In the simplest case, the new Expression Constraint table can be used to create the same standard constraint. In this case, the Flow Constraint record looks like the following. Note the use of “Define” and no Constraint Value value, as well as the Expression Name:

Expression Name	Period Name	Source Site	Destination Site	Product Name	Constraint Value	Constraint Type
Flow1	Period_001	DC1	CZ5	Product1		Define

You create an Expression Constraint that references the named Flow Constraint and sets the requirement:

Expression Name	Coefficient 1	Expression 1	Coefficient 2	Expression 2	Expression Type	Constraint Value
FlowConstraint	1	Flow1			Max	100

However, the true benefit comes when defining expression constraints. You can use the new Expression Constraints table to:

• Define a constraint made up of one or two expressions
• Define an expression that is a function of two other expressions

For example, assume that you want to limit the total production and flow quantity for Product1 at DC1 to 150 units in Period_001. This is actually a constraint across two different conditions that you combine using the Expression Constraints table. First, you create the Flow Constraint and Production Constraint records. Unlike the standard constraints, these will not have an actual quantity value specified, since the quantity is based on the combination of the two. In this case, the Constraint Type in each constraint record is “Define”, since the value will be determined by the Expression Constraint.

Here is the Flow Constraint record:

Expression Name	Period Name	Source Site	Destination Site	Product Name	Constraint Value	Constraint Type
Flow1	Period_001	DC1	CZ5	Product1		Define

And this is the Production Constraint record:

Expression Name	Period Name	Site Name	Product Name	Process Name	Constraint Value	Constraint Type
Prod1	Period_001	DC1	Product1			Define

To combine these using an Expression Constraint, the record looks like the following:

Expression Name	Coefficient 1	Expression 1	Coefficient 2	Expression 2	Expression Type	Constraint Value
LimitProd1	1	Flow1	1	Prod1	Max	150

This is defining that (1 * Flow1) + (1 * Prod1) must be less than or equal to 150. In other words, the sum of the flow of Product1 and the production of Product1 at DC1 cannot exceed 150 in Period_001.

You can use the coefficient fields to weight the expressions used in the Expression Constraint. You must take care with the signs that you place on the coefficients. These will be evaluated algebraically as shown in the following examples.

Example 1:

We can require that the Flow from DC_A is twice the Inventory at DC_B over the model horizon. The Inventory Constraint is:

Expression Name	Period Name	Site Name	Product Name	Constraint Value	Constraint Type
Inv2	(ALL_Periods) (All)	DC_B	(ALL_Products)		Define

The Flow Constraint is:

Expression Name	Period Name	Source Site	Destination Site	Product Name	Constraint Value	Constraint Type
Flow2	(ALL_Periods) (All)	DC_A	(ALL_Customers)	(ALL_Products)		Define

The Expression Constraint would then be:

Expression Name	Coefficient 1	Expression 1	Coefficient 2	Expression 2	Expression Type	Constraint Value
FlowInv2	2	Flow2	-1	Inv2	Fixed	0

That is:

(2 * Flow) + (-1 * Inventory) >= 0

Rearranging the equation gives:

(2 * Flow) = (1 * Inventory)

Example 2:

Another example is a requirement that the Total Flow Quantity out of DC_A must be less than 20% of the Total Inventory Quantity at DC_B during Period_003. In this case, the Flow Constraint is:

Expression Name	Period Name	Source Site	Destination Site	Product Name	Constraint Value	Constraint Type
Flow3	Period_003	DC_A	(ALL_Customers)	(ALL_Products)		Define

The Inventory Constraint is:

Expression Name	Period Name	Site Name	Product Name	Constraint Value	Constraint Type
Inv3	Period_003	DC_B	(ALL_Products)		Define

The Expression Constraint would then be:

Expression Name	Coefficient 1	Expression 1	Coefficient 2	Expression 2	Expression Type	Constraint Value
FlowInv3	0.2	Inv3	-1	Flow3	Min	0

That is:

[0.2 * InventoryDC_B] - [1 * FlowDC_A] >= 0

Rearranging the equation gives:

0.2 InventoryDC_B >= FlowDC_A

If the Inventory at DC_B is 1000:

200 >= FlowDC_A

The Flow at DC_A can be no more than 200 units in Period_003.

Let’s change this example to say that the Total Flow Quantity out of DC_A must be at least 20% of the Total Inventory Quantity at DC_B during Period_003. The coefficients must be changed to set up the appropriate evaluation and the Expression Constraint would then be:

Expression Name	Coefficient 1	Expression 1	Coefficient 2	Expression 2	Expression Type	Constraint Value
FlowInv3	-0.2	Inv3	1	Flow3	Min	0

That is:

[-0.2 * InventoryDC_B] + [1 * FlowDC_A] >= 0

Rearranging the equation gives:

FlowDC_A >= 0.2 InventoryDC_B

If the Inventory at DC_B is 1000:

FlowDC_A >= 200

The Flow at DC_A must be at least 200 units in Period_003.

Example 3:

In another example, we want to control the flow over 3 Modes:

• 50% must go LTL (fixed).
• If Air is used, no more than 30% can be Air.

• Ground can be used to serve what is not handled using LTL or Air.

We set up 4 Flow Constraint records:

Expression Name	Period Name	Source Site	Destination Site	Product Name	Mode	Constraint Value	Constraint Type
TotalFlow	(ALL_Periods) (All)	DC	CZ	P1			Define
LTL	(ALL_Periods) (All)	DC	CZ	P1	LTL		Define
Ground	(ALL_Periods) (All)	DC	CZ	P1	Ground		Define
Air	(ALL_Periods) (All)	DC	CZ	P1	Air		Define

The Expression Constraints would then be:

Expression Name	Coefficient 1	Expression 1	Coefficient 2	Expression 2	Expression Type	Constraint Value
LTL50	-0.5	TotalFlow	1	LTL	Fixed	0
Air30AtMost	0.3	TotalFlow	-1	Air	Min	0

Let’s assume that the total flow in this model is 1000. For the LTL 50 constraint, this is a Fixed value, so:

LTL = 500

In other words, if Ground is used, it must account for 200 or more units.

For the Air30AtMost constraint, this is evaluated as:

0.3 * [Total Flow] - [Air Flow] >= 0

0.3 Total - Air >= 0

300 >= Air

If Air is more expensive than Ground, it will not be used and all non-LTL will go by Ground. If less expensive, Air can incur up to 300 units.

The way you define Expression Constraints when using “Cond Min” as the Constraint Type will depend on whether your constraint uses a single expression or a summation. For example, assume you have the following Production Constraints:

Expression Name	Period Name	Site Name	Product Name	Process Name	Constraint Value	Constraint Type
TotalProduction	(ALL_Periods) (All)	MFGs	Product1			Define
ProductionMFG	(ALL_Periods) (All)	MFG	Product1			Define

Assume you want a constraint such that production from the single MFG and the total production is at least 110:

If the TotalProduction + ProductionMFG > 0, then TotalProduction + ProductionMFG > 110

The Expression Constraint would look like the following:

Expression Name	Coefficient 1	Expression 1	Coefficient 2	Expression 2	Constraint Type	Constraint Value
Cond Min Summation	1	TotalProduction	1	ProductionMFG	Cond Min	110

This structure forces a minimum value to a summation expression (the coefficients can be negative).

In another case, assume that you want to ensure that if there is production from the single MFG, the volume is at least 10:

If the ProductionMFG > 0, then ProductionMFG > 10

This structure forces a minimum value to a single expression.

Conditional minimum with non-zero constraint value

When working with conditional minimum as the Constraint Type, keep in mind how such a constraint is evaluated:

Conditional Minimum: x >= Minimum Amount or x = 0

For example, if you define a regular product constraint with a conditional minimum of 100, the production amount (x) has to be greater than or equal to 100 units (minimum amount) or the production will be 0 units.

This is significant because it means that the overall expression will meet either the Constraint Value minimum or a minimum of 0.

As an example, consider the following constraints:

First, we have Flow Constraint to define the flow of a product from all plants to all customers:

Expression Name	Period Name	Source Site	Destination Site	Product Name	Constraint Value	Constraint Type
Total_FG_1001	(ALL_Periods) (All)	All_Plant	All_CZ	FG_1001		Define

There is also a Production Constraint to define the production of the finished good product at one of the plants:

Expression Name	Period Name	Source Site	Product Name	Constraint Value	Constraint Type
Total_P104_FG_1001	(ALL_Periods) (All)	Plant_104	FG_1001		Define

In the first case, the goal of the Expression Constraint is to ensure that 5% of the total flow (Total_FG_1001) is greater than or equal to the production at Plant_104 (Total_P104_FG_1001). The Expression Constraint looks like this:

Expression Name	Coefficient 1	Expression 1	Coefficient 2	Expression 2	Constraint Type	Constraint Value
Exp_P104_FG_1001	0.05	Total_FG_1001	-1	Total_P104_FG_1001	Cond Min	0

Assume that the model has demand of 100. In this case:

• Total Flow (Total_FG_1001) = 100
• Production at Plant_104 (Total_P104_FG_1001) = 5
• and the Expression Constraint (Exp_P104_FG_1001) = 0

Now, let’s change the Expression Constraint so that we compare 5% of the flow to production at Plant_104 less 3 units. We change the Constraint Value to 3:

Expression Name	Coefficient 1	Expression 1	Coefficient 2	Expression 2	Constraint Type	Constraint Value
Exp_P104_FG_1001	0.05	Total_FG_1001	-1	Total_P104_FG_1001	Cond Min	3

Depending on the costs and other constraints in the model, the Constraint Value of 3 on the conditional minimum may not be enforced (remember that it is conditional). If the minimum is used, the Expression Constraint (Exp_P104_FG_1001) will be greater than or equal to 3. If it is not used the Expression Constraint is evaluated as 0 and the results will look the same as the initial example, where Constraint Value was 0.

If you use Min rather than Cond Min and using Plant_104 is cost-effective, you will see:

• Total Flow (Total_FG_1001) = 100
• Production at Plant_104 (Total_P104_FG_1001) = 2
• and the Expression Constraint (Exp_P104_FG_1001) = 3

If you use Min and using Plant_104 is not cost-effective, you will see:

• Total Flow (Total_FG_1001) = 100
• Production at Plant_104 (Total_P104_FG_1001) = 0
• and the Expression Constraint (Exp_P104_FG_1001) = 5

In this case, Production is 0 at Plant_104, but the Expression Constraint value is greater than the minimum of 3, so the Expression Constraint is satisfied.

If the minimum is over-constraining, the model will be infeasible.

You can also combine expression constraints. For example, let’s go back to our earlier example where we limit the total production and flow quantity for Product1 at DC1 to 150 units in Period_001. We now want to combine that with a constraint on the total production quantity at the MFG site for all products over the model horizon so that the total for all 3 considerations is 5000. The new Production Constraint at the MFG site would be:

Expression Name	Period Name	Site Name	Product Name	Process Name	Constraint Value	Constraint Type
MfgProd1	(ALL_Periods) (All)	MFG	(All)			Define

We add a second Expression Constraint that uses the existing Expression Constraint:

Expression Name	Coefficient 1	Expression 1	Coefficient 2	Expression 2	Expression Type	Constraint Value
LimitProd1	1	Flow1	1	Prod1	Max	150
LimitAll	1	LimitProd1	1	MfgProd1	Max	5000