ISE 501-Introduction to Operations Research
Final Project Report
Title:
A linear programming model for integrated
steel production and distribution planning
Student Names:
Aravind Jayalakshmi Stalin Babu
Student Id: 200493452
Arundathi Ranganathan
Student Id: 200450463
Rashi Thippareddy
Student Id: 200476012
Kiran Burugupalli
Student Id: 200490643
Manali Jadhav
Student Id: 200493767
Instructor:
Dr. S. Sebnem Ahiska King
Abstract
The project addresses the problem statement for steel production and distribution planning for
a Canadian steel-making production which is formulated and solved using a linear
programming model for purchasing, production, and distribution. Furthermore, a
computational analysis is performed to study the system’s behavior under several scenarios.
This will help in achieving the maximum total net profit by minimizing production cost
which includes the cost of production, through output rates at various stages of production,
raw material, semi-finished product purchasing costs, product distribution, and transportation
costs. Numerical experimentation is done by considering the impact of all these input
parameters on the optimality of the solution. The project addresses the problem statement for
steel production and distribution planning for a Canadian steel-making production which is
formulated and solved using a linear programming model for purchasing, production, and
distribution. Computation results and analysis show that high level financial benefit can be
achieved by using linear optimization planning methodology. The simplex algorithm and
excel solver is used for all the computations.
Furthermore, a computational analysis is performed to study the system’s behavior under
several scenarios. This will help in achieving the maximum total net profit by minimizing
production cost under different circumstances. It includes the cost of production, through
output rates at various stages of production, raw material, semi-finished product purchasing
costs, product distribution, and transportation costs. Numerical experimentation is done by
considering the impact of all these input parameters on the optimality of the solution. All the
findings are utilized to gain some managerial insights for decision making related to
production.
Introduction
The Canadian steel-making company has one central plant and several finishing factories in
other locations. Different factories, material suppliers in various territories, and customers in
different geographic regions form an interactive material flow network. The different costs
include production costs, throughput rates at various stages of production, raw material and
semi-finished product purchasing costs, product distribution, and transportation costs. The
selling prices of finished products depend on customer regions and may have different selling
prices in different regions for the same product. The goal is to formulate the linear
programming model to maximize the profit so that to get insights into production and
distribution planning.
The approach is to find the optimal solution by formulating a linear programming model for
purchasing, production, and distribution. The objective in solving the planning problem is to
maximize the total net profit, which can be expressed as linear programming. Total net profit
= Total revenue - Total cost. Here total revenue is simply equal to total selling income whereas
total cost includes Raw material purchasing cost, fixed cost, variable cost, semi-finished
product purchasing cost, raw material transportation cost, semi-finished product transportation
cost from the central plant to other factories, semi-finished product transportation cost from
supplier’s territories to other factories, finished product transportation cost. We have used
Solver to optimize the schedule and costing of production to increase profitability.
Problem description and mathematical formulation
Problem statement:
This case demonstrates how, while determining production capacity, a planner must take cost
and profit into account in addition to output rate. It is desirable to employ appropriately
established mathematical models and computer information systems for efficient production
planning in a complex steel manufacturing environment with a significant amount of
information on a wide variety of products and facilities. The problem is finding the optimal
production planning to optimize the cost and schedule to increase the factory's profitability.
To address the issues with integrated production planning involving purchasing, production,
and distribution, we created a formulation based on linear programming. The goal of the
concept is to combine several planning operations into a single planning process. All raw
materials are shipped to the central steel-producing plant because it is the only facility in the
corporation that can make semi-finished goods. The finishing factories either purchase semifinished goods directly from suppliers on the outside or get them from the central facility. In
the latter scenario, the suppliers will send the semi-finished goods directly to the demanding
factories. The cost function's linearity in the planning model is consistent with how the
corporation does business. ss. An inventory of raw materials, semi-finished goods, and
finished goods is not included in the model because at this point the company is more
interested in a one-time integrated planning model.
Indices:
i = index of factories, i = {0 ,1,2}; i = 0 refers to the central plant
j = index of raw material supplying territories, j = {1,2,3,4,5}
l = index of customer regions, l = {1,2,3}
k = index of territories of semi-finished product purchasing, k = {1, 2,3}
m = index of product groups, m = 1
nm = index of product items in group m, nm {1,2,3}
Parameters:
(1) Raw material supply
Costs:
RCj = unit raw materials purchasing cost from territory j
Capacities:
Lj = supply capacity in raw material territory j
(2) Production
Production costs:
FCnm = fixed costs for product nm in the central plant's steelmaking process (product item n
in group m). The estimated consumption values of a company's fixed facility by various
products are known as fixed costs. They function in a comparable manner to variable costs
VCnm = Unit variable cost for product nm in the central plant's steelmaking process
FFnm = fixed cost of product nm for finished good production
VFnm = unit variable cost of product nm for finished good production
Production rates:
PSnm = production rate for semi-finished goods that are utilized to make final goods in nm
PFnm = production rate of finished product nm
YRnm = yield percentage from raw materials to semi-finished product of product nm
YSnm = yield percentage from semi-finished product to finished product nm
Production capacity:
tc = available production hour for steel making at the central plant
tfim= available production time for finished product group m at factory i
(3) Semi-finished product purchasing cost:
CSk,nm= unit purchasing cost of semi-finished product corresponding to finished product nm in
territory k
Sales
Price:
𝑃𝑅𝐼,𝑛𝑚 = unit selling price of product nm in customer region l.
Customer demand:
𝐷𝐶𝐼,𝑛𝑚 = amount of core business for product nm in customer region l.
𝐷𝐹𝐼,𝑛𝑚 = amount of sales forecast for product nm in customer region l.
Transportation cost:
𝑇𝑅𝑗𝑐 = unit transportation cost of raw materials from territory j to the central plant
𝑇𝑆𝑖𝑐 = unit transportation cost of semi-finished products from the central plant to factory i
𝑓
𝑇𝑆𝑖,𝑘 = unit transportation cost of semi-finished products from supplier’s territory k to
factory i
𝑇𝐹𝑙𝑖 = unit transportation cost of finished products from factory i to customer region l
DECISION VARIABLES
𝑥𝑖𝑙,𝑛𝑚 = amount of product 𝑛𝑚 produced in factory i for customers in region l by using semifinished products from the central plant.
𝑦𝑖𝑙,𝑛𝑚 = amount of product 𝑛𝑚 produced in factory i for customers in region l by using
purchased semi-finished product.
𝑢𝑖,𝑛𝑚 = amount of semi-finished product purchased from territory k and used in factory i for
producing product 𝑛𝑚 .
𝑤𝑗 = Quantity of raw materials to purchase from territory j.
Additional terms used in presenting the objective and constraint functions of the model are
introduced as needed. To determine an integrated optimal production plan, the following
constraint conditions in the system are considered.
MODEL CONSTRAINTS
(1) Raw material supply:
The amount of raw materials 𝑤𝑗 purchased from territory j should not exceed the supplier’s
capacity Lj in that territory. This can be expressed by:
𝑤𝑗 ≤ 𝐿𝑗 , ∀𝑗 .
Let 𝑤
̅ 𝑖𝑙,𝑛𝑚 be the quantity of raw materials used to produce 𝑥𝑖𝑙,𝑛𝑚 tons of product 𝑛𝑚 .
Thus:
𝑥
𝑤
̅ 𝑖𝑙,𝑛𝑚 = 𝑌𝑅 𝑖𝑙,𝑛𝑚
𝑌𝑆
𝑛𝑚
𝑛𝑚
Total amount of raw materials purchased for production cannot exceed those
purchased from all territories. This is expressed as
𝐼
𝐿
𝑁𝑚
𝑀
𝐽
∑∑ ∑ ∑𝑤
̅ 𝑖𝑙,𝑛𝑚 ≤ ∑ 𝑤𝑗
𝑖=0 𝑙=1 𝑚=1 𝑛=1
𝑗=1
i.e.
𝐼
𝐿
𝑀
𝑁𝑚
𝐽
𝑥𝑖𝑙,𝑛𝑚
∑∑ ∑ ∑
≤ ∑ 𝑤𝑗 .
𝑌𝑅𝑛𝑚 𝑌𝑆𝑛𝑚
𝑖=0 𝑙=1 𝑚=1 𝑛=1
𝑗=1
(2) Steelmaking capacity in the central plant:
Let 𝑥̅𝑖𝑙,𝑛𝑚 be the amount of semi-finished products to produce 𝑥𝑖𝑙,𝑛𝑚 tons of finished
product 𝑛𝑚 , then:
𝑥̅𝑖𝑙,𝑛𝑚 =
𝑥𝑖𝑙,𝑛𝑚
𝑌𝑆𝑛𝑚
The time required to produce semi-finished products 𝑥̅𝑖𝑙,𝑛𝑚 at the central plant is:
𝑥̅ 𝑖𝑙,𝑛𝑚
𝑥𝑖𝑙,𝑛𝑚
=
𝑃𝑆𝑛𝑚
𝑌𝑆𝑛𝑚 . 𝑃𝑆𝑛𝑚
The amount of semi-finished products produced at the central plant is limited by its
rolling capacity 𝑡𝑐 . This is expressed by:
𝐼
𝐿
𝑀
𝑁𝑚
∑∑ ∑ ∑
𝑖=0 𝑙=1 𝑚=1 𝑛=1
𝑥𝑖𝑙,𝑛𝑚
≤ 𝑡𝑐
𝑌𝑆𝑛𝑚 . 𝑃𝑆𝑛𝑚
(3) Semi-finished product purchasing and production:
Let 𝑦̅𝑖𝑙,𝑛𝑚 be the corresponding amount of semi-finished products to produce 𝑦𝑖𝑙,𝑛𝑚
tons of finished products. Then:
𝑦̅𝑖𝑙,𝑛𝑚 =
𝑦𝑖𝑙,𝑛𝑚
𝑌𝑆𝑛𝑚
The total amount,
𝐿
∑ 𝑦̅𝑖𝑙,𝑛𝑚
𝑖𝑙=1
should be less than the corresponding purchased amount,
𝐿
𝐾
𝑦𝑖𝑙,𝑛𝑚
∑
≤ ∑ 𝑢𝑖𝑘,𝑛𝑚 ,
𝑌𝑆𝑛𝑚
𝑙=1
∀𝑖, ∀𝑛, ∀𝑚 .
𝑘=1
(4) Production capacities for finished products:
Let t,nm be the corresponding finishing production time
𝐿
𝑡𝑖,𝑛𝑚 = ∑
𝑙=1
𝐿
𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑃𝐹𝑛𝑚
𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑓
∑
≤ 𝑡𝑖𝑚
𝑃𝐹𝑛𝑚
𝑙=1
(5) Customer demands:
Core business demands must be satisfied. In other words, the total
amount of products nm produced for customer region l must be greater
than or equal to the corresponding core business, i.e.
𝐼
∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 ≥ 𝐷𝐶𝑙,𝑛𝑚
𝑖=0
On the other hand, total production should not exceed forecasted total
demand, i.e.
𝐼
∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 ≤ 𝐷𝐹𝑙,𝑛𝑚
𝑖=0
Objective function
The objective in solving the planning problem is to maximize the total net
profit, which can be usually expressed as:
Total net profit = Total revenue –Total cost.
In this model, the total revenue is simply the total selling income:
𝑁𝑚
𝐿
𝑀
𝐼
∑
∑
∑ 𝑅𝑃𝑙,𝑛𝑚 . ∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑙=1
𝑚=1
𝑛=1
𝑖=0
Total cost contains more factors as discussed below:
Raw material purchasing cost:
𝐽
∑ 𝑅𝐶𝑗 . 𝑤𝑗
𝑗=1
Fixed cost:
𝑁𝑚
𝐼
𝐿
𝑀
∑
∑
∑
𝐶
∑((𝐹𝐶𝑛𝑚
+ 𝐹𝐹𝑛𝑚 ). 𝑥𝑖𝑙,𝑛𝑚 + 𝐹𝐹𝑛𝑚 . 𝑦𝑖𝑙,𝑛𝑚 )
𝑖=0
𝐿=1
𝑚=1
𝑛=1
𝐼
𝐿
𝑀
𝑁𝑚
∑
∑
∑
𝐶
∑((𝑉𝐶𝑛𝑚
+ 𝑉𝐹𝑛𝑚 ). 𝑥𝑖𝑙,𝑛𝑚 + 𝑉𝐹𝑛𝑚 . 𝑦𝑖𝑙,𝑛𝑚 )
𝑖=0
𝐿=1
𝑚=1
𝑛=1
Variable cost:
Semi-finished product purchasing cost:
𝑁𝑚
𝐾
𝑀
𝐼
∑
∑
∑ 𝐶𝑆𝑘,𝑛𝑚 ∑ 𝑢𝑖𝑘,𝑛𝑚
𝑘=1
𝑚=1
𝑛=1
𝑖=0
Raw material transportation cost:
𝐽
∑ 𝑇𝑅𝑗𝑐 . 𝑤𝑗
𝑗=1
Semi-finished product transportation cost from central plant to other
factories:
𝐼
𝐿
∑ 𝑇𝑆𝑖𝐶 ∑
𝑖=1
𝐿=1
𝑀
𝑁𝑚
∑
∑
𝑚=1
𝑛=1
𝑥𝑖𝑙,𝑛𝑚
𝑌𝑆𝑛𝑚
Semi-finished product transportation cost from supplier’s territories to
other factories:
𝐼
𝐾
𝑁𝑚
𝑀
𝑓
∑
∑ 𝑇𝑆𝑖,𝑘 ∑
∑ 𝑢𝑖𝑘,𝑛𝑚
𝑖=0
𝑘=1
𝑛=1
𝑚=1
Finished product transportation cost:
𝐼
𝐿
𝑀
𝑁𝑚
∑
∑ 𝑇𝐹𝑖,𝑙 ∑
∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑖=0
𝑙=1
𝑛=1
𝑚=1
Summarizing the above discussed constraint and objective functions, the
complete linear programming model can be expressed by:
𝐿
𝑁𝑚
𝑀
𝐽
𝐼
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 = ∑
∑
∑ 𝑅𝑃𝑙,𝑛𝑚 . ∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 − ∑ 𝑅𝐶𝑗 . 𝑤𝑗 −
𝑚=1
𝑁𝑚
𝑛=1
𝐿
𝑙=1
𝑀
∑
∑
𝐶
∑((𝐹𝐶𝑛𝑚
+ 𝐹𝐹𝑛𝑚 ). 𝑥𝑖𝑙,𝑛𝑚 + 𝐹𝐹𝑛𝑚 . 𝑦𝑖𝑙,𝑛𝑚 ) −
𝐿=1
𝑚=1
𝑛=1
𝑖=0
𝑗=1
𝑁𝑚
𝐼
𝐿
𝑀
∑
∑
∑
𝐶
∑((𝑉𝐶𝑛𝑚
+ 𝑉𝐹𝑛𝑚 ). 𝑥𝑖𝑙,𝑛𝑚 + 𝑉𝐹𝑛𝑚 . 𝑦𝑖𝑙,𝑛𝑚 ) −
𝑖=0
𝐿=1
𝐾
𝑚=1
𝑀
𝑛=1
𝑁𝑚
∑
∑
∑ 𝐶𝑆𝑘,𝑛𝑚 ∑ 𝑢𝑖𝑘,𝑛𝑚 − ∑ 𝑇𝑅𝑗𝑐 . 𝑤𝑗 −
𝑘=1
𝐿
𝑚=1
𝑀
𝑛=1
𝑁𝑚
𝐼
∑ 𝑇𝑆𝑖𝐶 ∑
∑
𝑖=1
𝑚=1
𝐼
𝐿=1
𝐽
𝐼
𝑖=0
𝐼
𝑗=1
𝑥𝑖𝑙,𝑛𝑚
∑
− ∑
𝑌𝑆𝑛𝑚
𝑛=1
𝐿
𝑀
𝐾
𝑀
𝑓
∑ 𝑇𝑆𝑖,𝑘 ∑
𝑖=0
𝑘=1
𝑁𝑚
𝑚=1
∑
∑ 𝑇𝐹𝑖,𝑙 ∑
∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑖=0
𝑙=1
𝑛=1
𝑚=1
𝑁𝑚
∑ 𝑢𝑖𝑘,𝑛𝑚 −
𝑛=1
Subject to:
𝑊𝑗 ≤ Lj
∑1𝑖=0
∑1𝑖=0
∑𝑙𝑙=1
∑𝑚
𝑚=1
∑𝑙𝑙=1
∑𝑚
𝑚=1
∑𝑛𝑛=1 𝑋𝑖𝑙𝑛𝑚/(𝑌𝑅𝑛𝑚 ∗ 𝑌𝑆𝑛𝑚)≤ ∑𝑗𝑗=1 𝑊𝑗
∑𝑛𝑛=1 𝑋𝑖𝑙𝑛𝑚/(𝑌𝑅𝑛𝑚 ∗ 𝑃𝑆𝑛𝑚)≤ Tc
∑𝑙𝑙=1 𝑌𝑖𝑙𝑛𝑚/(𝑌𝑆𝑛𝑚)≤ ∑𝑘𝑘=1 𝑈𝑖𝑘 𝑛𝑚
,
∑𝑙𝑙=1
∑𝑛𝑛=1(𝑋𝑖𝑙𝑛𝑚 + 𝑌𝑖𝑙𝑛𝑚)/𝑌𝑆𝑛𝑚≤ Tf im
∑𝑙𝑖=1(𝑋𝑖𝑙𝑛𝑚 + 𝑌𝑖𝑙𝑛𝑚)≤ DF l,nm
∑𝑙𝑖=1(𝑋𝑖𝑙𝑛𝑚 + 𝑌𝑖𝑙𝑛𝑚)≥ SF l,nm
𝑋𝑖𝑙𝑛𝑚, 𝑌𝑖𝑙𝑛𝑚, 𝑈𝑖𝑘𝑛𝑚 ≥ 0
V i, l, k, n, m
Numerical study
As we are producing products for a fixed set of customers and meeting a constrained demand
for one time, we are not considering the change in raw material costs due to Inflation, labor
cost changes, or change in selling prices over time. Indeed, we would like to experiment by
considering real-time problems like limited in-house production capabilities. One more
scenario is to work on improving the defective finished product's quality by reworking them
to set the right quality. We considered the increase in costs due to reworking and then sold
the reworked products at discount prices. We even planned to sell the scrap to the raw
material supplier for a cheaper price so he can recycle it along with a percentage of virgin
materials to produce new raw materials.
Experimentation 1
Here, we considered a scenario where we cannot produce product 3 semi-finished parts as it
needs a special tool and currently it will be a huge investment for the company to buy the
tool. Instead, we planned to purchase the semi-finished parts of product 3 from suppliers
directly. In this case, we are changing the decision variables and restricting the X values of
product 3 to be ‘0’ which means we are not producing finished products from in-house
produced semi-finished parts of product 3. So, we get only Y values for product 3 as we are
buying all the semi-finished products from suppliers across the territories k = {1, 2,3}
LP Model:
MODEL CONSTRAINTS
(1) Raw material supply:
The amount of raw materials 𝑤𝑗 purchased from territory j should not exceed the supplier’s
capacity Lj in that territory. This can be expressed by:
𝑤𝑗 ≤ 𝐿𝑗 , ∀𝑗 .
Let 𝑤
̅ 𝑖𝑙,𝑛𝑚 be the quantity of raw materials used to produce 𝑥𝑖𝑙,𝑛𝑚 tons of product 𝑛𝑚 .
Thus:
𝑥
𝑤
̅ 𝑖𝑙,𝑛𝑚 = 𝑌𝑅 𝑖𝑙,𝑛𝑚
𝑌𝑆
𝑛𝑚
𝑛𝑚
Total amount of raw materials purchased for production cannot exceed those
purchased from all territories. This is expressed as
𝐼
𝐿
𝑀
𝑁𝑚
𝐽
∑∑ ∑ ∑𝑤
̅ 𝑖𝑙,𝑛𝑚 ≤ ∑ 𝑤𝑗
𝑖=0 𝑙=1 𝑚=1 𝑛=1
𝑗=1
i.e.
𝐼
𝐿
𝑀
𝑁𝑚
𝐽
𝑥𝑖𝑙,𝑛𝑚
∑∑ ∑ ∑
≤ ∑ 𝑤𝑗 .
𝑌𝑅𝑛𝑚 𝑌𝑆𝑛𝑚
𝑖=0 𝑙=1 𝑚=1 𝑛=1
𝑗=1
(2) Steelmaking capacity in the central plant:
Let 𝑥̅𝑖𝑙,𝑛𝑚 be the amount of semi-finished products to produce 𝑥𝑖𝑙,𝑛𝑚 tons of finished
product 𝑛𝑚 , then:
𝑥̅𝑖𝑙,𝑛𝑚 =
𝑥𝑖𝑙,𝑛𝑚
𝑌𝑆𝑛𝑚
The time required to produce semi-finished products 𝑥̅𝑖𝑙,𝑛𝑚 at the central plant is:
𝑥̅ 𝑖𝑙,𝑛𝑚
𝑥𝑖𝑙,𝑛𝑚
=
𝑃𝑆𝑛𝑚
𝑌𝑆𝑛𝑚 . 𝑃𝑆𝑛𝑚
The amount of semi-finished products produced at the central plant is limited by its
rolling capacity 𝑡𝑐 . This is expressed by:
𝐼
𝐿
𝑁𝑚
𝑀
∑∑ ∑ ∑
𝑖=0 𝑙=1 𝑚=1 𝑛=1
𝑥𝑖𝑙,𝑛𝑚
≤ 𝑡𝑐
𝑌𝑆𝑛𝑚 . 𝑃𝑆𝑛𝑚
(3) Semi-finished product purchasing and production:
Let 𝑦̅𝑖𝑙,𝑛𝑚 be the corresponding amount of semi-finished products to produce 𝑦𝑖𝑙,𝑛𝑚
tons of finished products. Then:
𝑦̅𝑖𝑙,𝑛𝑚 =
𝑦𝑖𝑙,𝑛𝑚
𝑌𝑆𝑛𝑚
The total amount,
𝐿
∑ 𝑦̅𝑖𝑙,𝑛𝑚
𝑖𝑙=1
should be less than the corresponding purchased amount,
𝐿
𝐾
𝑦𝑖𝑙,𝑛𝑚
∑
≤ ∑ 𝑢𝑖𝑘,𝑛𝑚 ,
𝑌𝑆𝑛𝑚
𝑙=1
∀𝑖, ∀𝑛, ∀𝑚 .
𝑘=1
(4) Production capacities for finished products:
Let t,nm be the corresponding finishing production time
𝐿
𝑡𝑖,𝑛𝑚 = ∑
𝐿
∑
𝑙=1
𝑙=1
𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑃𝐹𝑛𝑚
𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑓
≤ 𝑡𝑖𝑚
𝑃𝐹𝑛𝑚
(5) Customer demands:
Core business demands must be satisfied. In other words, the total
amount of products nm produced for customer region l must be greater
than or equal to the corresponding core business, i.e.
𝐼
∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 ≥ 𝐷𝐶𝑙,𝑛𝑚
𝑖=0
On the other hand, total production should not exceed forecasted total
demand, i.e.
𝐼
∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 ≤ 𝐷𝐹𝑙,𝑛𝑚
𝑖=0
Objective function
The objective in solving the planning problem is to maximize the total net
profit, which can be usually expressed as:
Total net profit = Total revenue –Total cost.
In this model, the total revenue is simply the total selling income:
𝑁𝑚
𝐿
𝑀
𝐼
∑
∑
∑ 𝑅𝑃𝑙,𝑛𝑚 . ∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑙=1
𝑚=1
𝑛=1
𝑖=0
Total cost contains more factors as discussed below:
Raw material purchasing cost:
𝐽
∑ 𝑅𝐶𝑗 . 𝑤𝑗
𝑗=1
Fixed cost:
𝑁𝑚
𝐼
𝐿
𝑀
∑
∑
∑
𝐶
∑((𝐹𝐶𝑛𝑚
+ 𝐹𝐹𝑛𝑚 ). 𝑥𝑖𝑙,𝑛𝑚 + 𝐹𝐹𝑛𝑚 . 𝑦𝑖𝑙,𝑛𝑚 )
𝑖=0
𝐿=1
𝑚=1
𝑛=1
𝐼
𝐿
𝑀
𝑁𝑚
∑
∑
∑
𝐶
∑((𝑉𝐶𝑛𝑚
+ 𝑉𝐹𝑛𝑚 ). 𝑥𝑖𝑙,𝑛𝑚 + 𝑉𝐹𝑛𝑚 . 𝑦𝑖𝑙,𝑛𝑚 )
𝑖=0
𝐿=1
𝑚=1
𝑛=1
Variable cost:
Semi-finished product purchasing cost:
𝐾
𝑀
𝑁𝑚
𝐼
∑
∑
∑ 𝐶𝑆𝑘,𝑛𝑚 ∑ 𝑢𝑖𝑘,𝑛𝑚
𝑘=1
𝑚=1
𝑛=1
Raw material transportation cost:
𝑖=0
𝐽
∑ 𝑇𝑅𝑗𝑐 . 𝑤𝑗
𝑗=1
Semi-finished product transportation cost from central plant to other
factories:
𝐼
𝐿
∑ 𝑇𝑆𝑖𝐶 ∑
𝑖=1
𝐿=1
𝑀
𝑁𝑚
∑
∑
𝑚=1
𝑛=1
𝑥𝑖𝑙,𝑛𝑚
𝑌𝑆𝑛𝑚
Semi-finished product transportation cost from supplier’s territories to
other factories:
𝐼
𝑁𝑚
𝐾
𝑀
𝑓
∑ 𝑇𝑆𝑖,𝑘 ∑
𝑘=1
𝑚=1
∑
𝑖=0
∑ 𝑢𝑖𝑘,𝑛𝑚
𝑛=1
Finished product transportation cost:
𝐼
𝐿
𝑁𝑚
𝑀
∑
∑ 𝑇𝐹𝑖,𝑙 ∑
∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑖=0
𝑙=1
𝑛=1
𝑚=1
Summarizing the above discussed constraint and objective functions, the
complete linear programming model can be expressed by:
𝐿
𝑁𝑚
𝑀
𝐽
𝐼
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 = ∑
∑
∑ 𝑅𝑃𝑙,𝑛𝑚 . ∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 − ∑ 𝑅𝐶𝑗 . 𝑤𝑗 −
𝑚=1
𝑁𝑚
𝑛=1
𝐿
𝑙=1
𝑀
∑
∑
𝐶
∑((𝐹𝐶𝑛𝑚
+ 𝐹𝐹𝑛𝑚 ). 𝑥𝑖𝑙,𝑛𝑚 + 𝐹𝐹𝑛𝑚 . 𝑦𝑖𝑙,𝑛𝑚 ) −
𝐿=1
𝑚=1
𝑛=1
𝑖=0
𝑗=1
𝑁𝑚
𝐼
𝐿
𝑀
∑
∑
∑
𝐶
∑((𝑉𝐶𝑛𝑚
+ 𝑉𝐹𝑛𝑚 ). 𝑥𝑖𝑙,𝑛𝑚 + 𝑉𝐹𝑛𝑚 . 𝑦𝑖𝑙,𝑛𝑚 ) −
𝑖=0
𝐿=1
𝐾
𝑚=1
𝑀
𝑛=1
𝑁𝑚
∑
∑
∑ 𝐶𝑆𝑘,𝑛𝑚 ∑ 𝑢𝑖𝑘,𝑛𝑚 − ∑ 𝑇𝑅𝑗𝑐 . 𝑤𝑗 −
𝑘=1
𝐿
𝑚=1
𝑀
𝑛=1
𝑁𝑚
𝐼
∑ 𝑇𝑆𝑖𝐶 ∑
∑
𝑖=1
𝑚=1
𝐼
𝐿=1
𝐽
𝐼
𝑖=0
𝐼
𝑗=1
𝑥𝑖𝑙,𝑛𝑚
∑
− ∑
𝑌𝑆𝑛𝑚
𝑛=1
𝐿
𝑀
𝑖=0
𝑁𝑚
𝐾
𝑀
𝑓
∑ 𝑇𝑆𝑖,𝑘 ∑
𝑘=1
𝑚=1
∑
∑ 𝑇𝐹𝑖,𝑙 ∑
∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑖=0
𝑙=1
𝑛=1
𝑚=1
Subject to:
𝑊𝑗 ≤ Lj
𝑁𝑚
∑ 𝑢𝑖𝑘,𝑛𝑚 −
𝑛=1
∑1𝑖=0
∑1𝑖=0
∑𝑙𝑙=1
∑𝑛𝑛=1 𝑋𝑖𝑙𝑛𝑚/(𝑌𝑅𝑛𝑚 ∗ 𝑌𝑆𝑛𝑚)≤ ∑𝑗𝑗=1 𝑊𝑗
∑𝑚
𝑚=1
∑𝑙𝑙=1
∑𝑚
𝑚=1
∑𝑛𝑛=1 𝑋𝑖𝑙𝑛𝑚/(𝑌𝑅𝑛𝑚 ∗ 𝑃𝑆𝑛𝑚)≤ Tc
∑𝑙𝑙=1 𝑌𝑖𝑙𝑛𝑚/(𝑌𝑆𝑛𝑚)≤ ∑𝑘𝑘=1 𝑈𝑖𝑘 𝑛𝑚
,
∑𝑙𝑙=1
∑𝑛𝑛=1(𝑋𝑖𝑙𝑛𝑚 + 𝑌𝑖𝑙𝑛𝑚)/𝑌𝑆𝑛𝑚≤ Tf im
∑𝑙𝑖=1(𝑋𝑖𝑙𝑛𝑚 + 𝑌𝑖𝑙𝑛𝑚)≤ DF l,nm
∑𝑙𝑖=1(𝑋𝑖𝑙𝑛𝑚 + 𝑌𝑖𝑙𝑛𝑚)≥ SF l,nm
𝑋𝑖𝑙𝑛𝑚 = 0 V n =3
𝑋𝑖𝑙𝑛𝑚, 𝑌𝑖𝑙𝑛𝑚, 𝑈𝑖𝑘𝑛𝑚 ≥ 0
RESULT:
V i, l, k, n, m
Experimentation 2
We plan to rework the defective finished products and sell them at a discounted price of 70%
when compared to the actual selling price. This will help us to increase the profit margin for
the company. But we need to spend some additional amount on reworking the products in this
case, which is of course little compared to the discounted selling price. Also, we spoke with
raw material vendors and negotiated to sell our scrap metal produced while making semifinished products from raw materials. But they agreed to buy for a ridiculously cheap price,
for 15% of the actual raw material cost. So, considering these two situations we added a few
more equations to our model to include all the cost calculations.
LP Model:
MODEL CONSTRAINTS
Reworking:
• Price:
𝑅𝐸𝐼,𝑛𝑚 = unit selling price of product nm in customer region I is 70% of PRl,nm.
SRj,nm = unit selling price of scrap from semi-finished prod nm to territory J.
•
Reworking expenditure:
𝑅𝑋𝑛𝑚 = amount spent to rework on the defective products nm
(1) Raw material supply:
The amount of raw materials 𝑤𝑗 purchased from territory j should not exceed the supplier’s
capacity Lj in that territory. This can be expressed by:
𝑤𝑗 ≤ 𝐿𝑗 , ∀𝑗 .
Let 𝑤
̅ 𝑖𝑙,𝑛𝑚 be the quantity of raw materials used to produce 𝑥𝑖𝑙,𝑛𝑚 tons of product 𝑛𝑚 .
Thus:
𝑥
𝑤
̅ 𝑖𝑙,𝑛𝑚 = 𝑌𝑅 𝑖𝑙,𝑛𝑚
𝑌𝑆
𝑛𝑚
𝑛𝑚
Total amount of raw materials purchased for production cannot exceed those
purchased from all territories. This is expressed as
𝐼
𝐿
𝑀
𝑁𝑚
𝐽
∑∑ ∑ ∑𝑤
̅ 𝑖𝑙,𝑛𝑚 ≤ ∑ 𝑤𝑗
𝑖=0 𝑙=1 𝑚=1 𝑛=1
𝑗=1
i.e.
𝐼
𝐿
𝑀
𝑁𝑚
𝐽
𝑥𝑖𝑙,𝑛𝑚
∑∑ ∑ ∑
≤ ∑ 𝑤𝑗 .
𝑌𝑅𝑛𝑚 𝑌𝑆𝑛𝑚
𝑖=0 𝑙=1 𝑚=1 𝑛=1
𝑗=1
(2) Steelmaking capacity in the central plant:
Let 𝑥̅𝑖𝑙,𝑛𝑚 be the amount of semi-finished products to produce 𝑥𝑖𝑙,𝑛𝑚 tons of finished
product 𝑛𝑚 , then:
𝑥̅𝑖𝑙,𝑛𝑚 =
𝑥𝑖𝑙,𝑛𝑚
𝑌𝑆𝑛𝑚
The time required to produce semi-finished products 𝑥̅𝑖𝑙,𝑛𝑚 at the central plant is:
𝑥̅ 𝑖𝑙,𝑛𝑚
𝑥𝑖𝑙,𝑛𝑚
=
𝑃𝑆𝑛𝑚
𝑌𝑆𝑛𝑚 . 𝑃𝑆𝑛𝑚
The amount of semi-finished products produced at the central plant is limited by its
rolling capacity 𝑡𝑐 . This is expressed by:
𝐼
𝐿
𝑁𝑚
𝑀
∑∑ ∑ ∑
𝑖=0 𝑙=1 𝑚=1 𝑛=1
𝑥𝑖𝑙,𝑛𝑚
≤ 𝑡𝑐
𝑌𝑆𝑛𝑚 . 𝑃𝑆𝑛𝑚
(3) Semi-finished product purchasing and production:
Let 𝑦̅𝑖𝑙,𝑛𝑚 be the corresponding amount of semi-finished products to produce 𝑦𝑖𝑙,𝑛𝑚
tons of finished products. Then:
𝑦̅𝑖𝑙,𝑛𝑚 =
𝑦𝑖𝑙,𝑛𝑚
𝑌𝑆𝑛𝑚
The total amount,
𝐿
∑ 𝑦̅𝑖𝑙,𝑛𝑚
𝑖𝑙=1
should be less than the corresponding purchased amount,
𝐿
𝐾
𝑦𝑖𝑙,𝑛𝑚
∑
≤ ∑ 𝑢𝑖𝑘,𝑛𝑚 ,
𝑌𝑆𝑛𝑚
𝑙=1
∀𝑖, ∀𝑛, ∀𝑚 .
𝑘=1
(4) Production capacities for finished products:
Let t,nm be the corresponding finishing production time
𝐿
𝑡𝑖,𝑛𝑚 = ∑
𝐿
∑
𝑙=1
𝑙=1
𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑃𝐹𝑛𝑚
𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑓
≤ 𝑡𝑖𝑚
𝑃𝐹𝑛𝑚
(5) Customer demands:
Core business demands must be satisfied. In other words, the total
amount of products nm produced for customer region l must be greater
than or equal to the corresponding core business, i.e.
𝐼
∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 ≥ 𝐷𝐶𝑙,𝑛𝑚
𝑖=0
On the other hand, total production should not exceed forecasted total
demand, i.e.
𝐼
∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 ≤ 𝐷𝐹𝑙,𝑛𝑚
𝑖=0
Objective function
The objective in solving the planning problem is to maximize the total net
profit, which can be usually expressed as:
Total net profit = Total revenue –Total cost.
In this model, the total revenue is simply the total selling income:
From Good Finished products
𝑁𝑚
𝐿
𝑀
𝐼
∑
∑
∑ 𝑅𝑃𝑙,𝑛𝑚 . ∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑙=1
𝑚=1
𝑛=1
𝑖=0
From reworked Finished products
𝑁𝑚
𝐿
𝑀
𝐼
∑
∑
∑ 𝑅𝐸𝑙,𝑛𝑚 . ∑((𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 ) − (𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 /𝑌𝑆𝑛𝑚)
𝑙=1
𝑚=1
𝑛=1
𝑖=0
From selling the scrap to raw material territories
𝐿
𝑁𝑚
𝑀
𝐼
∑
∑
∑ 𝑆𝑅𝑙,𝑛𝑚 . ∑((𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 /𝑌𝑆𝑛𝑚) − (𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 /𝑌𝑅𝑛𝑚)
𝑙=1
𝑚=1
𝑛=1
𝑖=0
Total cost contains more factors as discussed below:
Raw material purchasing cost:
𝐽
∑ 𝑅𝐶𝑗 . 𝑤𝑗
𝑗=1
Fixed cost:
𝑁𝑚
𝐼
𝐿
𝑀
∑
∑
∑
𝐶
∑((𝐹𝐶𝑛𝑚
+ 𝐹𝐹𝑛𝑚 ). 𝑥𝑖𝑙,𝑛𝑚 + 𝐹𝐹𝑛𝑚 . 𝑦𝑖𝑙,𝑛𝑚 )
𝑖=0
𝐿=1
𝑚=1
𝑛=1
𝐼
𝐿
𝑀
𝑁𝑚
∑
∑
∑
𝐶
∑((𝑉𝐶𝑛𝑚
+ 𝑉𝐹𝑛𝑚 ). 𝑥𝑖𝑙,𝑛𝑚 + 𝑉𝐹𝑛𝑚 . 𝑦𝑖𝑙,𝑛𝑚 )
𝑖=0
𝐿=1
𝑚=1
𝑛=1
Variable cost:
Semi-finished product purchasing cost:
𝑁𝑚
𝐾
𝑀
𝐼
∑
∑
∑ 𝐶𝑆𝑘,𝑛𝑚 ∑ 𝑢𝑖𝑘,𝑛𝑚
𝑘=1
𝑚=1
𝑛=1
𝑖=0
Reworking cost:
𝐿
𝑀
𝐼
𝑁
∑
∑
∑
∑ 𝑅𝑋𝑛𝑚 . ((𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚/𝑌𝑆𝑛𝑚) )
𝑙=1
𝑚=1
𝑖=0
𝑛=1
𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 −
Raw material transportation cost:
𝐽
∑ 𝑇𝑅𝑗𝑐 . 𝑤𝑗
𝑗=1
Semi-finished product transportation cost from central plant to other
factories:
𝐼
𝐿
∑ 𝑇𝑆𝑖𝐶 ∑
𝑖=1
𝐿=1
𝑀
𝑁𝑚
∑
∑
𝑚=1
𝑛=1
𝑥𝑖𝑙,𝑛𝑚
𝑌𝑆𝑛𝑚
Semi-finished product transportation cost from supplier’s territories to
other factories:
𝐼
𝐾
𝑁𝑚
𝑀
𝑓
∑
∑ 𝑇𝑆𝑖,𝑘 ∑
∑ 𝑢𝑖𝑘,𝑛𝑚
𝑖=0
𝑘=1
𝑛=1
𝑚=1
Finished product transportation cost:
𝐼
𝐿
𝑀
𝑁𝑚
∑
∑ 𝑇𝐹𝑖,𝑙 ∑
∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑖=0
𝑙=1
𝑛=1
𝑚=1
Summarizing the above discussed constraint and objective functions, the
complete linear programming model can be expressed by:
𝐿
𝐿
𝑀
𝑁𝑚
𝐼
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 = ∑
∑
∑ 𝑅𝑃𝑙,𝑛𝑚 . ∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑙=1
𝐿
𝑚=1
𝑀
𝑛=1
𝑁𝑚
𝑀
𝑁𝑚
𝑖=0
𝐼
+ ∑
∑
∑ 𝑅𝐸𝑙,𝑛𝑚 . ∑((𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 ) − (𝑥𝑖𝑙,𝑛𝑚
𝑙=1
𝑚=1
𝑛=1
𝑖=0
+ 𝑦𝑖𝑙,𝑛𝑚 /𝑌𝑆𝑛𝑚)
𝐼
+∑
∑
∑ 𝑆𝑅𝑙,𝑛𝑚 . ∑((𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 /𝑌𝑆𝑛𝑚) − (𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 /𝑌𝑅𝑛𝑚)
𝑙=1
𝑚=1
𝑛=1
𝑖=0
𝐽
− ∑ 𝑅𝐶𝑗 . 𝑤𝑗 −
𝑗=1
𝐾
𝑀
∑
∑
𝑘=1
𝑚=1
𝑁𝑚
𝐿
𝑀
∑
∑
𝐶
∑((𝐹𝐶𝑛𝑚
+ 𝐹𝐹𝑛𝑚 ). 𝑥𝑖𝑙,𝑛𝑚 + 𝐹𝐹𝑛𝑚 . 𝑦𝑖𝑙,𝑛𝑚 ) −
𝐿=1
𝑚=1
𝑛=1
𝑁𝑚
𝐼
𝐿
𝑀
∑
∑
∑
𝐶
∑((𝑉𝐶𝑛𝑚
+ 𝑉𝐹𝑛𝑚 ). 𝑥𝑖𝑙,𝑛𝑚 + 𝑉𝐹𝑛𝑚 . 𝑦𝑖𝑙,𝑛𝑚 ) −
𝑖=0
𝑁𝑚
𝐿=1
𝑚=1
𝐼
𝑛=1
𝐿
𝑀
𝐼
∑ 𝐶𝑆𝑘,𝑛𝑚 ∑ 𝑢𝑖𝑘,𝑛𝑚 − ∑
∑
∑
∑ 𝑅𝑋𝑛𝑚 . ((𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚/𝑌𝑆𝑛𝑚) )
𝑛=1
𝑚=1
𝑖=0
𝑛=1
𝑖=0
𝑙=1
𝐽
𝑁
𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚 −
− ∑ 𝑇𝑅𝑗𝑐 . 𝑤𝑗 −
𝑗=1
𝐼
𝐿
𝑁𝑚
𝑀
∑ 𝑇𝑆𝑖𝐶 ∑
∑
𝑖=1
𝑚=1
𝐼
𝐿=1
𝐼
𝑥𝑖𝑙,𝑛𝑚
∑
− ∑
𝑌𝑆𝑛𝑚
𝑛=1
𝐿
𝑀
𝑖=0
𝑁𝑚
𝐾
𝑀
𝑓
∑ 𝑇𝑆𝑖,𝑘 ∑
𝑘=1
𝑚=1
∑
∑ 𝑇𝐹𝑖,𝑙 ∑
∑ 𝑥𝑖𝑙,𝑛𝑚 + 𝑦𝑖𝑙,𝑛𝑚
𝑖=0
𝑙=1
𝑛=1
𝑚=1
𝑁𝑚
∑ 𝑢𝑖𝑘,𝑛𝑚 −
𝑛=1
Subject to:
𝑊𝑗 ≤ Lj
∑1𝑖=0
∑1𝑖=0
∑𝑙𝑙=1
∑𝑚
𝑚=1
∑𝑙𝑙=1
∑𝑚
𝑚=1
∑𝑛𝑛=1 𝑋𝑖𝑙𝑛𝑚/(𝑌𝑅𝑛𝑚 ∗ 𝑌𝑆𝑛𝑚)≤ ∑𝑗𝑗=1 𝑊𝑗
∑𝑛𝑛=1 𝑋𝑖𝑙𝑛𝑚/(𝑌𝑅𝑛𝑚 ∗ 𝑃𝑆𝑛𝑚)≤ Tc
∑𝑙𝑙=1 𝑌𝑖𝑙𝑛𝑚/(𝑌𝑆𝑛𝑚)≤ ∑𝑘𝑘=1 𝑈𝑖𝑘 𝑛𝑚
,
∑𝑙𝑙=1
∑𝑛𝑛=1(𝑋𝑖𝑙𝑛𝑚 + 𝑌𝑖𝑙𝑛𝑚)/𝑌𝑆𝑛𝑚≤ Tf im
∑𝑙𝑖=1(𝑋𝑖𝑙𝑛𝑚 + 𝑌𝑖𝑙𝑛𝑚)≤ DF l,nm
∑𝑙𝑖=1(𝑋𝑖𝑙𝑛𝑚 + 𝑌𝑖𝑙𝑛𝑚)≥ SF l,nm
𝑋𝑖𝑙𝑛𝑚, 𝑌𝑖𝑙𝑛𝑚, 𝑈𝑖𝑘𝑛𝑚 ≥ 0
V i, l, k, n, m
RESULT:
Conclusion
When we solved this problem using solver, the optimal solution we got did not require us to
subcontract as we had met the necessary profit with 1200 factory hours. So, we considered a
scenario as when pandemic hit, there was minimum number of skilled workers which made
us to outsource everything for product 3. If we had the inventory or salary constraints, we
would have attempted to achieve maximum profit. Our second experimentation is rework of
defective products which we are reselling to scrap with 15% of the original price to retain
sustainability and for reduced cost.
The future scope for this project is that we can decrease the variable cost and cost of products
by adding an inventory constraint. The inventory constraint can be for the use for the stocking
of raw materials, as steel prices might increase due to inflation or other external factors. We
could stock the inventory with raw materials and increase the selling price of the finished
product to increase our profit. We could also add constraint of workers where we could use
overtime hours to increase production and meet demands rather than outsourcing or
subcontracting the products through which our profit can be improved.
References
Mingyuan Chen, Weimin Wang (1997), “A linear programming model for integrated steel
production and distribution planning”, International Journal of Operations & Production
Management, Vol. 17, pp. 592-610.
Appendices
Experimentation 3 - Rework
Price for different customers
Group
Prod
l=1
l=2
l=3
Customers
1
1
410
425
433
2
440
425
452
3
461
423
440
Number of finished products sold to the customer, produced from inhouse semi-finished products
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
1
2
0
0
0
1
2439
2094
1550
0
0
0
0
1
0
3420
8
2
2
0
0
0
0
0
0
0
1
0
1000
446
1
3
2
0
0
0
1
3
2
2200
0
2004
Number of finished products sold to the customer, produced from purchased semi-finished products
Group
Prod
Factory
l=1
l=2
l=3
Customers
1
0
0
0
1
2
11
6
0
0
0
0
0
1
0
0
0
2
2
1850
0
1642
0
0
0
0
1
0
0
0
1
2439
2094
1550
1
2
11
6
0
0
0
0
0
1
0
3420
8
2
2
1850
0
1642
0
0
0
0
1
0
1000
446
2
1850
3420
1650
3
2200
1000
2450
2
814000
1453500
745800
3
1014200
423000
1078000
0
0
0
0
Total amount of products sold (factory split up)
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
1
3
2
2200
0
2004
Total amount of products sold
Group
Prod
l=1
l=2
l=3
Customers
1
1
2450
2100
1550
Total revenue
Group
Prod
l=1
l=2
l=3
Customers
1
1
1004500
892500
671150
Yield
1
Group
Prod
Yield Semi
Yield finished
1
45
85
2
40
80
3
38
80
Semi-finished product used to produce inhouse finished
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
1
2869.411765
2463.529412
1823.529412
1
2
0
0
0
0
0
0
0
1
0
4275
10
2
2
0
0
0
0
0
0
0
1
0
1250
557.5
1
3
2
0
0
0
1
2
0
0
0
0
0
0
0
1
0
10687.5
25
2
2
0
0
0
0
1
0
0
0 3289.474
0 1467.105
1
3
2
0
0
0
1
0
0
0
0
2
2
2312.5
0
2052.5
4365
0
1
0
0
0
142.5
0 0.333333333
2
2
0
0
0
Raw materials used to produce in-house semifinished
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
Total Inhouse Semi-finished material used
13248.97059
Total Raw material Used
31372.34692
j=1
31372.34692
Rw mtl territories J
Raw mtl purchased from territory J
Total Raw material purchased
j=2
0
1
6376.470588
5474.509804
4052.287582
j=3
0
j=4
0
j=5
0
31372.34692
Raw material supply
capacity of Territory j
territory vending capacity
1
63000
Total supply
2
47000
3
55000
4
60000
5
50000
275000
Semifinished goods used to produce finished goods from purchased goods
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
0
Total
1
0
0
0
0
1
2
12.94117647
7.058823529
0
20
2
75
3
65
0
0
0
0
0
1
0
0
0
0
1
3
2
2750
0
2505
5255
0
1
0
0
0 50.60729
0 22.57085
1
3
2
0
0
0
0
0
0
0
0
Production rate of processing raw material to semi-finished for Product n
Group
Prod
Rate
1
1
56
Time of producing Inhouse semi finished products
Customers
Group
Prod
Factory
l=1
l=2
l=3
Total time to get semi finished prod
0
0
0
0
1
113.8655462
97.75910364
72.36227824
499.9983991
Available Hours
500
Production rate of semi to finished
Group
1
1
2
0
0
0
Prod
Rate
1
75
2
114
3
85
1
38.25882353
32.84705882
24.31372549
1
2
0.17254902
0.094117647
0
Time of producing finished products
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
Total time to get finished prod
2
0
1
2
0
0 20.2851
0
37.5
0
0 0.087719298 18.0044
1
3
0
1
2
0
0 32.35294
0 14.70588
0
0 6.558824 29.47059
254.6517028
Finishing Time spent in each factory for all products
Factory
Group
i=0
i=1
i=2
1
0
154.272
100.3797
Given the finishing time at the factory i
Factory
Group
i=0
i=1
i=2
1
0
620
390
Semifinished product of product n purchased from vendor k supplied to factory i
Vendor territory
Group
Prod
Factory
k=1
k=2
0
0
0
0
Total
1
2
13
7
20
0
0
0
0
1
0
0
0
2
2
4365
0
4365
0
0
0
0
1
0
0
0
1
3
2
5255
0
5255
1
2
12.94117647
7.058823529
0
0
0
0
0
1
0
4275
10
2
2
2312.5
0
2052.5
0
0
0
0
1
0
1250
557.5
1
3
2
2750
0
2505
0
0
0
0
1
0
855
2
2
2
462.5
0
410.5
0
0
0
0
1
0
250
111.5
1
3
2
550
0
501
0
0
0
0
1
0
6412.5
15
2
2
0
0
0
0
1
0
0
0 2039.474
0 909.6053
1
3
2
0
0
0
1
0
0
0
Computed Total semifinished prod used to produce finished products
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
1
2869.411765
2463.529412
1823.529412
Computed defect finished products, based on yield
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
1
430.4117647
369.5294118
273.5294118
1
2
1.941176471
1.058823529
0
1
432.3529
370.5882
273.5294
1076.471
2
462.5
855
412.5
1730
1
3
550
250
612.5
1412.5
2
142450
254362.5
130515
1
3
177485
74025
188650
Defect products
Group
Prod
l=1
l=2
l=3
Quantity
Revenue by selling reworked products at 70% cost of actual products
Group
Prod
l=1
l=2
l=3
1
124085.3
110250
82906.76
Computed defect semi finished products, based on yield
Customers
Group
Prod
Factory
l=1
l=2
l=3
1
3507.058824
3010.980392
2228.75817
1
2
0
0
0
1
1100
1540
980
2
1500
1250
1240
1
3
1250
1000
1200
1
2450
2100
1550
2
1850
3420
1650
1
3
2200
1000
2450
0
0
0
0
Core business demand, number of products demanded by customer
Customers
Group
Prod
l=1
l=2
l=3
Sales capacity forecast
Customers
Group
Prod
l=1
l=2
l=3
Objective function costs
Production costs
Group
Prod
Fixed cost 1
Fixed cost 2
Variable cost 1
Variable cost 2
Semi cost k=1
Semi cost k=2
1
1
3
2
30
50
121
105
200
200
30
54
135
110
200
200
30
45
125
106
200
200
Fixed cost of in-house produced finshed goods
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
1
195120
167520
124000
1
2
0
0
0
0
0
0
0
1
0
287280
672
2
2
0
0
0
0
0
0
0
1
0
75000
33450
1
3
2
0
0
0
1
2
550
300
0
0
0
0
0
1
0
0
0
2
2
99900
0
88668
0
0
0
0
1
0
0
0
1
3
2
99000
0
90180
Fixed cost of finished goods from purchased Semi
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
Total fixed cost
1
0
0
0
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
1
2
550
300
0
0
0
0
0
1
0
287280
672
2
2
99900
0
88668
0
0
0
0
1
0
75000
33450
1
3
2
99000
0
90180
1
2
0
0
0
0
0
0
0
1
0
837900
1960
2
2
0
0
0
0
0
0
0
1
0
231000
103026
1
3
2
0
0
0
1
0
0
0
1
2
1155
630
0
0
0
0
0
1
0
0
0
2
2
203500
0
180620
0
0
0
0
1
0
0
0
1
3
2
233200
0
212424
1
551214
473244
350300
1
2
1155
630
0
0
0
0
0
1
0
837900
1960
2
2
203500
0
180620
0
0
0
0
1
0
231000
103026
1
3
2
233200
0
212424
1
0
0
2
873000
0
1
195120
167520
124000
Variable cost of in-house produced finshed goods
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
1
551214
473244
350300
Variable cost of finished goods from purchased Semi
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
Total variable cost
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
Semi finished purchase cost
Territories
1
Prod
Factory
k=1
k=2
0
0
0
1
0
0
2
2600
1400
0
0
0
0
0
0
1
0
0
2
1051000
0
Unit Raw mtl purchase cost
Territory j
Unit cost
1
12
2
16
3
19
4
23
5
28
1
376468.1631
2
0
3
0
4
0
5
0
1
8
2
10
3
15
4
21
5
27
1
250978.7754
2
0
3
0
4
0
5
0
1
14
2
23
1
2
0
0
0
0
0
0
0
1
0
59850
140
2
2
0
0
0
0
0
0
0
1
0
17500
7805
1
3
2
0
0
0
0
0
0
1
0
0
2
2
78570
0
0
0
0
1
0
0
1
3
2
94590
0
1
0
54720
216
2
2
33300
0
47618
0
0
0
0
1
0
16000
12042
1
3
2
39600
0
58116
Total raw mtl Purchase cost
Territory j
Unit cost
Unit Raw mtl trans cost
Territory j
Unit cost
Total raw mtl trans cost
Territory j
Unit cost
Semi transportation cost from central to others
Factories
Unit cost
0
0
Transportation Cost of semi-finished products from Central plant to others
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
1
40171.76471
34489.41176
25529.41176
Semifinished trans cost from territories to factories
Factory
k=1
k=2
Territories unit cost
0
25
21
1
29
24
2
18
28
Purchased Semi finished prod transportation cost
From supplier k
Group
Prod
Factory
k=1
k=2
0
0
0
1
0
0
1
2
234
196
0
15
25
19
1
22
16
27
2
18
21
29
0
0
0
0
1
53658
33504
41850
1
2
198
126
0
0
0
0
0
1
40
2
30
3
33
1
43058.82353
2
51900
3
46612.5
4
0
5
0
Unit cost for delivering Finished prod to customer
Customers
Factory
l=1
l=2
l=3
Total Delivery cost to customer
Customers
Group
Prod
Factory
l=1
l=2
l=3
Rework Costs for each product
Prod
Rework cost
Total rework cost
Product
Total cost
Total defect semi-finished products sold for recycling
Territory j
cost
Total revenue
Reworked prod revenue
Selling the mtrl for recycling
Fixed cost
Variable cost
SF purchase
RM Purchase
1
32622.0774
8096650
1284730
32622
1261640
3380173
1928000
376468.1631
2
0
3
0
RM trans
SF trans from central
SF trans
Finished trans
Rework cost
Profit
250978.7754
185485.5882
173590
231115
141571
1484980
Experimentation 1 - Product 3 limitation
Price for different customers
Group
Prod
l=1
l=2
l=3
Customers
1
1
410
425
433
2
440
425
452
3
461
423
440
Number of finished products sold to the customer, produced from inhouse semi-finished products
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
1
2
0
0
0
1
2450
2100
1550
0
0
0
0
1
1850
1250
1650
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
0
0
0
Number of finished products sold to the customer, produced from purchased semi-finished products
Group
Prod
Factory
l=1
l=2
l=3
Customers
1
0
0
0
1
2
0
0
0
0
0
0
0
1
0
0
0
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
2200
1000
1200
1
2450
2100
1550
1
2
0
0
0
0
0
0
0
1
1850
1250
1650
2
2
0
0
0
0
0
0
0
1
0
0
0
2
2200
1000
1200
2
1850
1250
1650
3
2200
1000
1200
2
814000
531250
745800
3
1014200
423000
528000
0
0
0
0
Total amount of products sold (factory split up)
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
1
3
Total amount of products sold
Group
Prod
l=1
l=2
l=3
Customers
1
1
2450
2100
1550
Total revenue
Group
Prod
l=1
l=2
l=3
Customers
1
1
1004500
892500
671150
Yield
1
Group
Prod
Yield Semi
Yield finished
1
45
85
2
40
80
3
38
80
Semi-finished product used to produce inhouse finished
Group
Prod
Factory
l=1
l=2
l=3
Customers
1
2
0
0
0
0
0
0
0
1
2312.5
1562.5
2062.5
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
0
0
0
1
2
0
0
0
0
0
0
0
1
5781.25
3906.25
5156.25
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
0
0
0
0
0
0
0
0
1
0
0
0
0
2
2
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
3
2
2750
1250
1500
5500
0
0
0
0
1
77.0833333
52.0833333
68.75
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
0
0
0
0
0
0
0
1
20.2850877
13.7061404
18.0921053
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
32.3529
14.7059
17.6471
1
2
0
0
0
0
0
0
0
1
0
0
0
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
5500
0
5500
1
2882.352941
2470.588235
1823.529412
1
2
0
0
0
0
0
0
0
1
2312.5
1562.5
2062.5
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
2750
1250
1500
1
1100
1540
980
2
1500
1250
1240
1
3
1250
1000
1200
1
2450
2100
1550
2
1850
3420
1650
1
3
2200
1000
2450
0
0
0
0
1
2882.352941
2470.588235
1823.529412
Raw materials used to produce in-house semifinished
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
Total Inhouse Semi-finished material used
1
6405.228758
5490.196078
4052.287582
13113.97059
Total Raw material Used
30791.46242
j=1
30791.46242
Rw mtl territories J
Raw mtl purchased from territory J
Total Raw material purchased
j=2
0
j=3
0
j=4
0
j=5
0
30791.46242
Raw material supply
capacity of Territory j
territory vending capacity
1
63000
Total supply
2
47000
3
55000
4
60000
5
50000
275000
Semifinished goods used to produce finished goods from purchased goods
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
0
Total
1
0
0
0
0
1
2
0
0
0
0
2
75
3
65
Production rate of processing raw material to semi-finished for Product n
Group
Prod
Rate
1
1
56
Time of producing Inhouse semi finished products
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
Total time to get semi finished prod
1
114.379085
98.03921569
72.36227824
1
2
0
0
0
482.6972456
Available Hours
500
Production rate of semi to finished
Group
Prod
Rate
1
1
75
2
114
3
85
1
38.43137255
32.94117647
24.31372549
1
2
0
0
0
Time of producing finished products
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
Total time to get finished prod
212.4754902
Finishing Time spent in each factory for all products
Factory
Group
i=0
i=1
i=2
1
0
147.77
64.7059
Given the finishing time at the factory i
Factory
Group
i=0
i=1
i=2
1
0
620
390
Semifinished product of product n purchased from vendor k supplied to factory i
Vendor territory
Group
Prod
Factory
k=1
k=2
0
0
0
0
Total
1
0
0
0
Computed Total semifinished prod used to produce finished products
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
Core business demand, number of products demanded by customer
Customers
Group
Prod
l=1
l=2
l=3
Sales capacity forecast
Customers
Group
Prod
l=1
l=2
l=3
Objective function costs
Production costs
Group
1
Prod
Fixed cost 1
Fixed cost 2
Variable cost 1
Variable cost 2
Semi cost k=1
Semi cost k=2
1
30
50
121
105
200
200
3
2
30
54
135
110
200
200
30
45
125
106
200
200
Fixed cost of in-house produced finshed goods
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
1
2
0
0
0
0
0
0
0
1
155400
105000
138600
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
0
0
0
1
0
0
0
1
2
0
0
0
0
0
0
0
1
0
0
0
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
99000
45000
54000
1
196000
168000
124000
1
2
0
0
0
0
0
0
0
1
155400
105000
138600
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
99000
45000
54000
1
2
0
0
0
0
0
0
0
1
453250
306250
404250
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
0
0
0
1
0
0
0
1
2
0
0
0
0
0
0
0
1
0
0
0
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
233200
106000
127200
1
553700
474600
350300
1
2
0
0
0
0
0
0
0
1
453250
306250
404250
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
233200
106000
127200
1
196000
168000
124000
Fixed cost of finished goods from purchased Semi
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
Total fixed cost
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
Variable cost of in-house produced finshed goods
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
1
553700
474600
350300
Variable cost of finished goods from purchased Semi
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
Total variable cost
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
Semi finished purchase cost
Territories
1
Prod
Factory
k=1
k=2
0
0
0
1
0
0
2
0
0
0
0
0
1
0
0
2
0
0
0
0
0
1
0
0
2
1100000
0
Unit Raw mtl purchase cost
Territory j
Unit cost
1
12
2
16
3
19
4
23
5
28
1
369497.549
2
0
3
0
4
0
5
0
1
8
2
10
3
15
4
21
5
27
1
246331.6993
2
0
3
0
4
0
5
0
1
14
2
23
1
2
0
0
0
0
0
0
0
1
32375
21875
28875
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
0
0
0
0
0
0
1
0
0
2
2
0
0
0
0
0
1
0
0
1
3
2
99000
0
0
0
0
0
1
40700
20000
44550
2
2
0
0
0
0
0
0
0
1
0
0
0
1
3
2
39600
21000
34800
Total raw mtl Purchase cost
Territory j
Unit cost
Unit Raw mtl trans cost
Territory j
Unit cost
Total raw mtl trans cost
Territory j
Unit cost
Semi transportation cost from central to others
Factories
Unit cost
0
0
Transportation Cost of semi-finished products from Central plant to others
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
1
40352.94118
34588.23529
25529.41176
Semifinished trans cost from territories to factories
Factory
k=1
k=2
Territories unit cost
0
25
21
1
29
24
2
18
28
Purchased Semi finished prod transportation cost
From supplier k
Group
Prod
Factory
k=1
k=2
0
0
0
1
0
0
1
2
0
0
0
15
25
19
1
22
16
27
2
18
21
29
1
53900
33600
41850
1
2
0
0
0
Unit cost for delivering Finished prod to customer
Customers
Factory
l=1
l=2
l=3
Total Delivery cost to customer
Customers
Total revenue
Fixed cost
Variable cost
SF purchase
RM Purchase
RM trans
SF trans from central
SF trans
Finished trans
Profit
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
6624400
1085000
3008750
1100000
369497.549
246331.6993
183595.5882
99000
330000
202225.1634
Experimentation 2 - Rework
Price for different customers
Group
Prod
l=1
l=2
l=3
Customers
1
1
410
425
433
2
440
425
452
3
461
423
440
Number of finished products sold to the customer, produced from inhouse semi-finished products
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
1
2
0
0
0
1
2439
2094
1550
0
0
0
0
1
0
3420
8
2
2
0
0
0
0
0
0
0
1
0
1000
446
1
3
2
0
0
0
1
3
2
2200
0
2004
Number of finished products sold to the customer, produced from purchased semi-finished products
Group
Prod
Factory
l=1
l=2
l=3
Customers
1
0
0
0
1
2
11
6
0
0
0
0
0
1
0
0
0
2
2
1850
0
1642
0
0
0
0
1
0
0
0
1
2439
2094
1550
1
2
11
6
0
0
0
0
0
1
0
3420
8
2
2
1850
0
1642
0
0
0
0
1
0
1000
446
2
1850
3420
1650
3
2200
1000
2450
2
814000
1453500
745800
3
1014200
423000
1078000
0
0
0
0
Total amount of products sold (factory split up)
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
1
3
2
2200
0
2004
Total amount of products sold
Group
Prod
l=1
l=2
l=3
Customers
1
1
2450
2100
1550
Total revenue
Group
Prod
l=1
l=2
l=3
Customers
1
1
1004500
892500
671150
Yield
1
Group
Prod
Yield Semi
Yield finished
1
45
85
2
40
80
3
38
80
Semi-finished product used to produce inhouse finished
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
1
2869.411765
2463.529412
1823.529412
1
2
0
0
0
0
0
0
0
1
0
4275
10
2
2
0
0
0
0
0
0
0
1
0
1250
557.5
1
3
2
0
0
0
1
2
0
0
0
0
0
0
0
1
0
10687.5
25
2
2
0
0
0
0
1
0
0
0 3289.474
0 1467.105
1
3
2
0
0
0
1
0
0
0
0
2
2
2312.5
0
2052.5
4365
0
1
0
0
0
142.5
0 0.333333333
2
2
0
0
0
Raw materials used to produce in-house semifinished
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
Total Inhouse Semi-finished material used
13248.97059
Total Raw material Used
31372.34692
j=1
31372.34692
Rw mtl territories J
Raw mtl purchased from territory J
Total Raw material purchased
j=2
0
1
6376.470588
5474.509804
4052.287582
j=3
0
j=4
0
j=5
0
31372.34692
Raw material supply
capacity of Territory j
territory vending capacity
1
63000
Total supply
2
47000
3
55000
4
60000
5
50000
275000
Semifinished goods used to produce finished goods from purchased goods
Group
Prod
Factory
l=1
l=2
l=3
Customers
0
0
0
0
0
Total
1
0
0
0
0
1
2
12.94117647
7.058823529
0
20
2
75
3
65
0
0
0
0
0
1
0
0
0
0
1
3
2
2750
0
2505
5255
0
1
0
0
0 50.60729
0 22.57085
1
3
2
0
0
0
0
0
0
0
0
Production rate of processing raw material to semi-finished for Product n
Group
Prod
Rate
1
1
56
Time of producing Inhouse semi finished products
Customers
Group
Prod
Factory
l=1
l=2
l=3
Total time to get semi finished prod
0
0
0
0
1
113.8655462
97.75910364
72.36227824
499.9983991
Available Hours
500
Production rate of semi to finished
Group
1
1
2
0
0
0
Prod
Rate
1
75
2
114
3
85
1
38.25882353
32.84705882
24.31372549
1
2
0.17254902
0.094117647
0
Time of producing finished products
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
Total time to get finished prod
2
0
1
2
0
0 20.2851
0
37.5
0
0 0.087719298 18.0044
1
3
0
1
2
0
0 32.35294
0 14.70588
0
0 6.558824 29.47059
254.6517028
Finishing Time spent in each factory for all products
Factory
Group
i=0
i=1
i=2
1
0
154.272
100.3797
Given the finishing time at the factory i
Factory
Group
i=0
i=1
i=2
1
0
620
390
Semifinished product of product n purchased from vendor k supplied to factory i
Vendor territory
Group
Prod
Factory
k=1
k=2
0
0
0
0
Total
1
2
13
7
20
0
0
0
0
1
0
0
0
2
2
4365
0
4365
0
0
0
0
1
0
0
0
1
3
2
5255
0
5255
1
2
12.94117647
7.058823529
0
0
0
0
0
1
0
4275
10
2
2
2312.5
0
2052.5
0
0
0
0
1
0
1250
557.5
1
3
2
2750
0
2505
0
0
0
0
1
0
855
2
2
2
462.5
0
410.5
0
0
0
0
1
0
250
111.5
1
3
2
550
0
501
0
0
0
0
1
0
6412.5
15
2
2
0
0
0
0
1
0
0
0 2039.474
0 909.6053
1
3
2
0
0
0
1
0
0
0
Computed Total semifinished prod used to produce finished products
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
1
2869.411765
2463.529412
1823.529412
Computed defect finished products, based on yield
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
1
430.4117647
369.5294118
273.5294118
1
2
1.941176471
1.058823529
0
1
432.3529
370.5882
273.5294
1076.471
2
462.5
855
412.5
1730
1
3
550
250
612.5
1412.5
2
142450
254362.5
130515
1
3
177485
74025
188650
Defect products
Group
Prod
l=1
l=2
l=3
Quantity
Revenue by selling reworked products at 70% cost of actual products
Group
Prod
l=1
l=2
l=3
1
124085.3
110250
82906.76
Computed defect semi finished products, based on yield
Customers
Group
Prod
Factory
l=1
l=2
l=3
1
3507.058824
3010.980392
2228.75817
1
2
0
0
0
1
1100
1540
980
2
1500
1250
1240
1
3
1250
1000
1200
1
2450
2100
1550
2
1850
3420
1650
1
3
2200
1000
2450
0
0
0
0
Core business demand, number of products demanded by customer
Customers
Group
Prod
l=1
l=2
l=3
Sales capacity forecast
Customers
Group
Prod
l=1
l=2
l=3
Objective function costs
Production costs
Group
Prod
Fixed cost 1
Fixed cost 2
Variable cost 1
Variable cost 2
Semi cost k=1
Semi cost k=2
1
1
3
2
30
50
121
105
200
200
30
54
135
110
200
200
30
45
125
106
200
200
Fixed cost of in-house produced finshed goods
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
1
195120
167520
124000
1
2
0
0
0
0
0
0
0
1
0
287280
672
2
2
0
0
0
0
0
0
0
1
0
75000
33450
1
3
2
0
0
0
1
2
550
300
0
0
0
0
0
1
0
0
0
2
2
99900
0
88668
0
0
0
0
1
0
0
0
1
3
2
99000
0
90180
Fixed cost of finished goods from purchased Semi
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
Total fixed cost
1
0
0
0
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
1
2
550
300
0
0
0
0
0
1
0
287280
672
2
2
99900
0
88668
0
0
0
0
1
0
75000
33450
1
3
2
99000
0
90180
1
2
0
0
0
0
0
0
0
1
0
837900
1960
2
2
0
0
0
0
0
0
0
1
0
231000
103026
1
3
2
0
0
0
1
0
0
0
1
2
1155
630
0
0
0
0
0
1
0
0
0
2
2
203500
0
180620
0
0
0
0
1
0
0
0
1
3
2
233200
0
212424
1
551214
473244
350300
1
2
1155
630
0
0
0
0
0
1
0
837900
1960
2
2
203500
0
180620
0
0
0
0
1
0
231000
103026
1
3
2
233200
0
212424
1
0
0
2
873000
0
1
195120
167520
124000
Variable cost of in-house produced finshed goods
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
1
551214
473244
350300
Variable cost of finished goods from purchased Semi
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
Total variable cost
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
Semi finished purchase cost
Territories
1
Prod
Factory
k=1
k=2
0
0
0
1
0
0
2
2600
1400
0
0
0
0
0
0
1
0
0
2
1051000
0
Unit Raw mtl purchase cost
Territory j
Unit cost
1
12
2
16
3
19
4
23
5
28
1
376468.1631
2
0
3
0
4
0
5
0
1
8
2
10
3
15
4
21
5
27
1
250978.7754
2
0
3
0
4
0
5
0
1
14
2
23
1
2
0
0
0
0
0
0
0
1
0
59850
140
2
2
0
0
0
0
0
0
0
1
0
17500
7805
1
3
2
0
0
0
0
0
0
1
0
0
2
2
78570
0
0
0
0
1
0
0
1
3
2
94590
0
1
0
54720
216
2
2
33300
0
47618
0
0
0
0
1
0
16000
12042
1
3
2
39600
0
58116
Total raw mtl Purchase cost
Territory j
Unit cost
Unit Raw mtl trans cost
Territory j
Unit cost
Total raw mtl trans cost
Territory j
Unit cost
Semi transportation cost from central to others
Factories
Unit cost
0
0
Transportation Cost of semi-finished products from Central plant to others
Customers
Group
Prod
Factory
l=1
l=2
l=3
0
0
0
0
1
40171.76471
34489.41176
25529.41176
Semifinished trans cost from territories to factories
Factory
k=1
k=2
Territories unit cost
0
25
21
1
29
24
2
18
28
Purchased Semi finished prod transportation cost
From supplier k
Group
Prod
Factory
k=1
k=2
0
0
0
1
0
0
1
2
234
196
0
15
25
19
1
22
16
27
2
18
21
29
0
0
0
0
1
53658
33504
41850
1
2
198
126
0
0
0
0
0
1
40
2
30
3
33
1
43058.82353
2
51900
3
46612.5
4
0
5
0
Unit cost for delivering Finished prod to customer
Customers
Factory
l=1
l=2
l=3
Total Delivery cost to customer
Customers
Group
Prod
Factory
l=1
l=2
l=3
Rework Costs for each product
Prod
Rework cost
Total rework cost
Product
Total cost
Total defect semi-finished products sold for recycling
Territory j
cost
Total revenue
Reworked prod revenue
Selling the mtrl for recycling
Fixed cost
Variable cost
SF purchase
RM Purchase
1
32622.0774
8096650
1284730
32622
1261640
3380173
1928000
376468.1631
2
0
3
0
RM trans
SF trans from central
SF trans
Finished trans
Rework cost
Profit
250978.7754
185485.5882
173590
231115
141571
1484980