TPS distributes its products in two echelons distribution systems that are from warehouse to several
distributors. The purpose of formulating and optimizing the food distribution model is to provide TPS with a means
for comparing their strategy for moving food through their distribution network with the optimized strategy derived
from the formulated model. Therefore, the first objective of the project was to establish the scope of the model by
identifying the aspects of TPS distribution that would be studied. The second objective was to define the decision
variables, parameters, constraints, and performance measures necessary for formulating a model of TPS food
distribution operations. The third objective was to formulate a mathematical programming model of the
distribution activities. The fourth objective was to identify software for solving the defined mathematical
programming model. And final objective was to analyze the optimal distribution of food determined for the model
to identify improvements to the current distribution strategy used by TPS. TPS staff assisted in establishing the
problem definition; defining the model’s decision variables, parameters, constraints, and performance measures;
validating the model formulation; providing input parameters for the model; and analyzing test scenarios.
2. PROBLEM DESCRIPTION
The scope of the distribution network can depict as Figure 1. TPS produces many varieties of food are
classified in 10 product families and 88 product items. Transportation vehicles used consist of two types that are
truck owned by company shipper and truck from third- party logistics providers carriers with differences of
volume and distribution area. There are around 36 carriers becoming company partner.
Figure 1: Illustration of Company Distribution Network
APIEMS 2008
Pr oceedings of t he 9t h Asia Pasific Indust r ial Engineer ing Management Syst ems Confer ence
Nusa Dua, Bali – INDONESIA
December 3r d – 5t h, 2008
995
TPS classifies their transportation vehicles become thirteen classes by considering truck volume in
meter cubic, ownership and destination point distributor location. Two categories is property of TPS, while the
others are property of the carrier companies. The company has two-echelon distribution network that consist of one
warehouse and 59 distributors divided in five distribution areas. Demand quantities are based on forecasting methods
that established by TPS.
The distribution of goods is a dynamic activity. Therefore, the activities captured by the model must be
indexed over time. The model must take into consideration the planning horizon of interest to TPS. A planning
horizon was used for four weeks. Lead-time delivery each distributor is dependent on distance between distributors
and company warehouse.
As illustrated earlier, this study is primarily concerned with reducing total transportation cost in single-
warehouse multi-agent distribution network by considering delivery lead time and capacitated transportation facilities.
We have to decide product amount will be delivered to each distributor and number of transportation facilities will
be assigned to each distributor.
.
3. MODEL FORMULATION
We formulate the problem above in Integer Linear Programming model. Assumptions considered by
this model are: •
Number of trucks always available, •
Number of distributors is constant, •
Quantity of daily demand is known in advance, •
Delivery cost is constant along time, •
Delivery lead time is deterministic and static While none of these assumptions are perfectly
valid, we agreed that they were necessary for one of two reasons. First, it was agreed that a simpler model will be
beneficial for this study. Second, valid data sources necessary for relaxing these assumptions did not exist. The
model developed in this paper use the following notations: Decision variables:
•
X
ijk
: number of trucks i are assigned to distributor j on day k,
Parameters: •
C
ijk
: delivery cost for truck category i to distributor j on day k,
• D
jk
: demand of distributor j on day k, •
q
i
: volume of truck- i m
3
, •
t
i
: loading tolerance for truck- i, •
T
ik
: number of truck- i available , •
i : index for truck category 1, ..,13 •
j : index for distributor 1, 2...59 •
k : index for day 1,2...6
Objective Function:
This model is aimed to minimize variable transportation cost as follow:
∑ ∑
∑
∈ ∈
∈
=
I i
ijk ijk
J j
K k
c x
Z
min
1
Subject to a. Number of trucks constraint
k i
T x
x
ik j
k j
i k
i
∀ =
≤ +
∑
=
; 5
, 3
, 1
, 2
12 2
, ,
, 1
,
2
k T
x x
x x
x
k j
jk j
jk j
jk j
jk k
∀ ≤
+ +
+ +
∑ ∑
∑ ∑
= −
= −
= =
; 2
2 46
42 1
2 26
18 1
2 46
28 2
26 2
2 ,
1 ,
2
3
k i
T x
x x
x
ik k
i k
i j
ijk j
ijk
∀ =
≤ +
+ +
− −
= −
=
∑ ∑
; 12
, 10
, 6
, 4
,
3 ,
50 ,
2 ,
50 ,
50 42
1 50
28
4
k T
x x
k j
jk j
jk
∀ ≤
+
∑ ∑
= −
=
,
7 27
13 1
7 27
13 7
5
k i
T x
x
ik j
ijk j
ijk
∀ =
≤ +
∑ ∑
= −
=
; 11
, 8
,
55 51
1 55
51
6
k i
T x
x x
x x
ik k
i k
i j
ijk j
ijk j
ijk
∀ =
≤ +
+ +
+
− −
= −
= −
=
∑ ∑
∑
; 13
, 9
,
4 ,
59 ,
3 59
, 59
56 2
59 56
1 59
56
7 b. Truck volume constraint
k j
i D
t q
x
jk i
I i
i ijk
, ,
, ∀
≥ −
∑
∈
8 c. Allocation decision constraint
k j
i x
ijk
, ,
, ∀
≥
9
k j
i x
ijk
, ,
integer ∀
=
10
APIEMS 2008
Pr oceedings of t he 9t h Asia Pasific Indust r ial Engineer ing Management Syst ems Confer ence
Nusa Dua, Bali – INDONESIA
December 3r d – 5t h, 2008
996
Equation 1 is objective function that minimizes the sum of the costs to distribute goods from warehouse to
distributor-j using truck-i during k days. Constraint set equation 2-7 represents the number of truck class-i
assigned to deliver products to distributor area on day-k are not permitted exceeds number of truck available by
considering delivery lead time. Constraint set equation 8 ensures that all the demand of distributor-j day-k is
balanced by total volume of product that has been transported from warehouse. Constraint equation 9
enforces the non-negativity restriction on the decision variables used in this model. Constraint set equation 10
enforces the integer number of the decision variables
4. NUMERICAL RESULT AND ANALYSIS
