Targeting Aggregate Production Planning for an Energy Supply Chain

Article pubs.acs.org/IECR

Targeting Aggregate Production Planning for an Energy Supply Chain Nitin Dutt Chaturvedi and Santanu Bandyopadhyay* Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India ABSTRACT: Aggregate production planning for an energy supply chain is presented in this paper. The methodology is equally applicable to aggregate production planning of production as well as aggregate planning of input material supply. The proposed methodology determines the range of production at different times as well as calculates the production targets with least variation in production levels between different time periods to satisfy the demands of each time period while maintaining the limits of the inventory. Mathematically, it is shown that the graphical representation of the aggregate production planning problem is equivalent to the Euclidean shortest path problem in computational geometry. An algebraic procedure, based on the principles of Pinch Analysis, is applied to solve the problem. The proposed graphical representation provides significant physical understanding of the overall aggregate production planning problem by identifying different kinds of production bottlenecks (also known as pinch points). Applicability of the proposed methodology is demonstrated through illustrative examples related to energy supply chains.

1. INTRODUCTION Aggregate production planning (APP) is one important aspect and needs to be addressed while planning future production strategy to manage supply and demand. APP represents either planning for production to meet varying demand or planning for input material supply to meet required production.1 Various methodologies based on linear programming,2,3 decision rules,4,5 switching rules,6 and simulation,7 as well as other kinds of mathematical and heuristics based techniques, are proposed to address capacity planning for APP. Luss8 reviewed methods of APP related to capacity expansion. Van Mieghem9 reviewed methods available for capacity management for APP. Wu et al.10 reviewed various aspects of APP for high-tech industries. In this paper APP for energy supply chain planning is identified and addressed. In recent years, significant research efforts have been directed toward developing and applying techniques of process integration toward addressing APP as these methods provide graphical as well as algebraic solutions for APP with better physical understanding of the overall problem. Singhvi and Shenoy11 solved APP problems based on the principles of Pinch Analysis. Singhvi et al.12 proposed a graphical methodology based on Pinch Analysis for APP in which demand and supply data are represented as composite curves. Geldermann et al.13 proposed a holistic approach to process design, retrofitting, and APP which was further demonstrated through a case study.14 Foo et al.15 presented an algebraic technique for determining optimum production rate for APP. Ludwig et al.16 developed Pinch Analysis based methods for production planning for biomass use in dynamic and seasonal markets. However, these methods do not provide flexibility in production planning. Furthermore, these methods are restricted up to production management and do no focus on inventory management. An energy system employs different conversion techniques (e.g., combustion, etc.) to transform primary energy (e.g., biomass, coal, natural gas, crude oil, etc.) to secondary energy © 2015 American Chemical Society

(e.g., electricity, heat, transportation fuel, etc.) for various end uses in industrial, commercial, and residential sectors. Effective planning of energy system supply chain is extremely important for economic and reliable supply of secondary energy to end users. Sanderson17 presented a descriptive and analytical understanding of the supply chain for industrial electricity. Bok et al.18 presented a multiperiod optimization model for supply chain optimization in continuous flexible process networks. Tsiakis et al.19 proposed a mathematical model for multiechelon supply chain networks, which includes components related with production, facility location, product transportation, and distribution. McCormick and Kaberger20 identified supply chain coordination as one of the important aspects for bioenergy production. Iakovou et al.21 presented a review of development in design and management of waste biomass supply chains. Lam et al.22 developed a methodology for the synthesizing of biomass energy supply chain networks with regional scopes. Shi et al.23 presented a structural model of natural resource based green supply chain management. Vance et al.24 proposed a methodology for designing sustainable energy supply chains by utilizing the P-graph framework. Halldórsson and Svanberg25 applied the principles of supply chain management to address important conditions for the production, accessibility, and use of energy, from the point of origin to the point of consumption. Castillo et al.26 have introduced a heuristic algorithm, based on the concept of inventory pinch, along with a two-level decomposition of the gasoline blend-planning problem. Lee27 presented a stochastic programming approach for energy supply chain optimization. Development of various mathematical formulations related to appropriate planning and management of energy systems are Received: Revised: Accepted: Published: 6941

February 10, 2015 June 10, 2015 June 18, 2015 June 18, 2015 DOI: 10.1021/acs.iecr.5b00587 Ind. Eng. Chem. Res. 2015, 54, 6941−6949

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Industrial & Engineering Chemistry Research proposed in literature. Hashim et al.28 presented a mathematical formulation for energy planning of an electrical grid along with carbon dioxide (CO2) emission considerations. Kravanja29 presented a mathematical programming approach to sustainable system synthesis with special emphasis on energy systems. Muis et al.30 proposed a mixed integer linear programming (MILP) model for the optimal planning of electricity generation schemes for a nation in order to meet a specified CO2 emission target. Mirzaesmaeeli et al.31 presented a mathematical model for the power generation planning of electric systems with objectives of optimizing energy sources and pollutant mitigation options in order to meet electricity demand and CO2 emission targets at the minimum cost. Pek̨ ala et al.32 developed a mathematical model for planning of energy systems subject to carbon and land footprint constraints. Dong et al.33 proposed an inexact optimization modeling approach for energy systems planning and air pollution mitigation under uncertainty. Diamante et al.34 developed a graphical approach for energy planning to appropriately match CO2 sources with carbon storage options. Chang35 proposed a MILP-based model for composite power system expansion planning. AlQattan et al.36 developed a multiperiod model for water and energy supply planning. Theodosiou et al.37 proposed an optimization model for planning of energy systems integrating the environmental aspects with other parameters, such as financial cost, availability, capacity, location, etc. It may be noted that these methodologies cannot be directly applicable for optimizing energy supply chain. In this paper, it is demonstrated that the APP with inventory constraints is equivalent to the Euclidean shortest path problem in computational geometry. The Euclidean shortest path problem specifies the object to be moved between a pair of locations within a given polygon and a set of obstacles. A solution to the problem is a minimum-length path for the object that connects the two locations and avoids the obstacles. The shortest path problem finds its application in different domains such as, to determine shortest path in a graph,38 urban and transportation planning,39 watchman routes in simple polygons,40 minimizing interplant flow rate for total site involving two plants,41 etc. Several algorithms have been proposed to find the shortest paths inside a simple polygon. Guibas and Hershberger42 proposed a method to calculate shortest path in a simple polygon. Lee and Preparata43 proposed an algorithm to calculate shortest path in the presence of barriers. Reif and Storer44 proposed a methodology for determining shortest paths in euclidean space with polyhedral obstacles. Guibas et al.45 proposed a linear-time algorithm solving a collection of problems related to shortest paths. These algorithms may be used to calculate shortest path algebraically without significant physical insights. In this paper, a graphical methodology, based on the principles of Pinch Analysis, is proposed for energy supply chain planning in order to address production strategy for a company to manage supply and demand. To improve computational accuracy, an algebraic methodology, proposed to minimize interplant flow,41 is adopted in this paper. The methodology calculates the production/supply targets for level strategy in order to satisfy the demand of each time period and maintain the limits of inventory. The proposed methodology also enables the range of production variation to be determined at different levels by identifying planning bottlenecks, also known as pinch points. Physical significances of various pinch points are explained in this paper.

2. PROBLEM DEFINITION AND MATHEMATICAL FORMULATION The general problem of APP may be given as follows: • The demand forecast (Di) for each period in a planning horizon is given. and maximum Imax limits of inventory at • Minimum Imin i i the end of each time period are given. • The objective is to determine the optimal production or supply plan (Pi) that satisfies the demands of each time period and also maintains the limits of inventories with least variation in productions/supplies between two time periods. It may be noted that given inventory limits are imposed at the end of each time interval, and in between variations of inventory are neglected. Additionally, demand and production in any period are assumed to be distributed uniformly. If any significant fluctuations are envisaged during a time period, a shorter time interval should be considered for the analysis. Let H be the time horizon for which APP has to be carried out. Let i (i = 1, 2, ..., n) be a general time period in time horizon H. The mathematical model for APP comprises constraints related to mass balances, demand supply, and inventory limits detailed as follows: Mass Balances. Let Pi and Di denotes the production/ supply and demand of ith time period. Let Ii denotes the inventory at end of ith time period. Following equation can be written for the ith time period using the mass balance: Ii − 1 + Pi = Di + Ii

∀i

(1)

Inventory (Ii) can be calculated using eq 2: i

Ii =

i

∑ Pi − ∑ Di + I0 j=1

∀i (2)

j=1

where I0 is the initial inventory. Minimum Inventory Limits. In order to take care of uncertainty during APP, a minimum level of inventory is maintained. Production must satisfy the demand of each time period and maintain the minimum inventory limit (Imin i ). Equation 3 expresses this constraint: Ii ≥ Iimin

∀i

(3)

Maximum Inventory Limits. Due to physical as well as economic restrictions, there exists a maximum limit for the inventory. Inventory must not exceed the specified maximum limit (Imax i ): Ii ≤ Iimax

∀i

(4)

Initial and Final Inventory. Initial inventory constraint is already included in cumulative mass balance constraint (eq 2). Final inventory constraint is expressed as following equation: In = IH

(5)

where, IH is final inventory. Objective. The objective is to minimize the variation in production at different time periods: n

Objective =

∑ |Pi − Pi− 1| i=1

(6)

It can be observed from eq 6 that the objective function is piece-wise linear with nondifferentiability. However, this 6942

DOI: 10.1021/acs.iecr.5b00587 Ind. Eng. Chem. Res. 2015, 54, 6941−6949

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Industrial & Engineering Chemistry Research

Cumulative Demand Curve (CDC). A cumulative demand curve (CDC) is defined as the line joining the cumulative demand at each time point. Cumulative Production/Supply Curve (CPC). A cumulative production/supply curve (CPC) is defined as the line joining the cumulative productions/supplies at each time point including the initial inventory. It should be noted that end points of CPC are initial and final inventory points. Minimum Inventory Curve. A minimum inventory curve is defined as the line joining the cumulative demands added with the minimum inventory limits at each time point. Equation 17 gives the minimum inventory point (Imin i ) for ith time point on the minimum inventory curve:

nondifferentiability can be eliminated by introducing two new positive variables a+i and a−i as follows:

Pi = ai+ − ai− + Pi − 1

(7)

a−i

a+i

where, and are variables denoting production/supply variations The modified objective function, that is to be minimized, is given as follows: n

Objective =

∑ ai+ + ai−

(8)

i=1

The formulation can be solved as a linear programming problem to determine the optimal production/supply for each time interval.

i

Iimin =

Maximum Inventory Curve. A maximum inventory curve is defined as the line joining the cumulative demands added with the maximum inventory limits at each time point. Equation 18 gives the maximum inventory point (Imax i ) for ith time point on the maximum inventory curve:

i

∑ Dj (9)

j=1

Similarly, cumulative production/supply function defined as

(Pcum i )

(17)

j=1

3. MATHEMATICAL DEVELOPMENT AND GRAPHICAL INTERPRETATION Cumulative demand function (Dcum i ) can be defined as Dicum =

∑ Dj + Iimin

i

is

Iimax =

∑ Dj + Iimax (18)

j=1

i

Picum =

From eqs 10, 13, 14, and 16 it can be concluded that CPC starts from the initial inventory point and ends at the final inventory point as well as has to be contained within the minimum and the maximum inventory curves to give a feasible production/supply plan. It should be noted that as the cumulative production curve is contained within the minimum and the maximum inventory curves, both the minimum inventory and the maximum inventory limits are satisfied at the end of every time interval. Following lemma concludes the above discussions. Lemma 1: Cumulative production/supply curve is feasible if and only if it lies completely within the minimum and the maximum inventory curves. Figure 1 shows typical composite curves for APP. Figure 1 displays the cumulative demand curve along with the maximum as well as the minimum inventory curves. A feasible CPC, which lies completely within the maximum and the minimum

∑ Pj + I0 (10)

j=1

Equation 2 can be rewritten in terms of cumulative production/supply function and cumulative demand function as follows: Ii = Picum − Dicum

(11)

The minimum inventory limit, expressed in eq 3, can be applied in eq 11 to express the inequality as follows: Picum − Dicum ≥ Iimin

(12)

Equation 12 can be rearranged as Picum ≥ Dicum + Iimin

(13)

Equation 13 indicates that cumulative production/supply function is always greater than or equal to the sum of the cumulative demand with the minimum inventory limit of that time period. In a similar way, applying the maximum inventory limit on eq 10, following inequality can be derived: Picum ≤ Dicum + Iimax

(14)

Equation 14 indicates that the cumulative production/supply function is always less than or equal to the sum of the cumulative demand and the maximum inventory limit of each time period. Final inventory should be equal to IH (eq 4) and this condition is applied as IH = Pncum − Dncum IH +

Dncum

Pcum n

=

Pncum

(15) (16)

Dcum n

where and are cumulative production/supply and demand for the nth (last) period. For graphical interpretation of the above results following cumulative curves are defined. These composite curves are similar to the composite curves of Pinch Analysis.

Figure 1. Typical composite curves for APP. 6943


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Industrial & Engineering Chemistry Research inventory curves, is shown in Figure 1. Point A and point B are the initial and the final inventory points, respectively (see Figure 1). Theorem 1: A feasible cumulative production/supply curve, joining the initial and the f inal inventory points, with the shortest length represents a level production/supply plan, i.e., least variation in production/supply levels. Proof: A straight line joining initial inventory point and final inventory point has the minimum length. If it is feasible (according to lemma 1) then it represents zero variation in production levels, i.e., the best level production plan. However, if this line is not feasible, it has to be deviated from such that it lies completely in the feasible region. Furthermore, it may be observed from eq 8 that the objective of the formulation is to minimize the absolute sum of deviations in CPC. The term a+i and a−i are the deviations in production levels of two successive time points. Therefore, to minimize the variation in production/supply levels, the line joining initial and final inventory points has to be deviated from as little as possible. Triangle inequality states that the sum of any two sides of a triangle is larger than the third. On the basis of this inequality, it may be inferred that, due to deviation, the length of the cumulative production curve is bound to increase. Therefore, the feasible cumulative production curve with the shortest length represents the level production plan. This proves Theorem 1. Essentially Theorem 1 suggests that the shortest path inside a polygonal space, defined by the minimum and the maximum inventory curves, represents the optimal CPC. Several algorithms have been proposed to find the shortest paths inside a simple polygon.40−44 These algorithms may be adopted to draw CPC with the shortest length in the given feasible region. Sahu and Bandyopadhyay41 have proposed an algebraic procedure to determine the smallest length curve within a closed polygon and applied this methodology to calculate minimum interplant cross-flow. In this paper, methodology proposed by Sahu and Bandyopadhyay41 is adopted to generate the CPC with the shortest length (details are not provided due to brevity). It may be noted that algebraic procedures are needed for computational accuracy and graphical representations are helpful in obtaining physical insight.

Figure 2. Typical CPC with shortest length and various pinch points.

basis of number of pinch points on the CPC, an APP problem can be classified as follows: i. CPC has no pinch point (Figure 3a), ii. CPC has a single maximum production pinch point (Figure 3b), iii. CPC curve has a single minimum pinch point (Figure 3c), iv. CPC has two pinch points, one minimum production pinch and one maximum production pinch (Figure 3d), and v. CPC has more than two pinch points (not shown for brevity). Physical significances of these cases are discussed in detail. For case (i), the straight line joining the initial and the final inventory points is in the feasible region, and therefore, there is no pinch point, implying a constant production throughout. Absence of any pinch point signifies that slight change in production/supply plan as well as a small change in demand is not going to violate any of the inventory limits. For case (ii) with the maximum production pinch point, the inventory level reaches its maximum at the pinch point. In this case, there exists a limit of maximum production/supply below this pinch point and the production/supply cannot be increased further. It may be observed that, after the inventory reaching its maximum, production is not decreased; on the contrary, it is increased. However, the inventory level decreases. It may be noted that decrease of demand below the maximum production pinch leads to violation of maximum inventory limit. On the other hand, increase of demand below the maximum production pinch offers opportunity to increase the demand and thereby reduce the production fluctuation. For case (iii), where the pinch point is on the minimum inventory curve, inventory level reaches its minimum limit at the minimum production pinch point. In this case, there exists a limit of minimum production/supply below this pinch point, and the production/supply cannot be reduced further. Similar to the previous case, production level decreases after the minimum inventory limit is reached and the inventory level increases. It is very important to note that these observations differ from general production perceptions. It may be noted that increase of demand below the minimum production pinch leads to violation of minimum inventory limit. Contrary to case

4. ANALYSIS OF THE SHORTEST LENGTH CPC AND PINCH POINTS A feasible CPC, joining the initial and the final inventory points, with the shortest length is shown in Figure 2. According to Theorem 1, this CPC displays the level production plan, i.e., the minimum variation in production levels. It may be observed from Figure 2 that the shortest length curve touches the maximum and the minimum inventory curves at points “Pmax” and “Pmin”, respectively. The point “Pmax” is denoted as the maximum production pinch as at this point inventory level reaches its maximum limit and it cannot be further increased. Similarly, the point “Pmin” is denoted as the minimum production pinch as at this point inventory level reaches its minimum limit and it cannot be reduced further. It should be noted that violation of the minimum inventory pinch point signifies stock-out option, and violation of the maximum inventory pinch point signifies wastage of produced goods due to limitation of maximum inventory limits. It is assumed that these options are not exercised. Analyses of these pinch points are important when modification of CPC is required. On the 6944


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Figure 3. Various possible production plans, classified based on pinch points: (a) no pinch point, (b) maximum production pinch (single pinch), (c) minimum production pinch (single pinch), and (d) two pinch points.

(ii), decrease of demand below the minimum production pinch offers opportunity for reduction of production fluctuation. In case (iv), the sequence of pinch points suggests the limit of the production plan. Figure 3d shows a case where the first pinch point is the maximum production pinch and the next one is the minimum production pinch point. The first pinch point dictates the maximum production up to this pinch point, and production increases after this point. However, the increase in production is restricted to a minimum value by the second minimum production pinch point. Production decreases beyond this second pinch point. It may be observed that production is fixed between two pinch points. In the similar ways, cases with multiple pinch points can be analyzed. In this particular case (see Figure 3d), decrease of demand below the maximum production pinch and/or increase of demand below the minimum production pinch lead to violation of the inventory limit. These analyses are extremely important to handle small modifications in demand. There are certain measures (e.g., advertisements, marketing, pricing, etc.) by which demands can be controlled by a company. Hence, a company can decide whether there is scope of increasing/decreasing production or not according to the pinch points. For case (ii) production cannot be increased up to the pinch point. Similarly production cannot be decreased up to the pinch point in case (iii), and no modification in production is possible between two pinch points. It is important to note that the initial and the final inventories are starting and end points of CPC. The slope of CPC in the last segment always depends on the final inventory; if the final inventory is increased, production is to be increased, and if the final inventory is reduced, production has to be reduced. In some practical cases, final inventory may not be provided. In such cases, the CPC can be adjusted such that the variation in

the production can be minimized. Reduction or increase in production during the last phase of production can easily be determined based on the type of last pinch point. In the case of a maximum production pinch, production during the last phase may be reduced. Similarly, production during the last phase may be increased if the last pinch point is a minimum production pinch.

5. TARGETING ALGORITHM Based on Theorem 1, a graphical targeting algorithm is proposed to calculate production targets or input material supply forecast for a given demand forecast in inventory limits. Step 1. Generate CDC via plotting cumulative demand versus time. Step 2. Add minimum inventory limit to cumulative demand at each time point and plot the minimum inventory curve via joining these points on the same graph. Step 3. Similarly, add maximum inventory limit to the cumulative demand at each time point and plot the maximum inventory curve on the same graph. Step 4. Identify the initial inventory (I0) and final inventory + IH) points on starting and ending time, respectively, on (Dcum n the same graph. Step 5. The region between the minimum and maximum inventory curve is the feasible region, and any feasible curve joining the points of initial and final inventory and contained completely in this region represents a feasible cumulative production curve (Lemma 1). Step 6. Draw CPC with shortest length in the feasible region joining the initial and the final inventory points. The shortest length CPC between the initial and the final inventory point gives the level production/supply plan (Theorem 1). Step 7. The production curve can be varied as per other constraints, but it has to be within the region specified in step 5. 6945


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feasible production plan. This feasible production plan satisfies the production demand as well as the minimum and the maximum inventory limits. Now, a CPC with the minimum length is generated and shown in Figure 4. This curve represents the optimal cumulative production of level strategy as per Theorem 1. From this month-wise production is calculated as shown in Table 1. It may be observed from Figure 4 that the CPC touches the maximum inventory curve three time consecutively at the first, second, and third month (i.e., three maximum production pinch points). After that, CPC again touches the minimum inventory curve at the fourth month (i.e., a minimum production pinch point). It may be noted that these results are identical to the results reported by Foo et al.15 6.2. Example 2: Bioenergy Production. Fuel demand and inventory limit profile over the 6 year planning horizon for bioenergy production are given in Table 2.15 Inventory level at

To improve computational accuracy and scalability, an equivalent algebraic approach may be adopted. In this paper, methodology proposed by Sahu and Bandyopadhyay41 is applied.

6. ILLUSTRATIVE EXAMPLES The proposed algorithm may be applied to determine an optimal production plan. Applicability of the proposed algorithm is demonstrated through various illustrative examples. The first example demonstrates the applicability of the proposed algorithm in APP to general production planning problems. Other two examples demonstrate the applicability of the proposed algorithm in APP of the energy supply chain. 6.1. Example 1: Production Planning. This example does not represent an energy supply chain related problem. However, applicability of the proposed procedure for a generic supply chain problem is demonstrated. The monthly forecast of demand for the duration of six months is given in Table 1.15

Table 2. Data for Example 2: Bio-Energy Production Parameters

Table 1. Level Production Plan for Example 1 month

forecasted demand (units)

minimum inventory curve (units)

maximum inventory curve (units)

optimal CPC (units)

month-wise production (units)

1 2 3 4 5 6

1600 3000 3200 5060 1760 1760

1800 4800 8000 13060 14820 16580

3600 6600 9800 14860 16620 18380

3600 6600 9800 13060 14970 16880

2600 3000 3200 3260 1910 1910

The maximum and the minimum limits of inventory are given to be 200 and 2000 units, respectively. Initial and final inventory are assumed to be 1000 units and 500 units, respectively. Cumulative demand is plotted vs time to generate the CDC (Figure 4). The minimum limit (200 units) is added to

year

fuel demand (103 t)

minimum inventory (103 t)

maximum inventory (103 t)

1 2 3 4 5 6

300 300 300 600 600 600

75 75 75 150 150 150

500 500 500 750 750 750

the start of the planning horizon is given as 450 × 103 t. It may be noted that the final inventory is not specified in this problem. It may also be observed from Table 2 that the minimum and the maximum inventory limits are varying over time. There are two cases considered for this problem: (i) biomass production is uniformly distributed over a year, and (ii) biomass production is restricted over a small period, at the beginning of every year. In the original problem, Foo et al.15 had considered only the second case. In countries like India, multiple food crops are cultivated during the entire year. For such cases, biomass production may be assumed to be uniformly distributed over a year (case i). Cumulative demand is plotted vs time to generate the CDC. The minimum and the maximum inventory limits, respective to the time interval, are added to cumulative demand in order to generate the minimum and the maximum inventory curves (Figure 5). As per Theorem 1, a shortest length curve starting at the initial inventory level and lying within the maximum inventory and the minimum inventory curves is plotted to obtain the optimum cumulative production for level strategy (Figure 5). Annual productions are calculated from the cumulative production as 316.67 × 103 t/y for the first three years and after that 483.33 × 103 t/y for the remaining periods. Hence, the total variation in production is 166.67 × 103 t. Figure 5 shows the demand curve, the minimum inventory curve, the maximum inventory curve, and the optimal CPC for the problem. It can be observed from Figure 5 that there is a single pinch point on CPC which is on the maximum inventory curve. Hence, the maximum production up to the maximum production pinch point (i.e., 316.67 × 103 t/y) is dictated by this pinch point after which it is increased (i.e., 483.33 × 103 t/ y).

Figure 4. Various composite curves for APP of Example 1.

cumulative demand and plotted to generate the minimum inventory curve (Figure 4). Similarly, the maximum limit (2000 units) is added to the CDC and plotted on the same diagram to generate the maximum inventory curve (Figure 4). As per Lemma 1, any feasible curve, joining the initial and the final inventory points, that is contained completely between the minimum and the maximum inventory curves, represents a 6946


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mentioned that 10% of the demand of the interval must be maintained as the minimum inventory. Maximum inventory limit is given to be 3000 t of coal. A level supply forecast is to be calculated in order to ensure the availability of coal for 6 weeks and limits of inventories are also to be ensured. CDC is generated by plotting cumulative demand vs time. The minimum and the maximum inventory curves are plotted on same diagram (Figure 6). A month-wise production is

Figure 5. Various composite curves for APP of Example 2.

As described by Foo et al.,15 biomass is produced in some countries at the end of harvesting season over a period of few weeks only. In such cases, biomass availability may be assumed to be available at the start of the planning year (case ii). Just before the harvest, the inventory level reaches its minimum level, and just after harvesting, the biomass inventory level reaches its maximum level. This leads to a stepwise production curve, as shown by Foo et al.15 For such cases, eq 13 may be rewritten as min Picum ≥ Dicum − 1 + Ii − 1

Figure 6. Composite curves for continuous input material supply planning (Example 3).

(19)

calculated from the cumulative production (Table 3). It is important to note that if there are other constraints (such as, economic constraints) then the production curve can be varied from determined level strategy plan but it has be to within the maximum and the minimum inventory curve. Figure 6 shows the demand curve, the minimum inventory curve, the maximum inventory curve, and the optimal CPC for the problem. From Figure 6 as well from Table 3, it may be observed that there are multiple pinch points. In total there are three pinch points: first is a minimum production pinch (week 2), second is a maximum production pinch (week 3), and third is again a minimum production pinch (week 5). Hence, there is a minimum production up to the first pinch point after that the production is fixed based on the different pinch points as discussed. It should be noted that the flexibility in supply variation is very limited. This is not a significant problem as a coal based power plant operates as a base-load power plant without significant variation in power production. However, a tight supply scheme severely limits the overall flexibility and reliability of the power plant. It is suggested to increase the maximum inventory to increase flexibility in its supply plan and thereby increase flexibility and reliability of the power plant itself.

As the constraint 13 is modified, the minimum inventory curve may be modified as i−1

Iimin =

∑ Dj + Iimin −1 (20)

j=1

Once these transformations are made, the proposed procedure can be applied directly to obtain the optimal production plan under level strategy (not shown for brevity). From the shortest length CPC, annual biomass productions are calculated to be 325 × 103 t/y for the first two years, 550 × 103 t/y for the third year, and 600 × 103 t/y for the remaining years (Table 3). As expected, the total variation in production is increased to 275 × 103 t, a 65% increase compared to case i. It may be noted that the production plan reported by Foo et al.15 consists of 300 × 103, 350 × 103, and 550 × 103 t/y for first three years and 600 × 103 t/y for the remaining three years. This leads to a total variation in production of 300 × 103 t. The proposed methodology reduces the production variation by 9.1%. 6.3. Example 3: Coal Inventory Management. Coal demand profile over the 6-week planning horizon for a 50 MW captive cogeneration power plant is given Table 3. It is Table 3. Calculation of Coal Supply Forecast for Example 3 week

demand forecast (t)

minimum inventory (t)

maximum inventory (t)

cumulative demand curve (t)

minimum inventory curve (t)

maximum inventory curve (t)

cumulative supply curve (t)

weekly supply (t)

1 2 3 4 5 6

7300 10600 6500 9100 10200 6900

730 1060 650 910 1020 690

3000 3000 3000 3000 3000 3000

7300 17900 24400 33500 43700 50600

8030 18960 25050 34410 44720 51290

9800 20400 26900 36000 46200 53100

9480 18960 (pinch) 26900 (pinch) 35810 44720 (pinch) 53100

9480 9480 7940 8910 8910 8380

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7. CONCLUSIONS Matching between energy production and demand are essential for a reliable energy system. Energy storage options are required to reduce the mismatch between the generation and utilization. Energy storage technology that employs energy conversion steps is inefficient due to efficiencies involved during energy conversion. For example, storage of electrical energy in battery involves both charging and discharging efficiencies and the overall cyclic efficiency is significantly low. On the other hand storage of energy or raw material without energy conversion is efficient. For example, storage of coal for a coal power plant is efficient. However, this involves proper management of coal inventory. An aggregate planning is required for such an energy system to ensure the availability of energy to consumers as well to maintain input material supply for the company. Aggregate production planning for an energy supply chain is presented in this paper which is equally applicable for production planning of supply demand as well as planning of input material supply. The graphical as well as algebraic methodologies determine the range of production at different levels and calculate the production at each level with least variation in production levels of different time periods to satisfy the demands of each time period and also to maintain the limits of inventory. On the basis of the mathematical features of this problem, it is shown that the graphical representation of the aggregate production planning problem is equivalent to the Euclidean shortest path problem in computational geometry. Also, it is shown that an APP problem may be classified according to number of pinch points, based on which production/supply may be deviated from level production plan. Present research is directed toward extending the APP to various capacity planning and inventory optimization problems involving economical aspects.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +91-22-25767894. Fax: +91-22-25726875. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Authors would like to thank the Industrial Research and Consultancy Centre, Indian Institute of Technology Bombay, for the financial support given towards the project “Development of Process Integration Methodologies with Multiple Qualities” (13IRAWD004).

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NOTATION a−i Variables denoting production/supply variations Di Demand in ith time period Dcum Cumulative demand at end of ith time period i I0 Initial inventory IH Final inventory Ii Inventory at end of ith time period Imin Minimum inventory limit at end of ith time period i Maximum inventory at end of ith time period Imax i Pi Production/supply in ith time period Pcum Cumulative production/supply at end of ith time period i a+i ,

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Targeting Aggregate Production Planning for an Energy Supply Chain

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