Mathematical modeling / English

Transportation Problem - LP Formulation

Joshua Emmanuel · YouTube · 6:40

YouTube · Joshua EmmanuelOriginal video

Selected moment · 0:00 · Chapter markers are time references, not playback progress.

IN THIS VIDEO

01Transportation problem objective
02How to read a transportation matrix
03Network form: nodes and arcs

FROM THE LEARNING NOTES

\begin{array}{c|ccc|c}\text{From}\backslash\text{To}&\text{DC1}&\text{DC2}&\text{DC3}&\text{Supply}\\\hline\text{Boston}&5&6&4&300\\\text{Toronto}&6&3&7&500\\\hline\text{Demand}&200&300&250&\end{array}

How to read a transportation matrix

SELECTED MOMENT

Title and topic introduction

SUGGESTED PREPARATION

Basic idea of minimizing cost in an optimization model
Familiarity with tables/matrices
Elementary notion of variables in algebra
Basic linear programming notation
Reading a cost table
Interpreting a simple network diagram
Linear programming basics
Network flow interpretation
Summation notation for constraints
Basic Linear Programming Concepts
Network Flow Representation

READ & KEEP

The explanation, unpacked.

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Reviewed learning material · Video analysis · English

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This 120-second clip introduces the transportation problem as a linear-programming model for shipping goods from origins to destinations at minimum cost. It first defines the objective, then presents a concrete 2-by-3 example matrix with Boston and Toronto as plants and DC1, DC2, DC3 as distribution centers. The same data are redrawn as a bipartite network whose nodes carry supply or demand and whose arcs carry unit shipping costs. Finally, the clip defines one decision variable per route, counts the variables as $2\times 3=6$ , and introduces the compact indexed notation X_{ij} for larger models. No optimization solution is shown within this excerpt. This 120-second clip formulates a small transportation problem as a linear program. It defines route variables X_{ij}, builds a cost-minimization objective from the given unit shipping costs, writes supply inequalities at the two origins and demand equalities at the three destinations, adds nonnegativity, and then briefly begins an extension involving production costs without completing it. This 120-second clip explains a transportation problem as a linear program. It begins with a two-plant, three-destination network and writes the objective, supply constraints, demand constraints, and nonnegativity conditions. The narration then updates the objective by adding 20 to every Boston route cost and 18 to every Toronto route cost, producing the displayed coefficients 25, 26, 24, 24, 21, and 25. Next it distinguishes balanced from unbalanced transportation problems using the totals 800 and 750, then modifies DC1 demand to 300 so that total demand becomes 850 and exceeds supply by 50. To handle this demand-heavy imbalance, the clip introduces a dummy plant with supply 50 and zero cost, expands the model with $X_{D1}$ , $X_{D2}$ , $X_{D3}$ , and finally presents an alternative approach that changes supply inequalities to equalities because all real supply is used up. This video segment presents the complete linear programming formulation for a transportation problem involving two plants (Boston, Toronto) and three distribution centers. It details the objective function for minimizing shipping costs and the constraints for supply and demand, noting that demand constraints are inequalities because total supply is less than total demand. The segment then demonstrates how to handle a prohibited shipping route (from Toronto to DC3) by adding an explicit constraint setting the corresponding variable to zero, rather than removing it from all equations.

Use the learning inspector for key ideas and moments, or open the reading tabs for the complete notes.

Chapters

0:00Title and topic introduction 0:08Definition of the transportation problem 0:22Example transportation matrix 0:55Network representation with nodes and arcs 1:21Decision variables for each route 1:39Counting variables and indexed notation 2:00Decision variables 2:15Objective function 2:35Supply constraints 3:12Demand constraints 3:35Nonnegativity and complete LP model 3:51Start of production-cost extension 4:00Initial transportation LP and network data 4:18Updating the objective with source-specific costs 4:36Balanced versus unbalanced transportation problems 5:00Increasing DC1 demand and creating a shortage of 50 5:18Dummy-plant reformulation 5:42Alternative direct constraint modification 6:00Complete LP Formulation for Transportation Problem 6:17Handling Prohibited Routes with Zero Constraints

Learning script

Generated from the video's visuals and explanation; not verbatim speech.

The clip opens by naming the subject: the linear programming formulation of the transportation problem.

It then states the core idea: goods are produced at supply locations, called origins, and must be transported to demand locations, called destinations, so that total cost is minimized.

To make the idea concrete, the video introduces a transportation matrix. Rows are origins, columns are destinations, the right-hand column lists supplies, the bottom row lists demands, and the interior cells list unit shipping costs.

In the example, Boston has supply 300 and Toronto has supply 500. The three destinations DC1, DC2, and DC3 have demands 200, 300, and 250. One highlighted cost entry shows that shipping one unit from Boston to DC2 costs $6.

The same data are then redrawn as a network. The two plants become origin nodes, the three distribution centers become destination nodes, and each possible route becomes a directed arc from an origin to a destination.

Node labels carry quantities: supplies at the origins and demands at the destinations. Arc labels carry the unit shipping costs taken from the matrix.

Next the video defines the decision variables. Each route gets one variable measuring how many units are shipped along that route: $X_{B1}$ , $X_{B2}$ , $X_{B3}$ from Boston, and $X_{T1}$ , $X_{T2}$ , $X_{T3}$ from Toronto.

Because there is one variable per origin-destination pair, the total number of variables is the number of origins times the number of destinations. Here that gives $2\times 3=6$ .

Finally, instead of listing every route separately, the clip introduces compact indexed notation: X_{ij} denotes the number of units shipped from plant i to distribution center j. This prepares the transition from a small worked example to a general LP formulation.

The clip opens on a transportation network with two origins, Boston and Toronto, and three destinations, DC1, DC2, and DC3. The narrator defines the decision variables by route: $X_{ij}$ is the number of units shipped from plant $i$ to destination center $j$ , with $i=B$ for Boston and $i=T$ for Toronto, and $j=1,2,3$ for the three DCs. This establishes six variables total, one for each edge in the network.

The next step is the objective function. Because the goal is to minimize total shipping cost, each route variable is multiplied by its unit cost from the table and network diagram. The resulting expression is $\text{Min } 5X_{B1}+6X_{B2}+4X_{B3}+6X_{T1}+3X_{T2}+7X_{T3}$ . The narration stresses that every route cost appears exactly once in the sum.

The model then adds one constraint for each node. At the origin nodes, total outbound shipments cannot exceed available supply. For Boston this gives $X_{B1}+X_{B2}+X_{B3}\le 300$ , and for Toronto it gives $X_{T1}+X_{T2}+X_{T3}\le 500$ . The visual emphasis on outgoing edges makes clear that these are flow-out constraints.

A key modeling point is made explicit: total supply is $800$ while total demand is $750$ , so supply exceeds demand. Therefore the origin constraints are written with $\le$ rather than $=$ , because unused capacity is allowed.

At the destination nodes, the logic changes. Each DC must receive exactly the amount demanded, so inbound flows are set equal to the demand values. This yields $X_{B1}+X_{T1}=200$ , $X_{B2}+X_{T2}=300$ , and $X_{B3}+X_{T3}=250$ . The narrator explains that equality is appropriate because demand must be fully met and there is enough total supply to do so.

Finally, the clip adds nonnegativity restrictions. These may be written by listing all six variables or compactly as $X_{ij}\ge 0$ for all $i,j$ . With the objective, supply constraints, demand constraints, and nonnegativity together, the narrator identifies the complete LP model for the transportation problem.

In the last seconds, the video begins an extension: production costs of $20$ in Boston and $18$ in Toronto are introduced, and the narrator says these should be added to the shipping cost. The revised objective is not completed before the clip ends.

The clip opens on a fully written transportation LP. The decision variable is defined as $X_{ij}$ , the number of units shipped from plant $i$ to distribution center $j$ , with $i\in\{B,T\}$ and $j\in\{1,2,3\}$ . The network diagram and table give Boston supply 300, Toronto supply 500, and demands 200, 300, 250, while the route costs are 5, 6, 4 from Boston and 6, 3, 7 from Toronto.

The first mathematical operation is an update to the objective. The narration says to add 20 to every unit shipped from Boston and 18 to every unit shipped from Toronto. Because the objective is a sum of route cost times shipment quantity, this means adding 20 to each Boston coefficient and 18 to each Toronto coefficient.

Carrying out that arithmetic gives the displayed updated objective $\text{Min }25X_{B1}+26X_{B2}+24X_{B3}+24X_{T1}+21X_{T2}+25X_{T3}$ . The structure of the constraints does not change in this step; only the cost coefficients are revised.

The lesson then shifts from algebraic updating to model classification. It states that if total supply equaled total demand, all constraints could be written as equalities. That situation is named a balanced transportation problem.

Using the numbers on screen, total supply is $300+500=800$ and original total demand is $200+300+250=750$ . Since $800>750$ , the original instance is unbalanced with excess supply. The narration also notes the opposite possibility, where demand exceeds supply.

A new scenario is introduced: DC1 demand rises from 200 to 300. Recomputing totals gives demand $300+300+250=850$ against unchanged supply 800, so now $800<850$ and the shortage is 50 units. This flips the imbalance from supply-heavy to demand-heavy.

To handle the demand-heavy case, the first method adds a dummy plant $D$ with supply 50. The dummy is not real inventory, so its unit and shipping costs are set to zero. In the LP this appears as new variables $X_{D1},X_{D2},X_{D3}$ , new objective terms $+0X_{D1}+0X_{D2}+0X_{D3}$ , and a dummy supply constraint $X_{D1}+X_{D2}+X_{D3}\le 50$ .

The demand constraints are then expanded to include dummy shipments, for example $X_{B1}+X_{T1}+X_{D1}=300$ . Conceptually, any positive dummy shipment records demand that cannot truly be satisfied, because the dummy units do not exist. Nonnegativity still applies to all variables.

The clip then presents a second method without a dummy node. Since demand exceeds supply, the argument is that all real supply will be exhausted, so the supply constraints are changed from inequalities to equalities: $X_{B1}+X_{B2}+X_{B3}=300$ and $X_{T1}+X_{T2}+X_{T3}=500$ .

Finally, because some destination demand cannot be fully met, the demand equalities are to be relaxed as well. The video explicitly announces this change, but the clip ends before the complete final demand-side inequality form is fully shown on screen.

The video displays the complete linear programming model for a transportation problem. The objective is to minimize the total cost: $\text{Min } 25X_{B1} + 26X_{B2} + 24X_{B3} + 24X_{T1} + 21X_{T2} + 25X_{T3}$ . The constraints include supply limits for Boston ( $X_{B1} + X_{B2} + X_{B3} = 300$ ) and Toronto ( $X_{T1} + X_{T2} + X_{T3} = 500$ ), and demand limits for the distribution centers. Because the total supply (800) is less than the total demand (850), the demand constraints are formulated as inequalities, such as $X_{B1} + X_{T1} \leq 200$ for DC1.

The narrator introduces a scenario where shipping from Toronto to DC3 is prohibited, perhaps due to high cost or distance. While one could simply remove the variable $X_{T3}$ from the model, this is inefficient for large models. A better approach is to add an explicit constraint: $X_{T3} = 0$ . This effectively eliminates the route without altering the structure of the other equations.

Knowledge cards

Transportation problem objective

A transportation problem seeks the least-cost way to move products from supply locations (origins) to demand locations (destinations). The clip states this goal before introducing any formulas.

How to read a transportation matrix

In the matrix shown, rows are origins, columns are destinations, the right column gives supplies, the bottom row gives demands, and the interior cells give unit shipping costs. The example has Boston supply 300, Toronto supply 500, and DC demands 200, 300, 250.

\begin{array}{c|ccc|c}\text{From}\backslash\text{To}&\text{DC1}&\text{DC2}&\text{DC3}&\text{Supply}\\\hline\text{Boston}&5&6&4&300\\\text{Toronto}&6&3&7&500\\\hline\text{Demand}&200&300&250&\end{array}

Network form: nodes and arcs

The matrix can be converted into a bipartite network. Circles are nodes: origins on one side and destinations on the other. Directed arrows are arcs representing possible shipping routes. Supplies and demands attach to nodes, while unit costs attach to arcs.

Route-based decision variables

For this 2-by-3 example, each possible route gets its own variable: $X_{B1}$ , $X_{B2}$ , $X_{B3}$ , $X_{T1}$ , $X_{T2}$ , $X_{T3}$ . Each variable means the number of units shipped from that origin to that distribution center.

X_{B1},\ X_{B2},\ X_{B3},\ X_{T1},\ X_{T2},\ X_{T3}

Counting transportation variables

When every origin can ship to every destination, the number of decision variables equals the number of origins times the number of destinations. The example therefore has $2\times 3=6$ variables.

Indexed notation X_ij

For larger problems, the video replaces the explicit list with a compact double-indexed variable. X_{ij} denotes the number of units shipped from plant i to distribution center j.

Route-based decision variables

A transportation model uses one variable for each origin-destination pair. In this example, $X_{ij}$ denotes units shipped from plant $i$ to DC $j$ , with $i\in\{B,T\}$ and $j\in\{1,2,3\}$ , giving six route variables.

X_{ij} = \#\text{ units shipped from Plant } i \text{ to DC } j

Cost-minimization objective

The objective function sums unit shipping cost times shipment quantity over all routes. The coefficients come directly from the cost table and network edges.

\text{Min } 5X_{B1} + 6X_{B2} + 4X_{B3} + 6X_{T1} + 3X_{T2} + 7X_{T3}

Supply constraints are inequalities when supply exceeds demand

Each origin constraint limits total outbound flow by available capacity. Because total supply is larger than total demand, unused capacity is allowed, so the constraints use $\le$ .

X_{B1}+X_{B2}+X_{B3} \le 300,\quad X_{T1}+X_{T2}+X_{T3} \le 500

Demand constraints are equalities

Each destination constraint requires total inbound flow to match exact demand. Equality is used because demand must be fully satisfied.

X_{B1}+X_{T1}=200,\quad X_{B2}+X_{T2}=300,\quad X_{B3}+X_{T3}=250

Nonnegativity closes the LP formulation

All shipment variables must be nonnegative. This can be written variable by variable or compactly with indexed notation.

X_{ij} \ge 0 \quad \text{for all } i,j

Complete transportation LP model

The full linear program combines the minimization objective, origin supply inequalities, destination demand equalities, and nonnegativity restrictions.

\begin{aligned} \text{Min } & 5X_{B1} + 6X_{B2} + 4X_{B3} + 6X_{T1} + 3X_{T2} + 7X_{T3}\\ \text{s.t. } & X_{B1}+X_{B2}+X_{B3} \le 300\\ & X_{T1}+X_{T2}+X_{T3} \le 500\\ & X_{B1}+X_{T1}=200\\ & X_{B2}+X_{T2}=300\\ & X_{B3}+X_{T3}=250\\ & X_{ij} \ge 0 \end{aligned}

Extension begins: adding production costs

Near the end, the video introduces origin-specific production costs of $20$ in Boston and $18$ in Toronto and says they should be added to shipping cost, but the revised objective is not shown completely within this clip.

Transportation problem LP structure

The clip formulates a transportation problem with decision variables $X_{ij}$ , a minimization objective over route costs, supply constraints at plants, demand constraints at destinations, and nonnegativity restrictions. The network picture and the algebra are kept synchronized so each arc corresponds to one variable.

\text{Min }\sum c_{ij}X_{ij}\quad\text{s.t. supply, demand, }X_{ij}\ge 0

Meaning of X_{ij}

$X_{ij}$ is the number of units shipped from plant $i$ to distribution center $j$ . In this example $i$ ranges over Boston and Toronto, while $j$ ranges over DC1, DC2, and DC3.

Source-level cost update in the objective

When an extra per-unit cost is attached to all shipments from one source, every outgoing-route coefficient for that source increases by the same amount. Here Boston gets +20 and Toronto gets +18, producing the updated objective shown on screen.

\text{Min }25X_{B1}+26X_{B2}+24X_{B3}+24X_{T1}+21X_{T2}+25X_{T3}

Balanced transportation problem

A transportation problem is balanced when total supply equals total demand. In that case the model can use equality constraints on both the supply and demand sides.

\sum_i s_i=\sum_j d_j

Unbalanced transportation problem

The example is unbalanced because aggregate supply and aggregate demand differ. The clip first shows supply exceeding demand, then creates the opposite case by raising DC1 demand.

800>750,\quad 800<850

Dummy plant for demand-heavy imbalance

When demand exceeds supply, a dummy plant can be added with supply equal to the shortage. Its cost is zero because the dummy units do not really exist; shipments from it represent unmet demand.

X_{D1}+X_{D2}+X_{D3}\le 50,\quad c_{Dj}=0

Expanded demand constraints with dummy variables

After adding the dummy source, each destination equation includes a dummy term. This keeps the algebraic structure of a transportation model while allowing part of the demand to be satisfied only artificially.

X_{B1}+X_{T1}+X_{D1}=300

Alternative direct reformulation

Instead of a dummy node, the clip proposes changing the supply constraints to equalities because all real supply is used when demand exceeds supply. It also states that the demand equalities must then be relaxed, though the final demand-side form is not fully shown before the clip ends.

X_{B1}+X_{B2}+X_{B3}=300,\quad X_{T1}+X_{T2}+X_{T3}=500

Transportation Problem LP Model

The standard linear programming formulation for a transportation problem minimizes total shipping cost subject to supply and demand constraints. When total supply is less than total demand, demand constraints are written as inequalities ( $\leq$ ) to reflect that not all demand can be met.

\begin{aligned} &\text{Min } \sum_{i}\sum_{j} c_{ij}X_{ij} \\ &\text{Subject to:} \\ &\sum_{j} X_{ij} = S_i \quad \forall i \\ &\sum_{i} X_{ij} \leq D_j \quad \forall j \\ &X_{ij} \geq 0 \end{aligned}

Handling Prohibited Routes in LP

To prohibit a specific route in a transportation model, instead of deleting the corresponding variable from all equations, it is more efficient to add a constraint setting that variable to zero. This maintains the model's structural consistency while enforcing the restriction.

X_{T3} = 0

Detailed learning notes

Explore conditions, steps and evidence. Supplementary explanations are labeled separately from content shown in the video.

Symbols · 52

$X_{B1}$

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide lists $X_{B1}$ = # units shipped from Boston to DC 1.
Audio
Observation
The narrator says, “Let XB1 equal the number of units shipped from Boston to DC1.”

Symbol

$X_{B1}$

Meaning

Number of units shipped from Boston to distribution center DC1.

Domain

Nonnegative shipment quantity in the transportation model.

$X_{B2}$

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide lists $X_{B2}$ = # units shipped from Boston to DC 2.
Audio
Observation
The narrator says, “Then XB2 from Boston to DC2.”

Symbol

$X_{B2}$

Meaning

Number of units shipped from Boston to distribution center DC2.

Domain

Nonnegative shipment quantity in the transportation model.

$X_{B3}$

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide lists $X_{B3}$ = # units shipped from Boston to DC 3.
Audio
Observation
The narrator says, “and XB3 to DC3.”

Symbol

$X_{B3}$

Meaning

Number of units shipped from Boston to distribution center DC3.

Domain

Nonnegative shipment quantity in the transportation model.

$X_{T1}$

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide lists $X_{T1}$ = # units shipped from Toronto to DC 1.
Audio
Observation
The narrator says, “Similarly for Toronto we can use XT1, XT2 and XT3.”

Symbol

$X_{T1}$

Meaning

Number of units shipped from Toronto to distribution center DC1.

Domain

Nonnegative shipment quantity in the transportation model.

$X_{T2}$

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide lists $X_{T2}$ = # units shipped from Toronto to DC 2.
Audio
Observation
The narrator says, “Similarly for Toronto we can use XT1, XT2 and XT3.”

Symbol

$X_{T2}$

Meaning

Number of units shipped from Toronto to distribution center DC2.

Domain

Nonnegative shipment quantity in the transportation model.

$X_{T3}$

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide lists $X_{T3}$ = # units shipped from Toronto to DC 3.
Audio
Observation
The narrator says, “Similarly for Toronto we can use XT1, XT2 and XT3.”

Symbol

$X_{T3}$

Meaning

Number of units shipped from Toronto to distribution center DC3.

Domain

Nonnegative shipment quantity in the transportation model.

X_{ij}

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide shows X_{ij} = # units shipped from Plant i to DC j.
Audio
Observation
The narrator says, “Let Xij equal the number of units...” before the clip ends mid-sentence.

Uncertainties

The spoken definition is cut off at the end of the clip, but the on-screen formula is complete.

Symbol

X_{ij}

Meaning

General decision variable for the number of units shipped from plant i to distribution center j.

Domain

Nonnegative shipment quantity; i indexes origins/plants and j indexes destinations/distribution centers.

300

Clear evidence

Shown in the video

Evidence

Formula
Observation
The transportation matrix shows Boston with Supply = 300.
Audio
Observation
The narrator states that Boston has a supply or capacity of 300 units.

Symbol

300

Meaning

Supply/capacity at the Boston origin.

Domain

Units of product available to ship from Boston.

500

Clear evidence

Shown in the video

Evidence

Formula
Observation
The transportation matrix shows Toronto with Supply = 500.
Audio
Observation
The narrator states that Toronto has a supply or capacity of 500 units.

Symbol

500

Meaning

Supply/capacity at the Toronto origin.

Domain

Units of product available to ship from Toronto.

200

Clear evidence

Shown in the video

Evidence

Formula
Observation
The transportation matrix shows Demand for DC1 = 200.
Audio
Observation
The narrator states that the destinations have demands of 200, 300, and 250 respectively.

Symbol

200

Meaning

Demand at destination DC1.

Domain

Units of product required at DC1.

300

Clear evidence

Shown in the video

Evidence

Formula
Observation
The transportation matrix shows Demand for DC2 = 300.
Audio
Observation
The narrator states that the destinations have demands of 200, 300, and 250 respectively.

Symbol

300

Meaning

Demand at destination DC2.

Domain

Units of product required at DC2.

250

Clear evidence

Shown in the video

Evidence

Formula
Observation
The transportation matrix shows Demand for DC3 = 250.
Audio
Observation
The narrator states that the destinations have demands of 200, 300, and 250 respectively.

Symbol

250

Meaning

Demand at destination DC3.

Domain

Units of product required at DC3.

Knowledge points · 21

Goal of the transportation problem

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narrator defines the goal of a transportation problem as producing products at supply locations or origins and transporting them to destinations where they are demanded at minimum costs.
Diagram
Observation
The slide shows Supply locations (Origins) pointing to Demand locations (Destinations) with the label Minimum Costs.

Definition

Explanation

The transportation problem asks how to move goods from origins to destinations so that all demand is met while total shipping cost is minimized. In this clip the origins are supply locations and the destinations are demand locations.

Conditions

There are supply locations/origins.
There are demand locations/destinations.
The objective is minimum cost.

Transportation matrix data layout

Clear evidence

Shown in the video

Evidence

Formula
Observation
A table labeled From/To lists rows Boston and Toronto, columns DC1, DC2, DC3, a Supply column, and a Demand row.
Audio
Observation
The narrator says they will write an LP formulation for this transportation matrix and then reads out the origins, supplies, destinations, demands, and costs.

Definition

Explanation

The matrix organizes the problem into origins as rows, destinations as columns, unit shipping costs in the interior cells, supply capacities in the rightmost column, and demands in the bottom row. For this example, Boston has supply 300, Toronto has supply 500, and DC1, DC2, DC3 have demands 200, 300, 250 respectively.

Conditions

Rows represent origins/plants.
Columns represent destinations/distribution centers.
Interior entries are unit shipping costs.
Right column entries are supplies.
Bottom row entries are demands.

Prerequisites

Goal of the transportation problem

Network representation of a transportation problem

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narrator says, “These circles are called nodes for the origins... and these are the nodes for the destinations,” then later, “The arrows from the origins to destinations are called arcs.”
Diagram
Observation
The slide draws two origin nodes labeled Boston and Toronto, three destination nodes labeled DC 1, DC 2, DC 3, and directed arrows between every origin-destination pair with costs on the arrows.

Definition

Explanation

A transportation problem can be drawn as a bipartite network: origin nodes on one side, destination nodes on the other, and directed arcs from each origin to each destination. Node labels carry supply or demand values, and arc labels carry unit shipping costs.

Conditions

Origins and destinations are represented as separate node sets.
Each feasible route is represented by an arc.
Arc direction is from origin to destination.

Prerequisites

Transportation matrix data layout

Explicit decision variables for a small transportation model

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide lists $X_{B1}$ , $X_{B2}$ , $X_{B3}$ , $X_{T1}$ , $X_{T2}$ , $X_{T3}$ with definitions as units shipped from each origin to each DC.
Audio
Observation
The narrator explicitly defines each variable in words.

Definition

Explanation

For a concrete small instance, each possible shipment route gets its own variable. Here the six variables represent the quantities shipped from Boston to DC1, DC2, DC3 and from Toronto to DC1, DC2, DC3.

Conditions

Each origin-destination pair corresponds to one variable.
The variable measures units shipped along that route.

Prerequisites

Network representation of a transportation problem

Counting decision variables in a transportation model

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narrator says the number of decision variables is simply the number of origins multiplied by the number of destinations.
Formula
Observation
The slide shows $2 \times 3 = 6$ .

Formula

Explanation

If every origin can ship to every destination, the number of decision variables equals the number of origins times the number of destinations. In this example there are 2 origins and 3 destinations, so there are 6 variables.

Conditions

The model includes one variable for each origin-destination route.
All origin-destination pairs are allowed in the count shown here.

Prerequisites

Explicit decision variables for a small transportation model

Indexed general form of transportation decision variables

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narrator introduces a short form for when listing every variable is inconvenient or there are too many variables.
Formula
Observation
The slide shows X_{ij} = # units shipped from Plant i to DC j.

Uncertainties

The spoken sentence is truncated at the end of the clip, but the displayed definition is complete.

Definition

Explanation

Instead of naming each route separately, one can use a double-indexed variable X_{ij}, where i identifies the plant/origin and j identifies the distribution center/destination. This compact notation scales to larger transportation problems.

Conditions

i indexes origins/plants.
j indexes destinations/distribution centers.
X_{ij} denotes the shipment quantity on route (i,j).

Prerequisites

Explicit decision variables for a small transportation model
Counting decision variables in a transportation model

Decision variables for the transportation model

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide defines X_{ij} = # units shipped from Plant i to DC j and lists the six specific variables $X_{B1}$ , $X_{B2}$ , $X_{B3}$ , $X_{T1}$ , $X_{T2}$ , $X_{T3}$ .
Audio
Observation
The narrator explains that i indexes Boston and Toronto while j indexes DC1, DC2, and DC3.

Definition

Explanation

The transportation problem is modeled with one variable for each origin-destination pair. Here there are two origins (Boston, Toronto) and three destinations (DC1, DC2, DC3), so the model uses six shipment variables. The subscript notation encodes the route: the first index is the plant and the second index is the destination center.

Formula

X_{ij} = \#\text{ units shipped from Plant } i \text{ to DC } j

Conditions

$i \in$ \{B,T\}
$j \in$ \{1,2,3\}
Each variable represents a nonnegative shipment quantity on a specific route.

Objective function minimizing total shipping cost

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide writes Min $5X_{B1} + 6X_{B2} + 4X_{B3} + 6X_{T1} + 3X_{T2} + 7X_{T3}$ .
Audio
Observation
The narrator says the objective is to minimize cost and reads the first terms explicitly.
Diagram
Observation
The cost matrix and network edges show the same coefficients 5, 6, 4, 6, 3, 7.

Formula

Explanation

The objective function sums the unit shipping cost on each route multiplied by the corresponding shipment variable. The coefficients come directly from the cost table and the labeled network edges. The video emphasizes that every route cost is included once in the total.

Formula

\text{Min } 5X_{B1} + 6X_{B2} + 4X_{B3} + 6X_{T1} + 3X_{T2} + 7X_{T3}

Conditions

The model minimizes total transportation cost.
Coefficients are per-unit shipping costs for each origin-destination pair.

Prerequisites

Decision variables for the transportation model

Supply constraints at origin nodes

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide writes Subject to: $X_{B1}+X_{B2}+X_{B3} \le 300$ and $X_{T1}+X_{T2}+X_{T3} \le 500$ .
Audio
Observation
The narrator says the total amount shipped from each plant cannot exceed its capacity.
Diagram
Observation
Red circles highlight Boston and Toronto while outgoing edges are highlighted during the explanation.

Formula

Explanation

For each plant, the sum of shipments leaving that plant must not exceed the plant's available supply. Because total supply exceeds total demand, these constraints are inequalities rather than equalities.

Formula

X_{B1}+X_{B2}+X_{B3} \le 300,\quad X_{T1}+X_{T2}+X_{T3} \le 500

Conditions

One constraint per origin node.
Right-hand sides are the plant capacities.
Inequality form is used because supply is not required to be exhausted.

Prerequisites

Decision variables for the transportation model

Demand constraints at destination nodes

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide writes $X_{B1}+X_{T1}=200$ , $X_{B2}+X_{T2}=300$ , $X_{B3}+X_{T3}=250$ .
Audio
Observation
The narrator says equality is used because demand must be met and there is enough supply to meet it.
Diagram
Observation
Incoming edges to each DC are highlighted while the corresponding demand value is circled.

Formula

Explanation

For each destination center, the total incoming shipments must exactly satisfy the stated demand. This gives one equality constraint per DC.

Formula

X_{B1}+X_{T1}=200,\quad X_{B2}+X_{T2}=300,\quad X_{B3}+X_{T3}=250

Conditions

One constraint per destination node.
Equality form is used because demand must be fully satisfied.
Feasibility relies on total supply being at least total demand.

Prerequisites

Decision variables for the transportation model

Nonnegativity constraints

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide writes $X_{B1}$ , $X_{B2}$ , $X_{B3}$ , $X_{T1}$ , $X_{T2}$ , $X_{T3} \ge 0$ or X_{ij} $\ge 0$ .
Audio
Observation
The narrator says nonnegativity can be listed variable by variable or written compactly as X_{ij} $\ge 0$ for all i and j.

Formula

Explanation

Shipment quantities cannot be negative. The model may state this either by listing all six variables or by using the compact indexed form for all routes.

Formula

X_{ij} \ge 0 \quad \text{for all } i,j

Conditions

Applies to every decision variable.
Required for a valid linear programming formulation.

Prerequisites

Decision variables for the transportation model

Complete linear programming formulation of the transportation problem

Clear evidence

Shown in the video

Evidence

Formula
Observation
A red box encloses the full model: objective, supply constraints, demand constraints, and nonnegativity.
Audio
Observation
The narrator concludes, 'And that's the LP model.'

Method

Explanation

The complete LP model consists of a minimization objective over all route variables, supply inequalities at origins, demand equalities at destinations, and nonnegativity restrictions. The video presents this as the standard formulation workflow for a transportation problem.

Formula

\begin{aligned} \text{Min } & 5X_{B1} + 6X_{B2} + 4X_{B3} + 6X_{T1} + 3X_{T2} + 7X_{T3}\\ \text{s.t. } & X_{B1}+X_{B2}+X_{B3} \le 300\\ & X_{T1}+X_{T2}+X_{T3} \le 500\\ & X_{B1}+X_{T1}=200\\ & X_{B2}+X_{T2}=300\\ & X_{B3}+X_{T3}=250\\ & X_{ij} \ge 0 \end{aligned}

Conditions

Two origins and three destinations.
Total supply exceeds total demand.
All costs are given per unit shipped on each route.

Prerequisites

Decision variables for the transportation model
Objective function minimizing total shipping cost
Supply constraints at origin nodes
Demand constraints at destination nodes
Nonnegativity constraints

Claims and conditions · 7

Why supply constraints use inequalities

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narrator says total supply exceeds total demand, so capacity will not be exhausted.
Diagram
Observation
A red comparison at the bottom shows $800 > 750$ .

Proposition

Statement

When total supply exceeds total demand, origin-side constraints are written with $\le$ rather than = because not all capacity must be used.

Hypotheses

Total supply is greater than total demand.
Demand constraints require exact satisfaction.

Quantifiers

For the given transportation instance with supply 800 and demand 750.

Why demand constraints use equalities

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narrator says equality is used because demand must be met and there is enough supply to meet it.
Formula
Observation
The destination constraints are written as equalities with right-hand sides 200, 300, and 250.

Proposition

Statement

Destination-side constraints are written as equalities when the demand at each destination must be fully satisfied.

Hypotheses

Each DC has a fixed required demand.
There is sufficient total supply to satisfy all demands.

Quantifiers

For each destination node j in the model.

Equality constraints in the balanced case

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narration states that if total supply were to equal total demand, equality could be used for all constraints.
Formula
Observation
The slide text reads If Total Supply = Total Demand.

Proposition

Statement

If total supply equals total demand in a transportation problem, then the model can use equality constraints for all supply and demand constraints.

Hypotheses

The problem is a transportation problem.
Total supply equals total demand.

Quantifiers

For all supply and demand constraints in the balanced case.

The original numerical example is supply-heavy

Clear evidence

Shown in the video

Evidence

Diagram
Observation
The slide displays $800 > 750$ .
Audio
Observation
The narration says this problem is an unbalanced transportation example where supply exceeds demand.

Proposition

Statement

In the original numerical example, total supply 800 exceeds total demand 750, so the problem is unbalanced with excess supply.

Hypotheses

Boston supply is 300.
Toronto supply is 500.
DC demands are 200, 300, and 250.

Quantifiers

For the displayed numerical instance.

Raising DC1 demand creates a demand-heavy imbalance

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narration says suppose demand at distribution center one increases to 300 units.
Diagram
Observation
The slide later shows $800 < 850$ .
Audio
Observation
The narration says demand exceeds supply by 50 units.

Proposition

Statement

After DC1 demand increases from 200 to 300, total demand becomes 850 while total supply remains 800, so demand exceeds supply by 50 units.

Hypotheses

Original total supply is 800.
Original DC1 demand is 200.
DC1 demand is changed to 300.

Quantifiers

For the modified numerical instance.

Dummy shipments have zero cost

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narration says these 50 units do not exist, so their cost per unit is zero and their shipping cost is also zero since they are not going to be shipped at all.
Diagram
Observation
The dummy node is labeled $0.
Formula
Observation
The expanded objective includes + $0X_{D1}+0X_{D2}+0X_{D3}$ .

Proposition

Statement

In the dummy-plant formulation, the dummy supply represents nonexistent units, so the associated unit and shipping costs are set to zero.

Hypotheses

A dummy plant is introduced to cover a shortage of 50 units.
Dummy shipments stand for unmet demand rather than real transportation.

Quantifiers

For all routes from the dummy plant to the distribution centers.

All real supply is exhausted when demand exceeds supply

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narration says since demand exceeds supply, all available supply will be used up.
Formula
Observation
The supply constraints are shown being changed from $\le$ to =.

Uncertainties

The clip states the conclusion verbally and shows the supply-side change, but does not provide a formal proof.

Proposition

Statement

In the demand-exceeds-supply scenario, the alternative formulation treats all available real supply as fully used, so the supply constraints become equalities.

Hypotheses

Total demand exceeds total supply.
No dummy node is introduced in this alternative formulation.

Quantifiers

For the real plants in the modified example.

Derivations and proofs · 6

Derivation of the example’s variable count

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narrator explains that the number of decision variables is the number of origins multiplied by the number of destinations.
Formula
Observation
The slide displays $2 \times 3 = 6$ next to the six listed variables.

Intuitive argument

Steps

Explanation
Start from the rule stated in the video for counting route variables.
Justification
Directly stated by the narrator.
Shown in the video
Expression
$2\times 3=6$
Explanation
Substitute the example’s 2 origins and 3 destinations into the rule.
Justification
The matrix and network both show 2 origins and 3 destinations.
Shown in the video

Conclusion

This transportation instance has 6 decision variables, matching the six route-specific variables listed on screen.

Step-by-step construction of the transportation LP

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narrator proceeds step by step from decision variables to objective function, then constraints, then nonnegativity, and finally states that this is the LP model.
Formula
Observation
Each stage appears sequentially on the slide in the same order.

Intuitive argument

Steps

Expression
$X_{ij} = \#\text{ units shipped from Plant } i \text{ to DC } j$
Explanation
Define one decision variable for each origin-destination route.
Justification
This is the standard way to represent shipment quantities in a transportation model.
Shown in the video
Expression
$\text{Min } 5X_{B1} + 6X_{B2} + 4X_{B3} + 6X_{T1} + 3X_{T2} + 7X_{T3}$
Explanation
Multiply each route variable by its unit shipping cost and sum all terms to form the objective.
Justification
The objective is to minimize total transportation cost, and the coefficients are read from the cost table and network diagram.
Shown in the video
Expression
$X_{B1}+X_{B2}+X_{B3} \le 300$
Explanation
Write the Boston supply constraint by summing all shipments leaving Boston.
Justification
Total outbound flow from an origin cannot exceed its available capacity.
Shown in the video
Expression
$X_{T1}+X_{T2}+X_{T3} \le 500$
Explanation
Write the Toronto supply constraint by summing all shipments leaving Toronto.
Justification
Total outbound flow from an origin cannot exceed its available capacity.
Shown in the video
Expression
$X_{B1}+X_{T1}=200$
Explanation
Write the DC1 demand constraint by summing all shipments entering DC1.
Justification
Each destination's demand must be met exactly.
Shown in the video
Expression
$X_{B2}+X_{T2}=300$
Explanation
Write the DC2 demand constraint by summing all shipments entering DC2.
Justification
Each destination's demand must be met exactly.
Shown in the video
Expression
$X_{B3}+X_{T3}=250$
Explanation
Write the DC3 demand constraint by summing all shipments entering DC3.
Justification
Each destination's demand must be met exactly.
Shown in the video
Expression
$X_{ij} \ge 0$
Explanation
Add nonnegativity restrictions for all route variables.
Justification
Shipment quantities cannot be negative in a physical transportation model.
Shown in the video

Conclusion

The resulting linear program is a minimization objective subject to origin supply inequalities, destination demand equalities, and nonnegativity constraints.

Derivation of the updated objective function

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narration explains adding 20 to Boston route costs and 18 to Toronto route costs.
Formula
Observation
The original and updated objectives are both shown on screen.

Numerical verification

Steps

Expression
$5X_{B1}+6X_{B2}+4X_{B3}+6X_{T1}+3X_{T2}+7X_{T3}$
Explanation
Start from the original transportation cost objective shown on the slide.
Justification
Observed directly from the initial formula display.
Shown in the video
Expression
$(5+20)X_{B1}+(6+20)X_{B2}+(4+20)X_{B3}$
Explanation
Add 20 to each Boston-route coefficient because the extra cost applies to every unit shipped from Boston.
Justification
Stated in the audio and supported by the $20 label on Boston.
Shown in the video
Expression
$(6+18)X_{T1}+(3+18)X_{T2}+(7+18)X_{T3}$
Explanation
Add 18 to each Toronto-route coefficient because the extra cost applies to every unit shipped from Toronto.
Justification
Stated in the audio and supported by the $18 label on Toronto.
Shown in the video
Expression
$25X_{B1}+26X_{B2}+24X_{B3}+24X_{T1}+21X_{T2}+25X_{T3}$
Explanation
Simplify the arithmetic to obtain the updated objective shown on screen.
Justification
Direct numerical simplification of the previous line.
Shown in the video

Conclusion

The updated objective is Min $25X_{B1}+26X_{B2}+24X_{B3}+24X_{T1}+21X_{T2}+25X_{T3}$ .

Checking balance in the original instance

Clear evidence

Shown in the video

Evidence

Formula
Observation
The table lists supplies 300 and 500 and demands 200, 300, and 250.
Diagram
Observation
The slide later displays $800 > 750$ .

Numerical verification

Steps

Expression
$300+500=800$
Explanation
Sum the supplies at Boston and Toronto.
Justification
Read from the supply column of the table.
Shown in the video
Expression
$200+300+250=750$
Explanation
Sum the demands at DC1, DC2, and DC3.
Justification
Read from the demand row of the table.
Shown in the video
Expression
$800>750$
Explanation
Compare total supply and total demand.
Justification
Numerical comparison of the two totals.
Shown in the video

Conclusion

The original instance is unbalanced with supply exceeding demand by 50 units.

Checking balance after increasing DC1 demand

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narration says DC1 demand increases to 300 units.
Diagram
Observation
The slide later displays $800 < 850$ .

Numerical verification

Steps

Expression
$300+500=800$
Explanation
Total supply remains unchanged.
Justification
The plant supplies are not altered in the new scenario.
Shown in the video
Expression
$300+300+250=850$
Explanation
Recompute total demand after DC1 demand changes from 200 to 300.
Justification
Uses the updated DC1 demand stated in the audio and shown on screen.
Shown in the video
Expression
$800<850$
Explanation
Compare the unchanged supply total with the new demand total.
Justification
Numerical comparison.
Shown in the video
Expression
$850-800=50$
Explanation
Compute the shortage by which demand exceeds supply.
Justification
Difference of the two totals.
Derived from the video

Conclusion

The modified instance is unbalanced with demand exceeding supply by 50 units.

Expanding the LP with a dummy plant

Clear evidence

Shown in the video

Evidence

Diagram
Observation
A dummy node with supply 50 and cost $0 is added.
Formula
Observation
The expanded model shows new variables $X_{D1}$ , $X_{D2}$ , $X_{D3}$ , new objective terms, and a dummy supply constraint.

Uncertainties

The clip visually highlights the dummy additions but does not separately prove that the expanded model is equivalent to the original intent.

Visual argument

Steps

Expression
$\text{Add } D \text{ with supply } 50$
Explanation
Introduce a dummy source whose capacity equals the shortage.
Justification
Stated in the audio and shown in the network diagram.
Shown in the video
Expression
$X_{D1},\;X_{D2},\;X_{D3}$
Explanation
Add decision variables for shipments from the dummy source to each distribution center.
Justification
Shown in the expanded variable list and constraints.
Shown in the video
Expression
$+0X_{D1}+0X_{D2}+0X_{D3}$
Explanation
Append zero-cost terms to the objective function for the dummy routes.
Justification
Explicitly shown in the updated objective and explained in the audio.
Shown in the video
Expression
$X_{D1}+X_{D2}+X_{D3}\le 50$
Explanation
Add a supply constraint for the dummy node.
Justification
Shown on the slide as (Dummy's Supply).
Shown in the video
Expression
$X_{B1}+X_{T1}+X_{D1}=300,\; X_{B2}+X_{T2}+X_{D2}=300,\; X_{B3}+X_{T3}+X_{D3}=250$
Explanation
Update the demand constraints to include dummy shipments.
Justification
Shown in the expanded demand-constraint block.
Shown in the video

Conclusion

The dummy-plant formulation turns the demand-heavy instance into an expanded transportation model with zero-cost artificial supply representing unmet demand.

Worked examples · 6

Worked setup of a 2-by-3 transportation problem

Clear evidence

Shown in the video

Evidence

Formula
Observation
The matrix gives Boston supply 300, Toronto supply 500, demands 200, 300, 250, and cost entries 5,6,4; 6,3,7.
Audio
Observation
The narrator reads the data and then converts it into a network and decision-variable list.

Uncertainties

The clip only formulates the model setup; it does not solve for optimal shipments.

Problem

Given two plants, Boston and Toronto, and three distribution centers, DC1, DC2, DC3, set up the transportation problem data and decision variables for an LP formulation.

Given

Boston supply = 300.
Toronto supply = 500.
DC1 demand = 200.
DC2 demand = 300.
DC3 demand = 250.
Unit shipping costs: Boston→DC1 = 5, Boston→DC2 = 6, Boston→DC3 = 4, Toronto→DC1 = 6, Toronto→DC2 = 3, Toronto→DC3 = 7.

Goal

Represent the instance as a transportation matrix, then as a network, and define the decision variables needed for an LP formulation.

Steps

Expression
$\begin{array}{c|ccc|c}\text{From}\backslash\text{To}&\text{DC1}&\text{DC2}&\text{DC3}&\text{Supply}\\\hline\text{Boston}&5&6&4&300\\\text{Toronto}&6&3&7&500\\\hline\text{Demand}&200&300&250&\end{array}$
Explanation
Place unit costs in the interior, supplies in the right column, and demands in the bottom row.
Justification
This is the matrix format explicitly shown and described in the video.
Shown in the video
Expression
$\text{Nodes: Boston, Toronto, DC1, DC2, DC3; Arcs: }B\to DC1,\ B\to DC2,\ B\to DC3,\ T\to DC1,\ T\to DC2,\ T\to DC3$
Explanation
Convert the matrix into a bipartite network with one node per origin and destination and one directed arc per route.
Justification
The narrator draws exactly this network after introducing nodes and arcs.
Shown in the video
Expression
$X_{B1},X_{B2},X_{B3},X_{T1},X_{T2},X_{T3}$
Explanation
Define one variable for each route as the number of units shipped on that route.
Justification
The slide and narration explicitly assign these meanings.
Shown in the video
Expression
$2\times 3=6$
Explanation
Count the variables by multiplying the number of origins by the number of destinations.
Justification
The narrator states this counting rule and the slide displays the arithmetic.
Shown in the video
Explanation
Replace the route-by-route listing with a compact indexed notation.
Justification
The final slide introduces this short form for larger models.
Shown in the video

Answer

The instance is represented by the given $2\times 3$ cost/supply/demand matrix, equivalently by a bipartite network with 5 nodes and 6 arcs, and the LP uses 6 decision variables, either listed individually or written generally as X_{ij}.

Verification

The six listed variables correspond one-to-one with the six arcs in the network, and the count $2\times 3=6$ matches the number of origin-destination pairs in the matrix.

Worked formulation of a Boston-Toronto transportation instance

Clear evidence

Shown in the video

Evidence

Diagram
Observation
The slide presents a concrete two-origin, three-destination instance with cost table, supplies, demands, and network graph.
Audio
Observation
The narrator formulates this exact instance as an LP model.

Uncertainties

The final extension with production costs begins but is not completed within the clip.

Problem

Formulate a transportation problem with two plants, Boston and Toronto, and three destination centers, DC1, DC2, and DC3, using the given unit shipping costs, supplies, and demands.

Given

Unit shipping costs: Boston to DC1 = 5, Boston to DC2 = 6, Boston to DC3 = 4; Toronto to DC1 = 6, Toronto to DC2 = 3, Toronto to DC3 = 7.
Supplies: Boston = 300, Toronto = 500.
Demands: DC1 = 200, DC2 = 300, DC3 = 250.
Total supply = 800 and total demand = 750.

Goal

Write the complete linear programming model for minimizing total shipping cost.

Steps

Expression
$X_{ij} = \#\text{ units shipped from Plant } i \text{ to DC } j$
Explanation
Introduce six route variables: $X_{B1}$ , $X_{B2}$ , $X_{B3}$ , $X_{T1}$ , $X_{T2}$ , $X_{T3}$ .
Justification
Each possible shipment route needs its own decision variable.
Shown in the video
Expression
$\text{Min } 5X_{B1} + 6X_{B2} + 4X_{B3} + 6X_{T1} + 3X_{T2} + 7X_{T3}$
Explanation
Use the given unit costs as coefficients in the objective function.
Justification
Total cost is the sum of route cost times route shipment quantity.
Shown in the video
Expression
$X_{B1}+X_{B2}+X_{B3} \le 300$
Explanation
Constrain total shipments out of Boston by Boston's supply.
Justification
Outflow from an origin cannot exceed available supply.
Shown in the video
Expression
$X_{T1}+X_{T2}+X_{T3} \le 500$
Explanation
Constrain total shipments out of Toronto by Toronto's supply.
Justification
Outflow from an origin cannot exceed available supply.
Shown in the video
Expression
$X_{B1}+X_{T1}=200$
Explanation
Require total inflow to DC1 to equal its demand.
Justification
Demand must be fully satisfied at each destination.
Shown in the video
Expression
$X_{B2}+X_{T2}=300$
Explanation
Require total inflow to DC2 to equal its demand.
Justification
Demand must be fully satisfied at each destination.
Shown in the video
Expression
$X_{B3}+X_{T3}=250$
Explanation
Require total inflow to DC3 to equal its demand.
Justification
Demand must be fully satisfied at each destination.
Shown in the video
Expression
$X_{ij} \ge 0$
Explanation
Impose nonnegativity on all route variables.
Justification
Negative shipments are not meaningful.
Shown in the video

Answer

The LP model is: Min $5X_{B1}+6X_{B2}+4X_{B3}+6X_{T1}+3X_{T2}+7X_{T3}$ subject to $X_{B1}+X_{B2}+X_{B3}\le300$ , $X_{T1}+X_{T2}+X_{T3}\le500$ , $X_{B1}+X_{T1}=200$ , $X_{B2}+X_{T2}=300$ , $X_{B3}+X_{T3}=250$ , and X_{ij} $\ge0$ .

Verification

The coefficients match the displayed cost table and network edge labels; the right-hand sides match the displayed supplies and demands; the inequality/equality pattern matches the narrator's explanation that supply exceeds demand while demand must be met.

Initial transportation LP before cost update

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide shows the full initial LP with objective, supply constraints, demand constraints, and nonnegativity.
Diagram
Observation
The network diagram shows Boston and Toronto connected to DC1, DC2, DC3 with route costs.

Problem

Formulate the transportation problem for two plants and three distribution centers using the displayed costs, supplies, and demands.

Given

Boston supply = 300.
Toronto supply = 500.
DC1 demand = 200.
DC2 demand = 300.
DC3 demand = 250.
Route costs: Boston→DC1 = 5, Boston→DC2 = 6, Boston→DC3 = 4, Toronto→DC1 = 6, Toronto→DC2 = 3, Toronto→DC3 = 7.

Goal

Write the linear programming model.

Steps

Expression
$\text{Min } 5X_{B1}+6X_{B2}+4X_{B3}+6X_{T1}+3X_{T2}+7X_{T3}$
Explanation
Use route costs as coefficients of the shipment variables in the objective.
Justification
Directly read from the displayed model.
Shown in the video
Expression
$X_{B1}+X_{B2}+X_{B3}\le 300$
Explanation
Constrain total shipments out of Boston by its supply.
Justification
Shown as (Boston's Supply).
Shown in the video
Expression
$X_{T1}+X_{T2}+X_{T3}\le 500$
Explanation
Constrain total shipments out of Toronto by its supply.
Justification
Shown as (Toronto's Supply).
Shown in the video
Expression
$X_{B1}+X_{T1}=200$
Explanation
Meet DC1 demand exactly in the initial formulation.
Justification
Shown as (DC1's Demand).
Shown in the video
Expression
$X_{B2}+X_{T2}=300$
Explanation
Meet DC2 demand exactly in the initial formulation.
Justification
Shown as (DC2's Demand).
Shown in the video
Expression
$X_{B3}+X_{T3}=250$
Explanation
Meet DC3 demand exactly in the initial formulation.
Justification
Shown as (DC3's Demand).
Shown in the video
Expression
$X_{ij}\ge 0$
Explanation
Impose nonnegativity on all shipment variables.
Justification
Shown on the slide and stated in the audio.
Shown in the video

Answer

The initial LP is Min $5X_{B1}+6X_{B2}+4X_{B3}+6X_{T1}+3X_{T2}+7X_{T3}$ subject to the displayed supply, demand, and nonnegativity constraints.

Verification

The totals implied by the data are supply 800 and demand 750, matching the later on-screen comparison $800 > 750$ .

Updating the objective after source-specific cost increases

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narration says to add 20 to Boston product costs and 18 to Toronto product costs.
Formula
Observation
The updated objective is displayed as Min $25X_{B1}+26X_{B2}+24X_{B3}+24X_{T1}+21X_{T2}+25X_{T3}$ .

Problem

Adjust the transportation objective when each unit shipped from Boston gains an extra cost of 20 and each unit shipped from Toronto gains an extra cost of 18.

Given

Original Boston route costs are 5, 6, 4.
Original Toronto route costs are 6, 3, 7.
Extra Boston cost = 20 per unit.
Extra Toronto cost = 18 per unit.

Goal

Write the updated objective function.

Steps

Expression
$5+20=25,\;6+20=26,\;4+20=24$
Explanation
Add 20 to each Boston-route coefficient.
Justification
The extra cost applies uniformly to all Boston shipments.
Shown in the video
Expression
$6+18=24,\;3+18=21,\;7+18=25$
Explanation
Add 18 to each Toronto-route coefficient.
Justification
The extra cost applies uniformly to all Toronto shipments.
Shown in the video
Expression
$\text{Min } 25X_{B1}+26X_{B2}+24X_{B3}+24X_{T1}+21X_{T2}+25X_{T3}$
Explanation
Substitute the updated coefficients into the objective.
Justification
Matches the formula shown on screen.
Shown in the video

Answer

Min $25X_{B1}+26X_{B2}+24X_{B3}+24X_{T1}+21X_{T2}+25X_{T3}$ .

Verification

Each updated coefficient equals the corresponding original coefficient plus the stated source-level surcharge.

Modified instance with higher DC1 demand

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narration says suppose demand at distribution center one increases to 300 units.
Diagram
Observation
The slide shows $800 < 850$ after the change.

Problem

Determine the effect on balance when DC1 demand increases from 200 to 300 while supplies stay the same.

Given

Total supply = 800.
Original demands = 200, 300, 250.
New DC1 demand = 300.

Goal

Find the new total demand and the imbalance type.

Steps

Expression
$300+300+250=850$
Explanation
Recompute total demand after replacing 200 by 300 for DC1.
Justification
Uses the updated demand value stated in the clip.
Shown in the video
Expression
$800<850$
Explanation
Compare total supply with the new total demand.
Justification
Matches the on-screen comparison.
Shown in the video
Expression
$850-800=50$
Explanation
Compute the shortage.
Justification
Arithmetic difference between demand and supply.
Derived from the video

Answer

Total demand becomes 850, so demand exceeds supply by 50 units.

Verification

The clip explicitly states the shortage is 50 units and displays $800 < 850$ .

Dummy-plant reformulation of the demand-heavy instance

Clear evidence

Shown in the video

Evidence

Diagram
Observation
A dummy node with supply 50 and cost $0 is added to the network.
Formula
Observation
The expanded model shows + $0X_{D1}+0X_{D2}+0X_{D3}$ and $X_{D1}+X_{D2}+X_{D3}\le 50$ .

Uncertainties

The clip does not solve the LP numerically; it only formulates the dummy extension.

Problem

Reformulate the instance where demand exceeds supply by introducing a dummy plant.

Given

Shortage = 50 units.
Dummy supply = 50.
Dummy unit cost = 0.
Real route costs remain as in the updated objective.

Goal

Extend the LP with dummy variables, dummy cost terms, dummy supply constraint, and updated demand constraints.

Steps

Expression
$\text{Add } D \text{ with supply } 50$
Explanation
Create a dummy source equal to the shortage.
Justification
Stated in the audio and shown in the diagram.
Shown in the video
Expression
$+0X_{D1}+0X_{D2}+0X_{D3}$
Explanation
Add zero-cost objective terms for dummy shipments.
Justification
Shown in the expanded objective and explained as nonexistent units.
Shown in the video
Expression
$X_{D1}+X_{D2}+X_{D3}\le 50$
Explanation
Add the dummy supply constraint.
Justification
Displayed as (Dummy's Supply).
Shown in the video
Expression
$X_{B1}+X_{T1}+X_{D1}=300$
Explanation
Update DC1 demand to include dummy shipments.
Justification
Shown in the expanded demand constraints.
Shown in the video
Expression
$X_{B2}+X_{T2}+X_{D2}=300$
Explanation
Update DC2 demand to include dummy shipments.
Justification
Shown in the expanded demand constraints.
Shown in the video
Expression
$X_{B3}+X_{T3}+X_{D3}=250$
Explanation
Update DC3 demand to include dummy shipments.
Justification
Shown in the expanded demand constraints.
Shown in the video

Answer

The expanded model adds $X_{D1}$ , $X_{D2}$ , $X_{D3}$ , zero-cost objective terms, the constraint $X_{D1}+X_{D2}+X_{D3}\le 50$ , and dummy terms inside the demand equations.

Verification

The dummy supply 50 exactly matches the previously computed shortage $850-800=50$ .

Visual events · 16

Opening concept graphic

Clear evidence

Shown in the video

Evidence

Diagram
Observation
Title slide reads Linear Programming / Transportation Problem, then a simple graphic shows Supply locations (Origins) → Demand locations (Destinations) under Minimum Costs.
Audio
Observation
The narrator introduces the topic and states the goal of minimizing transport cost from origins to destinations.

Objects

Title text
Supply locations (Origins)
Demand locations (Destinations)
Arrow
Minimum Costs label

Changes

The slide first presents the title, then adds the origin-to-destination schematic.
The arrow visually links supply to demand under the cost-minimization objective.

Invariants

Origins remain on the left and destinations on the right.
The objective label remains Minimum Costs.

Interpretation

The visual establishes the transportation problem as moving goods from sources to sinks at least cost.

Matrix walkthrough with highlighted regions

Clear evidence

Shown in the video

Evidence

Diagram
Observation
A transportation matrix appears; red boxes and ovals successively highlight the Supply column, the DC headers, the Demand row, and the interior cost block.
Audio
Observation
The narrator reads supplies, demands, and then explains the meaning of the interior cost entries.

Objects

Transportation matrix
Boston row
Toronto row
DC1/DC2/DC3 columns
Supply column
Demand row
Interior cost cells

Changes

First the supply column is emphasized.
Then the destination headers are emphasized.
Then the demand row is emphasized.
Finally the interior cost block is boxed and one cell is singled out as an example.

Invariants

The matrix dimensions stay 2 origins by 3 destinations.
The numerical data do not change during highlighting.

Interpretation

The highlighting teaches how to read a transportation table: rows are origins, columns are destinations, side entries are supplies/demands, and interior entries are unit costs.

Construction of the transportation network

Clear evidence

Shown in the video

Evidence

Animation
Observation
Two origin circles appear first, then three destination circles, then supply/demand labels, then all six directed arcs with costs.
Audio
Observation
The narrator names the circles nodes and the arrows arcs while assigning capacities, demands, and shipping costs.

Objects

Boston node
Toronto node
DC 1 node
DC 2 node
DC 3 node
Supply labels
Demand labels
Six directed arcs
Cost labels on arcs

Changes

Origin nodes are drawn before destination nodes.
Supply and demand values are attached to the corresponding nodes.
Arcs are added one group at a time until every origin connects to every destination.

Invariants

The network remains bipartite with origins on one side and destinations on the other.
Each arc direction is from an origin to a destination.

Interpretation

The animation translates the tabular data into graph form, making routes and their costs explicit.

Decision-variable introduction synchronized with arc highlighting

Clear evidence

Shown in the video

Evidence

Diagram
Observation
A Decision Variables panel lists $X_{B1}$ through $X_{T3}$ ; the corresponding arc is highlighted in yellow as each variable is introduced.
Formula
Observation
The lower part of the slide later shows $2 \times 3 = 6$ and then X_{ij} = # units shipped from Plant i to DC j.

Uncertainties

The last spoken phrase is cut off, but the displayed indexed definition is complete.

Objects

Decision Variables panel
Six route variables
Network arcs
Yellow highlight
$2 \times 3 = 6$ equation
Indexed variable X_{ij}

Changes

Variables are added sequentially from Boston routes to Toronto routes.
Each new variable highlights its matching arc in the network.
After the full list, the slide summarizes the count and then replaces the list with indexed notation.

Invariants

There are always six origin-destination routes in the example.
The network structure does not change while variables are defined.

Interpretation

The visual pairing shows that each decision variable corresponds exactly to one arc, i.e., one possible shipping route.

Initial display of the transportation instance and variable definitions

Clear evidence

Shown in the video

Evidence

Diagram
Observation
The left side shows a cost/supply/demand table and a bipartite network with Boston and Toronto on the left and DC1, DC2, DC3 on the right.
Formula
Observation
The right side lists the decision-variable definitions.

Objects

Cost table
Network graph with two origins and three destinations
Decision-variable list

Changes

The general definition X_{ij} is highlighted.
The index sets for i and j are added beneath the general definition.

Invariants

The supplies, demands, and unit costs remain fixed on the left side.

Interpretation

The visual layout links each algebraic variable to a specific edge in the network and a specific entry in the cost table.

Objective-function terms synchronized with network edges

Clear evidence

Shown in the video

Evidence

Animation
Observation
As the objective function is written, individual terms such as $5X_{B1}$ and $6X_{B2}$ are highlighted in yellow.
Diagram
Observation
Corresponding network edges are emphasized while their coefficients are discussed.

Objects

Objective function line
Network edges from Boston and Toronto to DCs

Changes

Each term in the objective is highlighted in sequence.
Matching route edges in the network are visually emphasized.

Invariants

The numerical costs 5, 6, 4, 6, 3, 7 do not change.

Interpretation

The animation demonstrates that each coefficient in the objective comes from a specific route cost in the transportation network.

Origin-node highlighting and supply-versus-demand comparison

Clear evidence

Shown in the video

Evidence

Animation
Observation
Boston is circled in red and its outgoing edges are highlighted while the first supply constraint is written; then Toronto is circled and its outgoing edges are highlighted.
Diagram
Observation
A red comparison $800 > 750$ appears at the bottom after both supply constraints are shown.

Objects

Boston node
Toronto node
Outgoing edges
Bottom comparison annotation

Changes

Focus shifts from Boston to Toronto.
Outgoing edges are highlighted for each origin in turn.
A summary comparison of total supply and total demand appears.

Invariants

Node capacities 300 and 500 remain unchanged.

Interpretation

The visuals connect each supply constraint to the total outflow from one origin node and explain why those constraints are inequalities.

Destination-node highlighting for demand constraints

Clear evidence

Shown in the video

Evidence

Animation
Observation
DC1, then DC2, then DC3 are circled in red while incoming edges are highlighted and the corresponding equality constraints are boxed.

Objects

DC1 node
DC2 node
DC3 node
Incoming edges

Changes

Attention moves from one destination to the next.
Incoming edges to each destination are highlighted as its equality constraint is written.

Invariants

Demand values 200, 300, and 250 remain unchanged.

Interpretation

The animation shows that each demand constraint aggregates all inbound flows to one destination node.

Complete LP model enclosed as a final result

Clear evidence

Shown in the video

Evidence

Diagram
Observation
A large red rectangle encloses the full LP formulation on the right side of the slide.

Objects

Full objective function
All constraints
Nonnegativity statement

Changes

The separate lines are visually grouped into one complete model.

Invariants

The mathematical content of each line is unchanged.

Interpretation

The boxed display signals that the formulation process is complete for the basic transportation model.

Beginning of an extension adding production costs

Clear evidence

Shown in the video

Evidence

Diagram
Observation
Labels $20 and $18 appear next to Boston and Toronto.
Audio
Observation
The narrator introduces production costs and says to add them to the shipping cost.

Uncertainties

The revised objective function is not fully shown before the clip ends.

Objects

Boston node
Toronto node
New cost labels $20 and $18

Changes

New per-unit production costs are attached to the origin nodes.
The narration indicates the objective function will be modified.

Invariants

The original shipping-cost structure and constraints remain visible.

Interpretation

The clip starts to extend the pure transportation cost model by incorporating origin-specific production costs, but the completed reformulation is outside this segment.

Initial transportation network and data table

Clear evidence

Shown in the video

Evidence

Diagram
Observation
A bipartite network shows Boston and Toronto on the left and DC1, DC2, DC3 on the right with route labels 5,6,4,6,3,7.
Formula
Observation
A table lists supplies 300 and 500 and demands 200, 300, 250.

Objects

Boston node
Toronto node
DC1 node
DC2 node
DC3 node
route cost labels
supply/demand table

Changes

The route costs and node capacities establish the numerical instance.
Later overlays modify costs and demand values.

Invariants

The graph structure is bipartite from plants to distribution centers.
The decision variables correspond to arcs in the network.

Interpretation

The visual layout links each algebraic term X_{ij} to a specific arc from a plant to a destination, making the LP formulation readable directly from the network.

Highlighting source-level cost additions

Clear evidence

Shown in the video

Evidence

Diagram
Observation
Red boxes show $20 near Boston and $18 near Toronto.
Formula
Observation
Yellow highlights mark +20+ and +18+ over groups of objective terms.

Objects

Boston label $20
Toronto label $18
objective function terms

Changes

Boston-related coefficients are visually grouped for +20.
Toronto-related coefficients are visually grouped for +18.
The objective is rewritten with larger coefficients.

Invariants

The set of decision variables remains the same during this step.
Only cost coefficients change, not supplies or demands.

Interpretation

The animation shows that a uniform surcharge at a source translates into adding the same constant to every outgoing-route coefficient in the objective.

Misconceptions · 7

Confusing what is stored on nodes versus arcs

Clear evidence

Supplementary explanation

Evidence

Audio
Observation
The narrator explicitly distinguishes circles as nodes and arrows as arcs.
Diagram
Observation
The network labels nodes with supply/demand and arcs with costs.

Misconception

One might think the shipping costs belong to the nodes rather than the routes.

Clarification

In this representation, nodes carry supply or demand amounts, while arcs carry the unit shipping cost for moving goods along that route.

Counting variables by nodes instead of routes

Clear evidence

Supplementary explanation

Evidence

Audio
Observation
The narrator states that the number of decision variables is the number of origins multiplied by the number of destinations.
Formula
Observation
The slide shows $2 \times 3 = 6$ .

Misconception

A learner may count only the visible places (for example, 5 nodes) and conclude there are 5 variables.

Clarification

The variables correspond to routes (arcs), not nodes. With 2 origins and 3 destinations, there are $2\times 3=6$ route variables.

Mistaking supply constraints for equalities

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narrator explicitly notes that total supply exceeds total demand and that this is why the less-than-or-equal sign is used.
Diagram
Observation
The slide shows $800 > 750$ .

Misconception

One might think every node constraint in a transportation problem must be an equality.

Clarification

Origin constraints are inequalities here because excess supply may remain unused when total supply exceeds total demand.

Using inequalities for mandatory demand

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narrator says equality is used because demand must be met.
Formula
Observation
All destination constraints are written with = signs.

Misconception

One might write destination constraints with $\le$ or $\ge$ instead of =.

Clarification

Because each destination's demand must be satisfied exactly, the inbound-flow constraints are equalities.

Assuming all transportation constraints must always be equalities

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narration says if total supply were to equal total demand, we could have used equality for all constraints.
Formula
Observation
The current model initially uses $\le$ for supply and = for demand in the unbalanced case.

Misconception

One might think a transportation LP always uses equality constraints on both supply and demand sides.

Clarification

Equality on all sides is appropriate in the balanced case. When the instance is unbalanced, the formulation must distinguish between exhausted supply and unmet or excess demand.

Treating dummy shipments as real goods

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narration says these 50 units do not exist and are not going to be shipped at all.
Diagram
Observation
The dummy node is labeled with $0 cost.

Misconception

A dummy plant’s shipments could be mistaken for actual physical inventory being transported.

Clarification

Dummy shipments are a modeling device for unmet demand. Their zero cost reflects that they are not real shipments.

Assuming supply constraints are always binding

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narration says since demand exceeds supply, all available supply will be used up.
Formula
Observation
The supply constraints are changed from $\le$ to = only in that demand-heavy scenario.

Misconception

One might assume plant supply constraints should always be written as equalities.

Clarification

Whether supply constraints are inequalities or equalities depends on the balance situation and the modeling assumption about unused capacity.

Concept relations · 16

Goal of the transportation problem → Transportation matrix data layout

Clear evidence

Derived from the video

Evidence

Audio
Observation
After defining the goal, the narrator immediately says they will write an LP formulation for the transportation matrix.
Diagram
Observation
The abstract origins/destinations graphic is followed by a concrete matrix of supplies, demands, and costs.

Application

Explanation

The general goal of minimizing transport cost is instantiated in the matrix by specifying particular origins, destinations, supplies, demands, and unit costs.

Transportation matrix data layout → Network representation of a transportation problem

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narrator says, “Now let’s draw a transportation network for this problem” after explaining the matrix.
Animation
Observation
The same numerical data are redrawn as nodes, supplies/demands, and arcs with costs.

Equivalent

Explanation

The matrix and the bipartite network encode the same transportation data in two different representations.

Network representation of a transportation problem → Explicit decision variables for a small transportation model

Clear evidence

Shown in the video

Evidence

Animation
Observation
As each variable is named, the corresponding arc in the network is highlighted.
Audio
Observation
The narrator defines variables as units shipped from a specific origin to a specific destination.

Application

Explanation

Each arc in the network becomes one decision variable representing the shipment quantity on that route.

Explicit decision variables for a small transportation model → Indexed general form of transportation decision variables

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narrator first lists all variables, then says that if there are too many variables one can use a short form.
Formula
Observation
The slide moves from $X_{B1}$ ,..., $X_{T3}$ to X_{ij}.

Generalizes

Explanation

The indexed notation X_{ij} generalizes the explicit route-by-route listing to arbitrary numbers of plants and distribution centers.

Counting decision variables in a transportation model → Indexed general form of transportation decision variables

Clear evidence

Derived from the video

Evidence

Formula
Observation
The slide shows $2 \times 3 = 6$ just before introducing X_{ij}.
Audio
Observation
The narrator explains the count rule and then motivates compact notation for larger models.

Prerequisite

Explanation

Understanding that variables correspond to origin-destination pairs helps explain why a double index i,j is the natural compact notation.

Decision variables for the transportation model → Objective function minimizing total shipping cost

Clear evidence

Shown in the video

Evidence

Formula
Observation
The objective function is written only after the variables X_{ij} have been defined.
Audio
Observation
The narrator moves from defining variables to stating the objective function.

Prerequisite

Explanation

The objective function is expressed in terms of the route variables, so the variable definitions must come first.

Decision variables for the transportation model → Supply constraints at origin nodes

Clear evidence

Shown in the video

Evidence

Formula
Observation
Supply constraints are sums of the previously defined route variables.

Prerequisite

Explanation

Each supply constraint aggregates the variables corresponding to outgoing routes from one origin.

Decision variables for the transportation model → Demand constraints at destination nodes

Clear evidence

Shown in the video

Evidence

Formula
Observation
Demand constraints are sums of the previously defined route variables entering each destination.

Prerequisite

Explanation

Each demand constraint aggregates the variables corresponding to incoming routes to one destination.

Objective function minimizing total shipping cost → Complete linear programming formulation of the transportation problem

Clear evidence

Shown in the video

Evidence

Formula
Observation
The final boxed model contains the objective, supply constraints, demand constraints, and nonnegativity together.
Audio
Observation
The narrator concludes that the assembled lines form the LP model.

Contains

Explanation

The complete LP formulation contains the objective function as one of its main components.

Supply constraints at origin nodes → Demand constraints at destination nodes

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narrator contrasts inequality use at supply nodes with equality use at demand nodes.
Formula
Observation
Supply constraints use $\le$ while demand constraints use =.

Contrast

Explanation

Origin constraints limit outflow with inequalities, whereas destination constraints enforce exact inflow with equalities.

Transportation problem as a linear program → Decision variable X_{ij}

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide presents a transportation network together with an explicit linear objective and linear constraints.

Contains

Explanation

The transportation model contains the decision-variable definition X_{ij} as its basic modeling ingredient.

Balanced transportation problem → Unbalanced transportation problem

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narration contrasts balanced and unbalanced cases using total supply and total demand.
Diagram
Observation
The comparisons $800 > 750$ and $800 < 850$ are shown.

Contrast

Explanation

Balanced and unbalanced transportation problems are distinguished by whether total supply equals total demand.

Find an answer · 20

What is the transportation problem in linear programming?

Clear evidence

Supplementary explanation

Evidence

Audio
Observation
The narrator defines the transportation problem goal in words.
Diagram
Observation
The opening slide shows origins, destinations, and minimum costs.

Knowledge points

Goal of the transportation problem

How do you read a transportation matrix with supplies, demands, and shipping costs?

Clear evidence

Supplementary explanation

Evidence

Formula
Observation
The matrix displays rows, columns, supplies, demands, and costs.
Audio
Observation
The narrator explains each part of the matrix.

Knowledge points

Transportation matrix data layout

In a transportation network, what do nodes and arcs represent?

Clear evidence

Supplementary explanation

Evidence

Audio
Observation
The narrator says circles are nodes and arrows are arcs.
Diagram
Observation
The network diagram labels nodes and arcs differently.

Knowledge points

Network representation of a transportation problem

How are decision variables defined in a transportation LP model?

Clear evidence

Supplementary explanation

Evidence

Formula
Observation
The slide lists $X_{B1}$ through $X_{T3}$ and then X_{ij}.
Audio
Observation
The narrator defines each variable as units shipped on a route.

Knowledge points

Explicit decision variables for a small transportation model
Indexed general form of transportation decision variables

Why does a 2-origin, 3-destination transportation problem have 6 decision variables?

Clear evidence

Supplementary explanation

Evidence

Audio
Observation
The narrator states the multiplication rule for counting variables.
Formula
Observation
The slide shows $2 \times 3 = 6$ .

Knowledge points

Counting decision variables in a transportation model

What does X_{ij} mean in transportation problem notation?

Clear evidence

Supplementary explanation

Evidence

Formula
Observation
The slide shows X_{ij} = # units shipped from Plant i to DC j.
Audio
Observation
The narrator introduces a short form for larger models.

Uncertainties

The spoken sentence is truncated at the clip boundary.

Knowledge points

Indexed general form of transportation decision variables

What does X_{ij} mean in a transportation problem?

Clear evidence

Shown in the video

Evidence

Formula
Observation
The slide explicitly defines X_{ij}.

Knowledge points

Decision variables for the transportation model

How is the objective function built from unit shipping costs?

Clear evidence

Shown in the video

Evidence

Formula
Observation
The objective function is displayed term by term.
Audio
Observation
The narrator explains that the objective is to minimize cost.

Knowledge points

Objective function minimizing total shipping cost

Why are supply constraints written with $\le$ instead of =?

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narrator says total supply exceeds total demand, so capacity is not exhausted.
Diagram
Observation
The slide shows $800 > 750$ .

Knowledge points

Supply constraints at origin nodes
Why supply constraints use inequalities

Why are demand constraints written as equalities?

Clear evidence

Shown in the video

Evidence

Audio
Observation
The narrator says demand must be met and there is enough supply to meet it.
Formula
Observation
Destination constraints are written as equalities.

Knowledge points

Demand constraints at destination nodes
Why demand constraints use equalities

What is the full LP formulation of this transportation example?

Clear evidence

Shown in the video

Evidence

Formula
Observation
The full model is boxed on screen.
Audio
Observation
The narrator says, 'And that's the LP model.'

Knowledge points

Complete linear programming formulation of the transportation problem

How would production costs at Boston and Toronto modify the transportation objective?

Approximate timing

Shown in the video

Evidence

Diagram
Observation
Labels $20 and $18 appear next to the origins.
Audio
Observation
The narrator begins explaining that production costs should be added to shipping costs.

Uncertainties

The revised objective is not completed within the clip.

Knowledge points

Objective function minimizing total shipping cost

Coverage and review notes

Covered · Title card and spoken introduction to the topic; no separate mathematical statement beyond naming the subject.

Covered · Verbal definition of the transportation problem objective with a simple origins-to-destinations graphic.

Covered · Presentation and explanation of the $2\times 3$ transportation matrix, including supplies, demands, and unit costs.

Covered · Conversion of the matrix into a bipartite network with nodes, supplies/demands, arcs, and arc costs.

Covered · Explicit definition of the six route variables and derivation of the count $2\times 3=6$ .

Covered · Introduction of compact indexed notation X_{ij}; the spoken sentence is cut off but the displayed definition is complete.

Covered · Defines the indexed decision variables and their route interpretation.

Covered · Constructs the minimization objective from route costs.

Covered · Writes origin supply constraints and explains why they are inequalities.

Covered · Writes destination demand constraints and explains why they are equalities.

Covered · Adds nonnegativity and boxes the complete LP model.

Covered · Begins an extension with production costs $20 and $18, but the revised objective is not completed within this clip.

Covered · The entire clip is a continuous worked explanation of the transportation LP, including the initial model, cost update, balance discussion, dummy-plant reformulation, and the start of an alternative direct reformulation.

Covered · The entire clip covers the presentation of the LP model and the modification for a prohibited route.

Explore the knowledge in this video

Reviewed subject paths

Transportation problem

Transportation Problem - LP Formulation

The explanation, unpacked.

Chapters

Learning script

Knowledge cards

Transportation problem objective

How to read a transportation matrix

Network form: nodes and arcs

Route-based decision variables

Counting transportation variables

Indexed notation X_ij

Route-based decision variables

Cost-minimization objective

Supply constraints are inequalities when supply exceeds demand

Demand constraints are equalities

Nonnegativity closes the LP formulation

Complete transportation LP model

Extension begins: adding production costs

Transportation problem LP structure

Meaning of X_{ij}

Source-level cost update in the objective

Balanced transportation problem

Unbalanced transportation problem

Dummy plant for demand-heavy imbalance

Expanded demand constraints with dummy variables

Alternative direct reformulation

Transportation Problem LP Model

Handling Prohibited Routes in LP

Detailed learning notes

XB1X_{B1}XB1​

XB2X_{B2}XB2​

XB3X_{B3}XB3​

XT1X_{T1}XT1​

XT2X_{T2}XT2​

XT3X_{T3}XT3​

X_{ij}

300

500

200

300

250

Goal of the transportation problem

Transportation matrix data layout

Network representation of a transportation problem

Explicit decision variables for a small transportation model

Counting decision variables in a transportation model

Indexed general form of transportation decision variables

Decision variables for the transportation model

Objective function minimizing total shipping cost

Supply constraints at origin nodes

Demand constraints at destination nodes

Nonnegativity constraints

Complete linear programming formulation of the transportation problem

Why supply constraints use inequalities

Why demand constraints use equalities

Equality constraints in the balanced case

The original numerical example is supply-heavy

Raising DC1 demand creates a demand-heavy imbalance

Dummy shipments have zero cost

All real supply is exhausted when demand exceeds supply

Derivation of the example’s variable count

Step-by-step construction of the transportation LP

Derivation of the updated objective function

Checking balance in the original instance

Checking balance after increasing DC1 demand

Expanding the LP with a dummy plant

Worked setup of a 2-by-3 transportation problem

Worked formulation of a Boston-Toronto transportation instance

Initial transportation LP before cost update

Updating the objective after source-specific cost increases

Modified instance with higher DC1 demand

Dummy-plant reformulation of the demand-heavy instance

Opening concept graphic

Matrix walkthrough with highlighted regions

Construction of the transportation network

Decision-variable introduction synchronized with arc highlighting

Initial display of the transportation instance and variable definitions

Objective-function terms synchronized with network edges

Origin-node highlighting and supply-versus-demand comparison

Destination-node highlighting for demand constraints

$X_{B1}$

$X_{B2}$

$X_{B3}$

$X_{T1}$

$X_{T2}$

$X_{T3}$

Why are supply constraints written with $\le$ instead of =?