**5. Optimization-Based Approach**

Mathematical Programming (MP) is another broad category of PI methodologies [60]. MP enables the search to be automated, while finding the optimal solution among many design alternatives [61]. The simplest representation of the MP model is defined as a deterministic model (the model where all the parameters are constant and not uncertain or random), and is written in the form of a Linear Program (LP). One advantage over the previous two approaches is the flexibility to change the objective function to suit the context of the problem. For PI problems, one of the first problem formulations of MP was the transportation model [62].

The transportation model (its variants are also known as the source-sink model or resource-allocation problem) is based on the principle that a product is transported from a number of sources to a number of sinks (destinations) based on a given objective (minimum cost, maximum profit, etc.). Dealing with both mass and energy for CCEP problems as extensive properties, in contrast to temperature in Heat Integration, which is an intensive property, the transportation model is a suitable approach for solving CCEP problems. The transportation model, which is written in a generic way and could be applied to any transportation or planning problem, is shown in Equations (3)–(7). The sets, data and variables applied in the transportation model are described on the left.

$$\sum\_{j} x\_{ij} \le a\_{i\prime} \; \forall i \in I \tag{3}$$

$$\sum\_{i} x\_{i\bar{j}} = b\_{\bar{j}}, \; \forall j \in J \tag{4}$$

$$\sum\_{i} x\_{i\bar{j}} F\_{i} = E\_{j\bar{\prime}} \,\,\forall j \in J \tag{5}$$

$$w\_{ij} = \frac{x\_{ij}}{b\_j}, \forall i \in I, j \in J \tag{6}$$

$$\text{minim}z = \sum\_{ij} x\_{ij} P\_i \tag{7}$$
