationIntroductionLiterature ReviewAssumptionsTrade-offs and Congestion EffectsGeneral Model and Relaxed ModelProperties of Optimal Solution & Solution ProcedureComputational StudyExperimental Insights

Introduction

Motivation is to determine whether an item is to make to stock(MTS) or make to order(MTO) and to offer an inventory policy for the make to stock items.Characteristics of the production environment is multiple items,limited capacity and setups between the production of consecutive items.Objective is minimize inventory costs of MTS items while ensuring that orders for MTO items are fullfilled with a sepicified probability.

Literature Review

Popp(1965) made cost comparisons to make an item MTO or MTS for a single-item stochastic inventory model.Williams(1984) assumed lower demand items are MTO and higher demand items as MTS.Federgruen and Katalan(1995,1999) allowed the interruption of MTS items when MTO demand is realized.Carr and Duenyas(1998) focuses on criteria to accept or reject MTO items.Karmakar(1987) considers the queue length as a decision variable.

Assumptions

Stochastic stationary, uncorrelated demand,varying processing times and limited capacity.No inventory is carried for MTO items.(Q,R) inventory policy is used for MTS items.First come first served (FCFS) queue discipline. Production facility is approximated by M/G/1 queue discipline.

Assumptions

Setup and processing times are deterministic.Type 1() service level represents probability of no stockout.The distribution of demand during lead time is characterized by queue time,material handling times are ignored. Whenever there exists a demand for MTO in the time period,a production order is initiated for a batch size equal to demand quantity.

Trade-offs and Congestion Effects

Making an item to order: Decreases inventory, Congestion effect:

Making an item to stock: Decreasing the lot size reduces cycle stock but increases

number of setups and utilization so lead time increases.As a result more cyle and safety stock for MTO items and poorer service for MTO items.

More setups and higher utilization

Longer and variable lead times

Higher cycle and safety stock for MTS and poorer service for MTO

Model Parameters

MTS MTO Processing time

i+qi/i

i+(i/mi)/i

Number of batches per unit time

i/qi mi

General Model

MinST

isiiii

i zPGfqh ))(.),(2/(

)1(2

)()(

2

xEwE )1(3

)()()(

32

xEwEwv

2i

2ii v(w)μE(w)σv(w))(E(w),τ

]1,0[

0

)),(),((

i

i

o

z

q

TPwvwEH

Model without congestion effects

)2/( ii

i qhMinST

1)/))1(/(( iiiiiii

i mzqz

]1,0[

0

i

i

z

q

Properties of optimal solution

)i

/qi

μi

(mi

α

/2)i

(qi

h)(η

jγ

1j

z then 1k

z if and )(ηj

γ)(ηk

γ3)If

/2i

mi

/qi

μ if MTS is item2)An

0γ(η)

and i

mi

/qi

μthen solution optimalan in 1zi 1)If

capacity lincrementaby dividedinventory in Savings :η)(j

γ

Solution procedure1)Set zi=1 for all i and set

= maxi {mi2ihi/2i}

2)Compute i for all items, if i <= 0 set zi=0, arrange items in order of decreasing ratio i

3) Set zi=0 in the order determined above.Compute lot sizes and costs, check whether total cost declines.If cost decreases stop.For the heuristic with congestion effects the ratio i includes cost of safety stock and for a given value of zi, a non-linear program is solved.

Application of Model

Computational Study

The heuristic performance was evaluated relative to lower bounds. Average percentage duality gap between heuristic gap and lower bound was performance criteria.It is found that the heuristic works well.

Experimental Insights

The MTS/MTO decisions with and without considering congestion effects were similar.The lowest and highest demand items are MTO medium demand items are MTS since incremental capacity to make an item to MTO is concave in the average demand.In addition to items demand decision depends on processing times,unit holding cost and set uptime.As size of time bucket increases, number of MTO items increase and total cost decreases.However customer responsiveness decreases.