Sequencing situations with controllable processing

A sequencing situation with controllable processing times, or cps situation for short, is a tuple (N, σ₀, α, β, p,p). Here, N , σ¯ ₀, α and p have the same interpretation as in Section 6.2. For the sake of notational simplicity we

assume throughout this section that σ₀(i) = i for each i ∈ {1, . . . , |N |}.

However, in this section we assume that the processing times of the jobs are not fixed. In particular, the processing time p_i of i ∈ N can be reduced to at most ¯p_i, the crashed processing time of i. The amount of time by which the processing time of i is reduced is called the crash time of i. We assume that 0 ≤ ¯p_i ≤ pi for each i ∈ N . The cost of each agent is linear in the completion time of its job, and in the crash time of its job. That is, if t is the completion time and y the crash time of job i, then

C_i(t, y) = α_it+ β_iy.

Of course, αi and βi are both positive constants for each i ∈ N . Since crashing a job requires additional resources, we assume that α_i≤ βi for all i∈ N . That is, reducing the processing time of a job by one time unit costs more than the processing of that job by one time unit.

Since processing times are not fixed, a processing schedule is a pair (σ, x) with σ ∈ P r(N ) and x a vector of feasible processing times. At processing schedule (σ, x), the completion time of job i ∈ N is equal to P

j∈{1,...,|N |}:j≤σ⁻¹(i)x_σ(j), and its crash time is p_i− x_i. Hence, the cost of agent i ∈ N at processing schedule (σ, x) is

C_i(σ, x) = α_i( X

j∈{1,...,|N |}:j≤σ⁻¹(i)

x_σ(j)) + β_i(p_i− xi).

A processing schedule (σ, x) is called optimal if it minimises the sum of the costs of all agents, i.e. if

X

i∈N

C_i(σ, x) ≤X

i∈N

C_i(τ, y) for any processing schedule (τ, y).

Finding an optimal processing schedule for a cps situation falls into the class of NP-hard problems (Hoogeveen and Woeginger (2002)). The difficulty of this problem lies in finding optimal processing times. Once a vector of opti-mal processing times is known, it is straightforward to find a corresponding optimal processing order by applying the Smith-rule. Although finding op-timal processing schedules is difficult, the following lemma, due to Vickson

(1980a), is helpful for our purposes. This lemma states that there is an op-timal processing schedule such that the processing time of each job is either equal to its initial processing time, or its crashed processing time. We note that this result easily follows from the linearity of the cost functions of the agents.

Lemma 6.3.1 (Vickson (1980a)) Let (N, σ₀, α, β, p,p) be a cps situa-¯ tion. There exists an optimal processing schedule (σ, x) such that x_i ∈ {pi,p¯_i} for all i ∈ N .

From Lemma 6.3.1 it follows that an optimal processing schedule can be found by considering all 2^{|N |}possibilities for the processing times and apply-ing the Smith-rule for each of these possibilities. Without loss of generality we assume throughout this section that optimal processing schedules satisfy the property of Lemma 6.3.1, i.e. if (σ, x) is an optimal processing schedule, then x_i ∈ {pi,p¯_i} for all i ∈ N .

In the remainder of this section we introduce sequencing games with controllable processing times, or cps games for short. Let (N, σ₀, α, β, p,p)¯ be a cps situation. The characteristic function of a cps game will express the maximal cost savings each coalition can obtain by means of an admis-sible alteration of the initial processing schedule. For this we have to agree upon which processing schedules are admissible for a coalition. We call a processing schedule admissible for a coalition if it satisfies the following three properties. First, the processing times of the players belonging to the coalition should be feasible, i.e. in between the crashed processing time and the initial processing time. Secondly, the processing times of players out-side the coalition should remain unchanged. Finally, the schedule should be such that the jobs outside the coalition remain in their initial position and no jumps take place over players outside the coalition. Let AS(S) de-note the set of admissible schedules for coalition S ⊆ N . Mathematically, (σ, x) ∈ AS(S) if

¯

p_i ≤ xi≤ pi for all i∈ S (6.2)

x_i= p_i for all i∈ N \S (6.3)

and if σ satisfies (6.1). The cps game (N, v) is now defined by

v(S) =X

i∈S

C_i(σ₀, p) − min

(σ,x)∈AS(S)

X

i∈S

C_i(σ, x)

for each S ⊆ N . The following lemma shows that cps games are superaddi-tive.

Lemma 6.3.2 Let (N, σ₀, α, β, p,p) be a cps situation and let (N, v) be the¯ corresponding game. Then (N, v) is superadditive.

Proof: Let S, T ⊆ N be non-empty sets with S ∩ T = ∅. Let (σ^S, x^S) ∈ AS(S) be an optimal processing schedule for coalition S, and (σ^T, x^T) ∈ AS(T ) be an optimal processing schedule for coalition T . Now let x^S∪T be given by

x^S∪T_i =







x^S_i, if i ∈ S;

x^T_i , if i ∈ T ;

p_i, if i ∈ N \(S ∪ T );

Furthermore, let σ^S∪T ∈ P r(N ) be a “merger” between σ^S and σ^T, i.e. let σ^S∪T ∈ P r(N ) be such that

(σ^S∪T)⁻¹(i) =







(σ^S)⁻¹(i), if i ∈ S;

(σ^T)⁻¹(i), if i ∈ T ;

σ₀⁻¹(i), if i ∈ N \(S ∪ T );

Now observe that (σ^S∪T, x^S∪T) ∈ AS(S ∪ T ). It is easily verified that C_i(σ^S∪T, x^S∪T) ≤ C_i(σ^S, x^S) for each i ∈ S. Indeed, the position of player i∈ S is the same at both orders, and the crash time of i is the same as well.

However, the completion time of i at (σ^S∪T, x^S∪T) might be smaller than at (σ^S, x^S), since at (σ^S∪T, x^S∪T) player i might benefit from possible crashes of jobs corresponding to players in T . Similarly, C_i(σ^S∪T, x^S∪T) ≤ C_i(σ^T, x^T)

for each i ∈ T . It is straightforward to verify that

The first inequality is satisfied because (σ^S∪T, x^S∪T) need not be optimal

for S ∪ T . 2

Similarly to Lemma 6.3.1 it is straightforward to see that for each coalition S ⊆ N there is an optimal processing schedule with the processing time of each job either equal to its initial processing time, or to its crashed processing time. Therefore we assume throughout this section that optimal processing schedules satisfy this property. We now give an example of cps games. This example illustrates that cps games need not be chain-component additive with respect to σ₀ nor convex.

Example 6.3.1 Let (N, σ₀, α, β, p,p) be given by N = {1, 2, 3, 4}, α =¯ (1, 1, 1, 1), β = (2, 2, 2, 2), p = (10, 4, 3, 15), ¯p= (4, 3, 2, 5), and let (N, v) be the corresponding cps game. Now consider for instance coalition {1, 2, 3}.

An optimal schedule for this coalition is given by (σ, x) ∈ AS({1, 2, 3}) with σ = (3, 2, 1, 4) and x = (10, 4, 2, 15). This yields cost savings v({1, 2, 3}) = 15.

It can be verified that v({1}) = v({1, 3}) = v({3, 4}) = 0, v({1, 3, 4}) = 6 and v(N ) = 17. We conclude that v({1}) + v({3, 4}) = 0 < 6 = v({1, 3, 4}).

Hence, (N, v) is not chain-component additive with respect to σ₀. Further-more, v({1, 2, 3}) + v({1, 3, 4}) = 21 > 17 = v(N ) + v({1, 3}). This shows

that (N, v) is not convex. 3

Note that non-emptiness of the core of many extensions of standard sequenc-ing games is proved by means of chain-component additivity with respect to σ₀ (e.g. Borm, Fiestras-Janeiro, Hamers, S´anchez, and Voorneveld (2002) and Hamers, Borm, and Tijs (1995)). Since cps games need not be chain-component additive with respect to σ₀ nor convex, another approach is needed to establish non-emptiness of the core. This will be the main issue of the upcoming section.

6.3.2 Cores of sequencing games with controllable processing