Dynamics of Quality as a Strategic Variable in Complex Food Supply Chain Network Competition

Transcription

1 Dynamics of Quality as a Strategic Variable in Complex Food Supply Chain Network Competition Anna Nagurney 1, Deniz Besik 1 and Min Yu 2 1 Department of Operations and Information Management Isenberg School of Management University of Massachusetts Amherst, MA Pamplin School of Business Administration University of Portland University of Massachusetts Portland, Oregon NEDSI Conference in Providence, Rhode Island April 12-14, 2018

2 This presentation is based on the paper: Nagurney, A., Besik, D., and Yu M., Dynamics of Quality as a Strategic Variable in Complex Food Supply Chain Network Competition: The Case of Fresh Produce, to appear in the focus issue on Supply Network Dynamics in Chaos.

3 Background and Motivation Food supply chains, as noted in Yu and Nagurney (2013), are distinct from other product supply chains. Fresh produce is exposed to continuous and significant change in the quality of food products throughout the entire supply chain from the points of production/harvesting to points of demand/consumption. The quality of food products is decreasing with time, even with the use of advanced facilities and under the best processing, handling, storage, and shipment conditions (Sloof, Tijskens, and Wilkinson (1996) and Zhang, Habenicht, and Spieß (2003)).

4 Background and Motivation It has been discovered that the quality of fresh produce can be determined scientifically using chemical formulae, which include both time and temperature. The initial quality is also very important and food producers, such as farmers, have significant control over this important strategic variable at their production/harvesting sites. There are great opportunities for enhanced decision-making in this realm that can be supported by appropriate models and methodological tools.

5 Literature Review We note that early contributions focused on perishability and, in particular, on inventory management (see Ghare and Schrader (1963), Nahmias (1982, 2011) and Silver, Pyke, and Peterson (1998) for reviews). More recently, some studies have proposed integrating more than a single supply chain network activity (see, e.g., Zhang, Habenicht, and Spieß (2003), Widodo et al. (2006), Ahumada and Villalobos (2011), and Kopanos, Puigjaner, and Georgiadis (2012)). Yu and Nagurney (2013) have emphasized the need to bring greater realism to the underlying economics and competition on food supply chains. Monteiro (2007) studied the traceability in food supply chains theoretically. Additional modeling and methodological contributions in the food supply chain and quality domain have been made by Blackburn and Scudder (2009) and by Rong, Akkerman, and Grunow (2011). Besik and Nagurney (2017) formulate short fresh produce supply chains with the inclusion of the dynamics of quality, in the context of farmers markets.

6 Contribution We construct a competitive supply chain network model for fresh produce under oligopolistic competition among the food firms, who are profit-maximizers. The firms have, as their strategic variables, not only the product flows on the pathways of their supply chain networks from the production/harvesting locations to the ultimate points of demand, but also the initial quality of the produce that they grow at their production locations. The consumers at the retail outlets (demand points), differentiate the fresh produce from the distinct firms and reflect their preferences through the prices that they are willing to pay which depend on quantities of the produce as well as the average quality of the produce associated with the firm and retail outlet pair(s). Quality of the produce reaching a destination node depends on its initial quality and on the path that it took with each particular path consisting of specific links, with particular characteristics of physical features of time, temperature, etc.,.

7 Preliminaries on Quality Fresh foods deteriorate since they are biological products, and, therefore, lose quality over time. The rate of quality deterioration can be represented as a function of the microenvironment, the gas composition, the relative humidity, and the temperature (Taoukis and Labuza (1989)). Labuza (1984) demonstrated that the quality of a food attribute, Q, over time t, which can correspond, depending on the fruit or vegetable, to the color change, the moisture content, the amount of nutrition such as vitamin C, or the softening of the texture, can be formulated via the differential equation: Q t = kqn = Ae ( E/RT ) Q n. (1) In (1), k is the reaction rate and is defined by the Arrhenius formula, Ae ( E/RT ), A is a pre-exponential constant, T is the temperature, E is the activation energy, and R is the universal gas constant (cf. Arrhenius (1889)).

8 Preliminaries on Quality If the reaction order n is zero, that is, Q t = k, and the initial quality is denoted by Q 0, we can quantify the remaining quality Q t at time t (Tijskens and Polderdijk (1996)) according to: Q t = Q 0 kt. (2) Examples of fresh produce that follow a reaction order of zero include watermelons and spinach. If the reaction order is 1, known as a first order reaction, the quality decay function is then given by the expression: Q t = Q 0 e kt. (3) Popular fruits that follow first order kinetics include peaches, and strawberries, as well as vegetables such as: peas, beans, carrots, avocados, and tomatoes.

9 Network Topology M1 1 C1,1 1 C1,2 1 D1,1 1 Food Firm 1 Food Firm I 1 I Production/Harvesting M 1 n M M 1 I M I nm Shipment C 1 nc C1,1 I C I nc Processing C 1 nc C1,2 I C I nc Shipment D 1 nd D1,1 I D I nd Storage D1,2 1 D 1 nd D1,2 I D I nd Distribution Distribution R1 RnR Retail Outlets

10 Quality Over a Path We let L i denote the set of directed links in the supply chain network of food firm i; where i = 1,..., I, which consists of a set of production links, L i 1, and a set of post-harvest links, Li 2, that is, Li L i 1 Li 2. We allow for a distinct initial quality q0a i associated with each top-tiered production link a L i 1 ; i = 1,..., I, since distinct production sites of a firm may have different associated quality of the produce that is harvested because of soil conditions, investment in irrigation, types of pesticides, and fertilizers used, etc. We let β b denote the quality decay incurred on link b, for b L i 2, which is a factor that depends on the reaction order n, the reaction rate k b, and the time t b on link b, according to: k b t b,, if n = 0, b L i 2, i, β b (4) e k bt b, if n = 1, b L i 2, i. Here, the reaction rate is: k b = Ae ( E/RTb), b L i 2, i. (5)

11 Quality Over a Path We can have multiple paths from an origin node i to a destination node k, Pk i denotes the set of all paths that have origin i and destination k. The quality q p, over a path p, joining the origin node i, with a destination node k, with the incorporation of the quality deterioration of the fresh produce, is, hence: q i 0a + β b, if n = 0, p Pk i, i, k, b p L i 2 q p (6) q0a i β b, if n = 1, p Pk i, i, k. b p L i 2 Here, q0a i is the initial quality of the fresh produce on a top-most link a from an origin node i and in the path p under consideration.

12 The Competitive Fresh Produce Supply Chain Network Model with Quality Nonnegativity Constraint of the Path Flows For each path p, joining an origin node i with a destination node k, the following nonnegativity condition must hold: x p 0, p P i k; i = 1,..., I ; k = R 1,..., R nr. (7) Nonnegativity Constraint of the Initial Quality The initial quality of the fresh produce on the top-most links a of an origin node i, must be nonnegative, that is: Maximum Initial Quality q i 0a 0, a L i 1; i = 1,..., I. (8) We assume that the quality is bounded from above by a maximum value; hence, we have that: q i 0a q i 0a, a L i 1; i = 1,..., I. (9)

13 The Competitive Fresh Produce Supply Chain Network Model with Quality Link Flows The conservation of flow equations that relate the link flows of each food firm i; i = 1,..., I, to the path flows are given by: f l = p P x p δ lp, l L i ; i = 1,..., I, (10) where δ ap = 1, if link a is contained in path p, and 0, otherwise. Capacity over the Link Flows Link flows must satisfy capacity constraints, we have that: f l u l, l L. (11) Capacity over the Path Flows Observe that, in view of the conservation of flow equations (10), we can rewrite (11) in terms of path flows as: x p δ lp u l, l L. (12) p P

14 The Competitive Fresh Produce Supply Chain Network Model with Quality Average Quality The average quality product of firm i, at retail outlet k, is given by the expression: p P q k ˆq ik = i p x p, i = 1,..., I ; k = R 1,..., R nr. (13) x p p P i k Demand The demand for food firm i s fresh food product at retail outlet k is denoted by d ik and is equal to the sum of all the fresh produce flows on paths joining (i, k), so that: x p = d ik, i = 1,..., I ; k = R 1,..., R nr. (14) p P i k Demand Price Function We denote the demand price of food firm i s product at retail outlet k by: ρ ik = ρ ik (d, ˆq), i = 1,..., I ; k = R 1,..., R nr. (15)

15 The Competitive Fresh Produce Supply Chain Network Model with Quality Cost of Production/Harvesting The cost of production/harvesting at firm i s production site a depends, in general, on the initial quality q i 0a, and the product flow on the production/harvesting link, that is: ẑ a = ẑ a (f a, q i 0a), a L i 1; i = 1,..., I. (16) Operational Cost Function We define the operational cost functions associated with the remaining links in the supply chain network as: ĉ b = ĉ b (f ), b L i 2; i = 1,..., I. (17) Vector of Path Flow Strategies The vector of path flows associated with firm i; i = 1,..., I is: X i {{x p } p P i }} R n P i +, P i k=r1...,rn R P i k. (18) Vector of Initial Quality Strategies the vector of initial quality levels, associated with firm i; i = 1,..., I is: q i 0 {{q i 0a} a L i 1}} R n L i 1 +. (19)

16 The Competitive Fresh Produce Supply Chain Network Model with Quality Utility Function The utility of firm i; i = 1,..., I, is expressed as: R nr U i = ρ ik (d, ˆq)d ik ( k=r 1 Rewritten Demand Price Functions In view of (6), (13), and (14), we can rewrite (15) as: a L i 1 ẑ a (f a, q i 0a) + b L i 2 ĉ b (f )). (20) ˆρ ik (x, q 0 ) ρ ik (d, ˆq), i = 1,..., I ; k = R 1,..., R nr. (21) Vector of the Profits We can define the vector of the profits of all the firms for all firms i; i = 1,..., I, with the I -dimensional vector Û, that is: Û = Û(X, q 0 ). (22)

17 Supply Chain Network Nash Equilibrium with Fresh Produce Quality A fresh produce path flow pattern and initial quality level (X, q 0 ) K = I i=1 K i constitutes a supply chain network Nash Equilibrium with fresh produce quality if for each food firm i; i = 1,..., I : Û i (X i, X i, q i 0, q i 0 ) Û i (X i, X i, q i 0, q i 0 ), (X i, q i 0) K i, (23) where X i (X1,..., X i 1, X i+1,..., X I ), q i 0 (q0 1,..., qi 1 0, q0 i+1,..., q I and K i {(X i, q0 i ) X i R n P i +, q0 i Rn Li 1 +, (9) and (12) hold for l L i }. 0 ) An equilibrium is established if no food firm can unilaterally improve upon its profit by altering its product flows and initial quality at production sites in its supply chain network, given the product flows and initial quality decisions of the other firms.

18 Variational Inequality Formulation of the Governing Equilibrium Conditions Assume that, for each food firm i; i = 1,..., I, the profit function Ûi(X, q 0 ) is concave with respect to the variables X i and q0 i, and is continuously differentiable. Then (X, q0 ) K is a supply chain network Nash Equilibrium with fresh produce quality according to Definition 1 if and only if it satisfies the variational inequality: I I Xi Û i (X, q0 ), X i Xi q i 0 Û i (X, q0 ), q0 q i 0 i 0, (X, q 0 ) K, i=1 i=1 where, denotes the inner product in the corresponding Euclidean space. (24)

19 Variational Inequality Formulation of the Governing Equilibrium Conditions Xi Û i (X, q 0 ) denotes the gradient of Û i (X, q 0 ) with respect to X i and q i 0 Û i (X, q 0 ) denotes the gradient of Ûi(X, q 0 ) with respect to q0 i. The solution of variational inequality (22), in turn, is equivalent to the solution of the variational inequality: determine (x, q0, λ, γ ) K 1 satisfying: + I R nr i=1 k=r 1 p Pk i [x p x p ] + I i=1 a L i 1 Ẑ i (x, q i 0 ) x p + Ĉ i (x ) x p R nr + γl δ lp ˆρ ik (x, q ˆρ ij(x, 0 ) q x p l L i j=r 1 I Ẑ i (x, q0 i ) R nr q0a i + λ ˆρ ij(x, a q q i j=r 1 0a i=1 a L i 1 [ q i 0a q i 0a] [λa λ a]+ 0 ) r P i j x r 0 ) r P i j [q i 0a q i 0a] [ I u l xr δ lr ] [γ l γl ] 0, (x, q 0, λ, γ) K 1. l L i r P (25) i=1 x r

20 Variational Inequality Formulation of the Governing Equilibrium Conditions We have K 1 {(x, q 0, λ, γ) x R+ np, q 0 R nl 1 +, λ R nl 1 +, γ R+ nl } and for each path p; p Pk i ; i = 1,..., I ; k = R1,..., RnR : Ẑ i (x, q0 i ) ẑ a(f a, q0a i ) δ ap, (26a) x p f a a L i 1 For each a; a L i 1 ; i = 1,..., I, ˆρ ij(x, q 0) x p Ĉ i (x) x p b L i 2 ( ρij(d, ˆq) ˆq ik ρij(d, ˆq) + d ik ˆρ ij(x, q 0) q i 0a Ẑ i (x, q0 i ) q0a i Rn R h=r1 s Ph i l L i ĉ b(f ) f l δ lp, (26b) ) q p r P q k i r x r x r ( r P x k i r ) 2. (26c) r P i k ẑa(fa, qi 0a ) q0a i, (26d) x s r Ph i x r ρ ij(d, ˆq) ˆq ih q s q0a i. (26e) qs In particular, if link a is not included in path s, = 0; if link a is included in path s, following (6), we have: q0a i 1, if n = 0, q s q0a i = β b, if n = 1. b s L i 2 (26f )

21 Variational Inequality Formulation of the Governing Equilibrium Conditions Proof: (22) follows directly from Gabay and Moulin (1980); see also Dafermos and Nagurney (1987). Under the imposed assumptions, (22) holds if and only if (see, e.g., Bertsekas and Tsitsiklis (1989)) the following holds: For each path p; p Pk i, we have that Ûi xp = [ RnR ρij(d, ˆq)dij ( j=r1 e L ẑe(fe, q i 0e i ) + 1 b L ĉb(f ))] i 2 xp = [ RnR ρij(d, ˆq)dij] j=r1 [ e L ẑe(fe, q i 0e i )] 1 xp xp Rn R [ ] [ρij(d, ˆq)dij] dik [ρij(d, ˆq)dij] ˆqik = + dik xp ˆqik xp j=r1 ĉb(f ) fl fl xp b L i l L 2 i [ Rn R ρij(d, ˆq) ρij(d, ˆq) =ρik(d, ˆq) + dij + dik ˆqik j=r1 ˆqik + [ [ )] r P i xr ] k r Pk i xr ] xp a L i 1 [ ( Rn R ρij(d, ˆq) ρij(d, ˆq) =ρik(d, ˆq) + + dik ˆqik j=r1 ẑa(fa, q0a i ) δap ĉb(f ) fa fl a L i 1 b L i l L 2 i [ b L ĉb(f )] i 2 xp a L i 1 ẑe(fe, q0e i ) fa e L i 1 fa xp ( ˆqik dij [ [ r P i qr xr ] k r Pk i qr xr ] xp ẑa(fa, q i 0a ) fa qp r Pk i δap ĉb(f ) δlp fl b L i l L 2 i )] xr r Pk i qr xr ( r Pk i xr ) 2 dij δlp. (27)

22 Variational Inequality Formulation of the Governing Equilibrium Conditions For each link a; a L i 1, we know that: Û i q i 0a = [ RnR ρ ij(d, ˆq)d j=r1 ij ( e L ẑ i e (f e, q0e i ) + 1 b L ĉ i b (f ))] 2 q0a i = [ RnR ρ ij(d, ˆq)d j=r1 ij ] [ e L ẑ i e (f e, q0e i )] 1 Rn R = Rn R h=r1 j=r1 Rn R = Rn R h=r1 j=r1 Rn R = Rn R h=r1 j=r1 Rn R = Rn R q i 0a q i 0a [ρ ij (d, ˆq)d ij ] ˆq ih ˆq ih q0a i ẑ a(f a, q0a i ) q0a i ρ ij (d, ˆq) ˆq ih d ij ˆq ih [ [ r P q i r x r ] h r P q h i r x r ] q0a i ρ ij (d, ˆq) 1 d ij ˆq ih r P x h i r s Ph i h=r1 j=r1 s Ph i x s r Ph i x r ρ ij (d, ˆq) ˆq ih q s q i 0a [ r P q h i r x r ] q s [ b L ĉ i b (f )] 2 q0a i ẑ a(f a, q i 0a ) q i 0a q s q i 0a ẑ a(f a, q i 0a ) q i 0a d ij ẑ a(f a, q0a i ) q0a i. (28)

23 Simple Illustrative Examples Food Firm 1 Food Firm Production/Harvesting 7 M1 1 M1 2 C 1 1,1 C 1 1,2 D 1 1,1 2 Shipment 8 C1,1 2 3 Processing 9 C1,2 2 4 Shipment 10 D1, Storage D1,2 1 D1,2 2 Distribution 6 12 R1 Retail Outlet

24 Simple Illustrative Examples The cost of production/harvesting is higher for Food Firm 1 since it uses better machinery and has invested more into the necessary chemicals to maintain the soil quality, with the top-tiered link cost functions being: ẑ 1 (f 1, q 1 01) = f f 1 + 3q 1 01, ẑ 7 (f 7, q 2 07) = f q There are ten additional links, belonging to the sets L 1 2 and L2 2, in the supply chain network and their total link cost functions are: ĉ 2 (f 2 ) = 5f f 2, ĉ 3 (f 3 ) = 2f 2 3, ĉ 4 (f 4 ) = 2f 2 4 +f 4, ĉ 5 (f 5 ) = 3f 2 5, ĉ 6 (f 6 ) = f 2 6 +f 6, ĉ 8 (f 8 ) = f f 8, ĉ 9 (f 9 ) = 3f f 9, ĉ 10 (f 10 ) = 2f 2 10, ĉ 11 (f 11 ) = 6f f 11, ĉ 12 (f 12 ) = 6f f 12. The total link cost functions are constructed according to the assumptions made for Food Firm 1, Food Firm 2, and Retail Outlet R 1.

25 Quality Decay for the Illustrative Examples Link b Hours Temperature (Celsius) β b (n = 0) β b (n = 1) The shipment time is longer for Food Firm 1 than for Food Firm 2 because of their respective distances to their processing facilities, which can be seen from the time difference between link 2 and link 8.

26 Example 1a: Linear Quality Decay (Zero Order Kinetics) Since there exists one path for each food firm: d 11 = x p 1, d 21 = x p 2. In a zero order quality decay function, the reaction order n = 0, and the quality q p over a path p can be determined by the appropriate formula for n = 0. The initial quality variables are q 1 01 and q2 07. The demand price functions are: ˆρ 11 (x, q 0 ) ρ 11 (d, ˆq) = 2x p1 x p2 + q p 1 x p1 x p and ˆρ 21 (x, q 0 ) ρ 21 (d, ˆq) = 3x p2 x p1 + q p 2 x p2 + 90, x p2 with the path quality q p for the two paths constructed according to zero order decay function, for n = 0 for i = 1, 2, given by: q p1 = q β 2 + β 3 + β 4 + β 5 + β 6 = q , q p2 = q β 8 + β 9 + β 10 + β 11 + β 12 = q

27 Example 1a: Linear Quality Decay (Zero Order Kinetics) In order to obtain the equilibrium path flows and the equilibrium initial quality levels that satisfy variational inequality (25), the following expressions must be equal to 0: Ẑ 1 (x, q 1 01 ) x p1 + Ĉ 1 (x ) x p1 ˆρ 11 (x, q 0 ) ˆρ 11(x, q 0 ) x p1 x p 1 = 0, (29) Ẑ 1 (x, q 1 01 ) q 1 01 ˆρ 11(x, q0 ) q01 1 xp 1 = 0, (30) Ẑ 2 (x, q 2 07 ) x p2 + Ĉ 2 (x ) x p2 ˆρ 21 (x, q 0 ) ˆρ 21(x, q 0 ) x p2 x p 2 = 0, (31) Ẑ 2 (x, q 2 07 ) q 2 07 ˆρ 21(x, q0 ) q07 2 xp 2 = 0. (32)

28 Example 1a: Linear Quality Decay (Zero Order Kinetics) Grouping the terms above corresponding to each equation (29) (32) we obtain the following system of equations: 32x p 1 + x p 2 q 1 01 = , 3 x p 1 = 0, with solution: x p x p 2 q 2 07 = , 3 x p 2 = 0, x p 1 = 3, x p 2 = 3, q 1 01 = , q 2 07 = The path quality levels are: q p1 = 19 and q p2 = 49, the demand prices are: ρ 11 = 110 and ρ 21 = 127, with Food Firm 1 enjoying a profit (in dollars) of Û 1 (X, q 0 ) = and Food Firm 2 a profit of Û2(X, q 0 ) =

29 Example 1b: Exponential Quality Decay (First Order Kinetics) The product now has an exponential quality decay with a reaction order n = 1. The quality levels of the paths are constructed and the β b values in Table 1, for n = 1 for i = 1, 2, yielding: q p1 = q 1 01 β 2 β 3 β 4 β 5 β 6 = (q 1 01)(0.7359), q p2 = q 2 07 β 8 β 9 β 10 β 11 β 12 = (q 2 07)(0.9418). We now proceed to solve the equations (29) (32) for this example, with the following terms for paths p 1 and p 2 presented for completeness and convenience: ˆρ 11 (x, q0 ) = 2x + x (q1 01 )(0.7359)(x ) p1 p1 p2 x 100 = 2x + x p1 p2 (q1 01 )(0.7359) 100, p1 ˆρ 11(x, q0 ) q01 1 x p1 = x, p1 ˆρ 21 (x, q0 ) = 3x + x (q2 07 )(0.9418)(x ) p2 p2 p1 x 90 = 3x + x p2 p1 (q2 07 )(0.9418) 90, p2 ˆρ 21(x, q0 ) q07 2 x p2 = x. p2

30 Example 1b: Exponential Quality Decay (First Order Kinetics) We obtain the following system of equations: 32x p1 + x p q1 01 = 80, x p1 = 0, x p1 + 44x p q2 07 = 86, x p2 = 0. Straightforward calculations yield the following equilibrium path flows and equilibrium initial quality levels: x p1 = , x p2 = , q1 01 = , q 2 07 = The path quality levels are, q p1 = and q p2 = We obtain the following equilibrium demand prices at the retail outlet for Food Firm 1 and Food Firm 2, respectively: ρ 11 = , ρ 21 = The profits of the food firms are calculated, in dollars, as: Û 1 (X, q 0) = , Û2(X, q 0 ) =

31 The Algorithm - Euler Method Euler method, which is induced by the general iterative scheme of Dupuis and Nagurney (1993) is a solution methodology of the Variational Inequality Problem. Specifically, iteration τ of the Euler method is given by: X τ+1 = P K (X τ a τ F (X τ )), (33) The Euler method, the sequence {a τ } must satisfy: τ=0 a τ =, a τ > 0, a τ 0, as τ.

32 The Euler Method Explicit Formulae For each path p Pj i, i, j, compute: x τ+1 p RnR = max{0, xp τ + ατ (ˆρ ik(x τ, q0 τ ) + j=r1 ˆρ ij (x τ, q τ 0 ) x p xr τ r P i j Ẑ i (x τ, q iτ 0 ) x p Ĉ i (x τ ) x p l L i γ τ l δ lp )}. (34) For each initial quality level a L i 1, i, in turn, compute: q iτ+1 0a Rn R = max{0, q0a iτ + α τ ( j=r1 ˆρ ij (x τ, q τ 0 ) q i 0a r P i j x τ r Ẑ i (x τ, q0 iτ ) q0a i λ τ a )}. (35) The Lagrange multiplier for each top-most link a L i 1 ; i = 1,..., I, associated with the initial quality bounds is computed as: λ τ+1 a = max{0, λ τ a + α τ (q iτ 0a q i 0a)}. (36) The Lagrange multiplier for each link l L i ; i = 1,..., I, associated with the link capacities is computed according to: γ τ+1 l = max{0, γl τ + α τ ( xr τ δ lr u l )}. (37) r P

33 A Case Study of Peaches We focus on the peach market in the United States, specifically in Western Massachusetts. It is noted that, in 2015, the United States peach production was 825,415 tons in volume, and 606 million dollars in worth (USDA NASS (2016), Zhao et al. (2017)). We selected two orchards from Western Massachusetts: Apex Orchards and Cold Spring Orchard, located, respectively, in Shelburne, MA and Belchertown, MA. The orchards sell their peaches to two retailers, Whole Foods, located in Hadley, MA, and Formaggio Kitchen, located in Cambridge, MA. The mode of transportation for both of the orchards is trucks. The color change attribute of peaches, in the form of browning, follows a first-order, that is, an exponential decay function.

34 Network Topology of the Case Study Apex Orchards Cold Spring Orchard 1 2 Production/Harvesting M1 1 M2 1 M Shipment 8 C1,1 1 C2,1 1 C1, Processing C1,2 1 C2,2 1 C1, Shipment D1,1 1 D2,1 1 D1, Storage D1,2 1 D2,2 1 D1,1 2 Distribution R1 R2 Whole Foods Formaggio Kitchen

35 Parameters for the Calculation of Quality Decay Link b Hours Temperature (Celsius) β b (n = 1)

36 Cost Functions, Capacities and Upper Bounds for the Numerical Examples Table: Total Production / Harvesting Cost Functions, Link Capacities, and Upper Bounds on Initial Quality Link a ẑa(fa, q0a i ) ua qi 0a 1.002f (q1 01 ) f (q1 02 ) f (q1 03 ) Table: Total Operational Link Cost Functions and Link Capacities Link b ĉb(f ) ub 4.001f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f We report the total production / harvesting cost functions, the upper bounds on the initial quality, the total operational cost functions, and the link flow capacities. The Euler method is implemented in FORTRAN and a Linux system at the University of Massachusetts used for the computations. The data is gathered from Sumner and Murdock (2017) and Dris and Jain (2007), in which the authors made a sample cost analysis. The time horizon, under consideration, is that of a week. The Euler method is implemented in FORTRAN and a Linux system at the University of Massachusetts used for the computations. The sequence is, a τ = {1, 1 2, 1 2, 1 3,...}, with the convergence tolerance being 10 7.

37 Example 1 - Baseline It is known that both retailers sell high quality food products, with Formaggio Kitchen selling peaches at a higher price due to its emphasis on quality. Through conversations at the retailers, we concluded that Apex Orchards sell their peaches at a higher price. Demand Price Functions of Apex Orchards: ρ 11 =.02d 11.01d ˆq , ρ 12 =.02d 12.01d ˆq Demand Price Functions of Cold Spring Orchard: ρ 21 =.02d d ˆq , ρ 22 =.02d d ˆq

38 Equilibrium Flows, Equilibrium Initial Quality, and the Equilibrium Lagrange Multipliers Table: Equilibrium Link Flows and the Equilibrium Link Lagrange Multipliers Link a fa q0a i γa λ a Table: Equilibrium Link Flows, Equilibrium Initial Quality, and the Equilibrium Production Site Lagrange Multipliers Link b fb γb

39 Equilibrium Prices, Demands, Average Quality and Profits Equilibrium prices at the demand markets: ρ 11 = 17.67, ρ 12 = 19.00, ρ 21 = 15.47, ρ 22 = Equilibrium demands: d11 = , d 12 = , d21 = 47.80, d22 = Average Quality: ˆq 11 = 93.40, ˆq 12 = 92.56, ˆq 21 = 46.60, ˆq 22 = Profits: U 1 = 3, , U 2 =

40 Example 2 - Disruption Scenario 1 We now consider a disruption scenario in which a natural disaster has significantly affected the capacity of the orchard production sites of both orchards. Such an incident occurred in 2016 in the Northeast of the United States when extreme weather in terms of cold temperatures decimated the peach crop. We now have the following capacities on the production/harvesting links: Link a u 1 = 100, u 2 = 150, u 3 = 80. f a q i 0a γ a λ a Table: Equilibrium Link Flows, Equilibrium Initial Quality, and the Equilibrium Production Site Lagrange Multipliers

41 Equilibrium Link Flows and the Equilibrium Link Lagrange Multipliers Link b γb f b

42 Equilibrium Prices, Demands, Average Quality and Profits Equilibrium prices at the demand markets: ρ 11 = 18.12, ρ 12 = 19.40, ρ 21 = 15.74, ρ 22 = Equilibrium demands: d11 = , d 12 = , d21 = 37.67, d22 = Average Quality: ˆq 11 = 73.42, ˆq 12 = 72.68, ˆq 21 = 7.83, ˆq 22 = Profits: U 1 = 2, , U 2 =

43 Example 3 - Disruption Scenario 2 We consider a disruption that affects transportation in that the links 5 and 6 associated with the supply chain network of Apex Orchards are no longer available. This can occur and has occurred in western Massachusetts as a result of flooding. We now have the following capacities on the production/harvesting links: u 5 = 0, u 6 = 0. Link a f a q i 0a γ a λ a Table: Equilibrium Link Flows, Equilibrium Initial Quality, and the Equilibrium Production Site Lagrange Multipliers

44 Equilibrium Link Flows and the Equilibrium Link Lagrange Multipliers Link b γb f b

45 Equilibrium Prices, Demands, Average Quality and Profits Equilibrium prices at the demand markets: ρ 11 = 17.89, ρ 12 = 19.19, ρ 21 = 15.69, ρ 22 = Equilibrium demands: d11 = , d 12 = , d21 = 47.86, d22 = Average Quality: ˆq 11 = 82.32, ˆq 12 = 81.46, ˆq 21 = 46.59, ˆq 22 = Profits: U 1 = 3, , U 2 =

46 Conclusions We constructed a general framework for the modeling, analysis, and computation of solutions to competitive fresh produce supply chain networks in which food firm owners seek to maximize their profits while determining both the initial quality of the fresh produce with associated costs as well as the fresh produce flows along pathways of their supply chain network through the various activities of harvesting, processing, storage, and distribution. We utilize explicit formulae associated with quality deterioration on the supply chain network links which are a function of physical characteristics, including temperature and time. The governing Nash Equilibrium conditions are stated and alternative variational inequality formulations provided, along with existence results. Stylistic examples are provided to illustrate the framework and a case study on peaches, consisting of numerical examples under status quo and disruption scenarios, is then presented, along with the computed equilibrium patterns.

47 Some References Ahumada, O., Villalobos, J. R., A tactical model for planning the production and distribution of fresh produce. Annals of Operations Research 190, Besik, D., Nagurney, A., Quality in fresh produce supply chains with application to farmers markets. Socio-Economic Planning Sciences 60, Blackburn, J., Scudder, G., Supply chain strategies for perishable products: The case of fresh produce. Production and Operations Management 18(2), Ghare, P., Schrader, G., A model for an exponentially decaying inventory. Journal of Industrial Engineering 14, Kopanos, G.M., Puigjaner, L., Georgiadis, M.C., Simultaneous production and logistics operations planning in semicontinuous food industries. Omega 40, Monteiro, D.M.S., Theoretical and empirical analysis of the economics of traceability adoption in food supply chains. Ph.D. Thesis, University of Massachusetts Amherst, Amherst, MA. Nahmias, S., Perishable inventory theory: A review. Operations Research 30(4), Nahmias, S., Perishable Inventory Systems. Springer, New York. Silver, E.A., Pyke, D.F., Peterson, R., Inventory Management and Production Planning and Scheduling, third edition. John Wiley & Sons, New York. Sloof, M., Tijskens, L.M.M., Wilkinson, E.C., Concepts for modelling the quality of perishable products. Trends in Food Science & Technology 7(5), Widodo, K.H., Nagasawa, H., Morizawa, K., Ota, M., A periodical flowering-harvesting model for delivering agricultural fresh products. European Journal of Operational Research 170, Yu M, Nagurney A. Competitive food supply chain networks with application to fresh produce. European Journal of Operational Research 2013; 224(2): Zhang, G., Habenicht, W., Spieß, W.E.L., Improving the structure of deep frozen and chilled food chain with tabu search procedure. Journal of Food Engineering 60,

48 THANK YOU! For more information: Nagurney, Besik and Yu (UMass) Quality in Food Supply Chain Networks NEDSI 2018